1 /*
2 * Copyright (c) 1996, 2015, Oracle and/or its affiliates. All rights reserved.
3 * DO NOT ALTER OR REMOVE COPYRIGHT NOTICES OR THIS FILE HEADER.
4 *
5 * This code is free software; you can redistribute it and/or modify it
6 * under the terms of the GNU General Public License version 2 only, as
7 * published by the Free Software Foundation. Oracle designates this
8 * particular file as subject to the "Classpath" exception as provided
9 * by Oracle in the LICENSE file that accompanied this code.
10 *
11 * This code is distributed in the hope that it will be useful, but WITHOUT
12 * ANY WARRANTY; without even the implied warranty of MERCHANTABILITY or
13 * FITNESS FOR A PARTICULAR PURPOSE. See the GNU General Public License
14 * version 2 for more details (a copy is included in the LICENSE file that
15 * accompanied this code).
16 *
17 * You should have received a copy of the GNU General Public License version
18 * 2 along with this work; if not, write to the Free Software Foundation,
19 * Inc., 51 Franklin St, Fifth Floor, Boston, MA 02110-1301 USA.
20 *
21 * Please contact Oracle, 500 Oracle Parkway, Redwood Shores, CA 94065 USA
22 * or visit www.oracle.com if you need additional information or have any
23 * questions.
24 */
25
26 /*
27 * Portions Copyright (c) 1995 Colin Plumb. All rights reserved.
28 */
29
30 package java.math;
31
32 import java.io.IOException;
33 import java.io.ObjectInputStream;
34 import java.io.ObjectOutputStream;
35 import java.io.ObjectStreamField;
36 import java.util.Arrays;
37 import java.util.Objects;
38 import java.util.Random;
39 import java.util.concurrent.ThreadLocalRandom;
40
41 import jdk.internal.math.DoubleConsts;
42 import jdk.internal.math.FloatConsts;
43 import jdk.internal.HotSpotIntrinsicCandidate;
44
45 /**
46 * Immutable arbitrary-precision integers. All operations behave as if
47 * BigIntegers were represented in two's-complement notation (like Java's
48 * primitive integer types). BigInteger provides analogues to all of Java's
49 * primitive integer operators, and all relevant methods from java.lang.Math.
50 * Additionally, BigInteger provides operations for modular arithmetic, GCD
51 * calculation, primality testing, prime generation, bit manipulation,
52 * and a few other miscellaneous operations.
53 *
54 * <p>Semantics of arithmetic operations exactly mimic those of Java's integer
55 * arithmetic operators, as defined in <i>The Java Language Specification</i>.
56 * For example, division by zero throws an {@code ArithmeticException}, and
57 * division of a negative by a positive yields a negative (or zero) remainder.
58 * All of the details in the Spec concerning overflow are ignored, as
59 * BigIntegers are made as large as necessary to accommodate the results of an
60 * operation.
61 *
62 * <p>Semantics of shift operations extend those of Java's shift operators
63 * to allow for negative shift distances. A right-shift with a negative
64 * shift distance results in a left shift, and vice-versa. The unsigned
65 * right shift operator ({@code >>>}) is omitted, as this operation makes
66 * little sense in combination with the "infinite word size" abstraction
67 * provided by this class.
68 *
69 * <p>Semantics of bitwise logical operations exactly mimic those of Java's
70 * bitwise integer operators. The binary operators ({@code and},
71 * {@code or}, {@code xor}) implicitly perform sign extension on the shorter
72 * of the two operands prior to performing the operation.
73 *
74 * <p>Comparison operations perform signed integer comparisons, analogous to
75 * those performed by Java's relational and equality operators.
76 *
77 * <p>Modular arithmetic operations are provided to compute residues, perform
78 * exponentiation, and compute multiplicative inverses. These methods always
79 * return a non-negative result, between {@code 0} and {@code (modulus - 1)},
80 * inclusive.
81 *
82 * <p>Bit operations operate on a single bit of the two's-complement
83 * representation of their operand. If necessary, the operand is sign-
84 * extended so that it contains the designated bit. None of the single-bit
85 * operations can produce a BigInteger with a different sign from the
86 * BigInteger being operated on, as they affect only a single bit, and the
87 * "infinite word size" abstraction provided by this class ensures that there
88 * are infinitely many "virtual sign bits" preceding each BigInteger.
89 *
90 * <p>For the sake of brevity and clarity, pseudo-code is used throughout the
91 * descriptions of BigInteger methods. The pseudo-code expression
92 * {@code (i + j)} is shorthand for "a BigInteger whose value is
93 * that of the BigInteger {@code i} plus that of the BigInteger {@code j}."
94 * The pseudo-code expression {@code (i == j)} is shorthand for
95 * "{@code true} if and only if the BigInteger {@code i} represents the same
96 * value as the BigInteger {@code j}." Other pseudo-code expressions are
97 * interpreted similarly.
98 *
99 * <p>All methods and constructors in this class throw
100 * {@code NullPointerException} when passed
101 * a null object reference for any input parameter.
102 *
103 * BigInteger must support values in the range
104 * -2<sup>{@code Integer.MAX_VALUE}</sup> (exclusive) to
105 * +2<sup>{@code Integer.MAX_VALUE}</sup> (exclusive)
106 * and may support values outside of that range.
107 *
108 * The range of probable prime values is limited and may be less than
109 * the full supported positive range of {@code BigInteger}.
110 * The range must be at least 1 to 2<sup>500000000</sup>.
111 *
112 * @implNote
113 * BigInteger constructors and operations throw {@code ArithmeticException} when
114 * the result is out of the supported range of
115 * -2<sup>{@code Integer.MAX_VALUE}</sup> (exclusive) to
116 * +2<sup>{@code Integer.MAX_VALUE}</sup> (exclusive).
117 *
118 * @see BigDecimal
119 * @author Josh Bloch
120 * @author Michael McCloskey
121 * @author Alan Eliasen
122 * @author Timothy Buktu
123 * @since 1.1
124 */
125
126 public class BigInteger extends Number implements Comparable<BigInteger> {
127 /**
128 * The signum of this BigInteger: -1 for negative, 0 for zero, or
129 * 1 for positive. Note that the BigInteger zero <i>must</i> have
130 * a signum of 0. This is necessary to ensures that there is exactly one
131 * representation for each BigInteger value.
132 */
133 final int signum;
134
135 /**
136 * The magnitude of this BigInteger, in <i>big-endian</i> order: the
137 * zeroth element of this array is the most-significant int of the
138 * magnitude. The magnitude must be "minimal" in that the most-significant
139 * int ({@code mag[0]}) must be non-zero. This is necessary to
140 * ensure that there is exactly one representation for each BigInteger
141 * value. Note that this implies that the BigInteger zero has a
142 * zero-length mag array.
143 */
144 final int[] mag;
145
146 // The following fields are stable variables. A stable variable's value
147 // changes at most once from the default zero value to a non-zero stable
148 // value. A stable value is calculated lazily on demand.
149
150 /**
151 * One plus the bitCount of this BigInteger. This is a stable variable.
152 *
153 * @see #bitCount
154 */
155 private int bitCountPlusOne;
156
157 /**
158 * One plus the bitLength of this BigInteger. This is a stable variable.
159 * (either value is acceptable).
160 *
161 * @see #bitLength()
162 */
163 private int bitLengthPlusOne;
164
165 /**
166 * Two plus the lowest set bit of this BigInteger. This is a stable variable.
167 *
168 * @see #getLowestSetBit
169 */
170 private int lowestSetBitPlusTwo;
171
172 /**
173 * Two plus the index of the lowest-order int in the magnitude of this
174 * BigInteger that contains a nonzero int. This is a stable variable. The
175 * least significant int has int-number 0, the next int in order of
176 * increasing significance has int-number 1, and so forth.
177 *
178 * <p>Note: never used for a BigInteger with a magnitude of zero.
179 *
180 * @see #firstNonzeroIntNum()
181 */
182 private int firstNonzeroIntNumPlusTwo;
183
184 /**
185 * This mask is used to obtain the value of an int as if it were unsigned.
186 */
187 static final long LONG_MASK = 0xffffffffL;
188
189 /**
190 * This constant limits {@code mag.length} of BigIntegers to the supported
191 * range.
192 */
193 private static final int MAX_MAG_LENGTH = Integer.MAX_VALUE / Integer.SIZE + 1; // (1 << 26)
194
195 /**
196 * Bit lengths larger than this constant can cause overflow in searchLen
197 * calculation and in BitSieve.singleSearch method.
198 */
199 private static final int PRIME_SEARCH_BIT_LENGTH_LIMIT = 500000000;
200
201 /**
202 * The threshold value for using Karatsuba multiplication. If the number
203 * of ints in both mag arrays are greater than this number, then
204 * Karatsuba multiplication will be used. This value is found
205 * experimentally to work well.
206 */
207 private static final int KARATSUBA_THRESHOLD = 80;
208
209 /**
210 * The threshold value for using 3-way Toom-Cook multiplication.
211 * If the number of ints in each mag array is greater than the
212 * Karatsuba threshold, and the number of ints in at least one of
213 * the mag arrays is greater than this threshold, then Toom-Cook
214 * multiplication will be used.
215 */
216 private static final int TOOM_COOK_THRESHOLD = 240;
217
218 /**
219 * The threshold value for using Karatsuba squaring. If the number
220 * of ints in the number are larger than this value,
221 * Karatsuba squaring will be used. This value is found
222 * experimentally to work well.
223 */
224 private static final int KARATSUBA_SQUARE_THRESHOLD = 128;
225
226 /**
227 * The threshold value for using Toom-Cook squaring. If the number
228 * of ints in the number are larger than this value,
229 * Toom-Cook squaring will be used. This value is found
230 * experimentally to work well.
231 */
232 private static final int TOOM_COOK_SQUARE_THRESHOLD = 216;
233
234 /**
235 * The threshold value for using Burnikel-Ziegler division. If the number
236 * of ints in the divisor are larger than this value, Burnikel-Ziegler
237 * division may be used. This value is found experimentally to work well.
238 */
239 static final int BURNIKEL_ZIEGLER_THRESHOLD = 80;
240
241 /**
242 * The offset value for using Burnikel-Ziegler division. If the number
243 * of ints in the divisor exceeds the Burnikel-Ziegler threshold, and the
244 * number of ints in the dividend is greater than the number of ints in the
245 * divisor plus this value, Burnikel-Ziegler division will be used. This
246 * value is found experimentally to work well.
247 */
248 static final int BURNIKEL_ZIEGLER_OFFSET = 40;
249
250 /**
251 * The threshold value for using Schoenhage recursive base conversion. If
252 * the number of ints in the number are larger than this value,
253 * the Schoenhage algorithm will be used. In practice, it appears that the
254 * Schoenhage routine is faster for any threshold down to 2, and is
255 * relatively flat for thresholds between 2-25, so this choice may be
256 * varied within this range for very small effect.
257 */
258 private static final int SCHOENHAGE_BASE_CONVERSION_THRESHOLD = 20;
259
260 /**
261 * The threshold value for using squaring code to perform multiplication
262 * of a {@code BigInteger} instance by itself. If the number of ints in
263 * the number are larger than this value, {@code multiply(this)} will
264 * return {@code square()}.
265 */
266 private static final int MULTIPLY_SQUARE_THRESHOLD = 20;
267
268 /**
269 * The threshold for using an intrinsic version of
270 * implMontgomeryXXX to perform Montgomery multiplication. If the
271 * number of ints in the number is more than this value we do not
272 * use the intrinsic.
273 */
274 private static final int MONTGOMERY_INTRINSIC_THRESHOLD = 512;
275
276
277 // Constructors
278
279 /**
280 * Translates a byte sub-array containing the two's-complement binary
281 * representation of a BigInteger into a BigInteger. The sub-array is
282 * specified via an offset into the array and a length. The sub-array is
283 * assumed to be in <i>big-endian</i> byte-order: the most significant
284 * byte is the element at index {@code off}. The {@code val} array is
285 * assumed to be unchanged for the duration of the constructor call.
286 *
287 * An {@code IndexOutOfBoundsException} is thrown if the length of the array
288 * {@code val} is non-zero and either {@code off} is negative, {@code len}
289 * is negative, or {@code off+len} is greater than the length of
290 * {@code val}.
291 *
292 * @param val byte array containing a sub-array which is the big-endian
293 * two's-complement binary representation of a BigInteger.
294 * @param off the start offset of the binary representation.
295 * @param len the number of bytes to use.
296 * @throws NumberFormatException {@code val} is zero bytes long.
297 * @throws IndexOutOfBoundsException if the provided array offset and
298 * length would cause an index into the byte array to be
299 * negative or greater than or equal to the array length.
300 * @since 1.9
301 */
302 public BigInteger(byte[] val, int off, int len) {
303 if (val.length == 0) {
304 throw new NumberFormatException("Zero length BigInteger");
305 } else if ((off < 0) || (off >= val.length) || (len < 0) ||
306 (len > val.length - off)) { // 0 <= off < val.length
307 throw new IndexOutOfBoundsException();
308 }
309
310 if (val[off] < 0) {
311 mag = makePositive(val, off, len);
312 signum = -1;
313 } else {
314 mag = stripLeadingZeroBytes(val, off, len);
315 signum = (mag.length == 0 ? 0 : 1);
316 }
317 if (mag.length >= MAX_MAG_LENGTH) {
318 checkRange();
319 }
320 }
321
322 /**
323 * Translates a byte array containing the two's-complement binary
324 * representation of a BigInteger into a BigInteger. The input array is
325 * assumed to be in <i>big-endian</i> byte-order: the most significant
326 * byte is in the zeroth element. The {@code val} array is assumed to be
327 * unchanged for the duration of the constructor call.
328 *
329 * @param val big-endian two's-complement binary representation of a
330 * BigInteger.
331 * @throws NumberFormatException {@code val} is zero bytes long.
332 */
333 public BigInteger(byte[] val) {
334 this(val, 0, val.length);
335 }
336
337 /**
338 * This private constructor translates an int array containing the
339 * two's-complement binary representation of a BigInteger into a
340 * BigInteger. The input array is assumed to be in <i>big-endian</i>
341 * int-order: the most significant int is in the zeroth element. The
342 * {@code val} array is assumed to be unchanged for the duration of
343 * the constructor call.
344 */
345 private BigInteger(int[] val) {
346 if (val.length == 0)
347 throw new NumberFormatException("Zero length BigInteger");
348
349 if (val[0] < 0) {
350 mag = makePositive(val);
351 signum = -1;
352 } else {
353 mag = trustedStripLeadingZeroInts(val);
354 signum = (mag.length == 0 ? 0 : 1);
355 }
356 if (mag.length >= MAX_MAG_LENGTH) {
357 checkRange();
358 }
359 }
360
361 /**
362 * Translates the sign-magnitude representation of a BigInteger into a
363 * BigInteger. The sign is represented as an integer signum value: -1 for
364 * negative, 0 for zero, or 1 for positive. The magnitude is a sub-array of
365 * a byte array in <i>big-endian</i> byte-order: the most significant byte
366 * is the element at index {@code off}. A zero value of the length
367 * {@code len} is permissible, and will result in a BigInteger value of 0,
368 * whether signum is -1, 0 or 1. The {@code magnitude} array is assumed to
369 * be unchanged for the duration of the constructor call.
370 *
371 * An {@code IndexOutOfBoundsException} is thrown if the length of the array
372 * {@code magnitude} is non-zero and either {@code off} is negative,
373 * {@code len} is negative, or {@code off+len} is greater than the length of
374 * {@code magnitude}.
375 *
376 * @param signum signum of the number (-1 for negative, 0 for zero, 1
377 * for positive).
378 * @param magnitude big-endian binary representation of the magnitude of
379 * the number.
380 * @param off the start offset of the binary representation.
381 * @param len the number of bytes to use.
382 * @throws NumberFormatException {@code signum} is not one of the three
383 * legal values (-1, 0, and 1), or {@code signum} is 0 and
384 * {@code magnitude} contains one or more non-zero bytes.
385 * @throws IndexOutOfBoundsException if the provided array offset and
386 * length would cause an index into the byte array to be
387 * negative or greater than or equal to the array length.
388 * @since 1.9
389 */
390 public BigInteger(int signum, byte[] magnitude, int off, int len) {
391 if (signum < -1 || signum > 1) {
392 throw(new NumberFormatException("Invalid signum value"));
393 } else if ((off < 0) || (len < 0) ||
394 (len > 0 &&
395 ((off >= magnitude.length) ||
396 (len > magnitude.length - off)))) { // 0 <= off < magnitude.length
397 throw new IndexOutOfBoundsException();
398 }
399
400 // stripLeadingZeroBytes() returns a zero length array if len == 0
401 this.mag = stripLeadingZeroBytes(magnitude, off, len);
402
403 if (this.mag.length == 0) {
404 this.signum = 0;
405 } else {
406 if (signum == 0)
407 throw(new NumberFormatException("signum-magnitude mismatch"));
408 this.signum = signum;
409 }
410 if (mag.length >= MAX_MAG_LENGTH) {
411 checkRange();
412 }
413 }
414
415 /**
416 * Translates the sign-magnitude representation of a BigInteger into a
417 * BigInteger. The sign is represented as an integer signum value: -1 for
418 * negative, 0 for zero, or 1 for positive. The magnitude is a byte array
419 * in <i>big-endian</i> byte-order: the most significant byte is the
420 * zeroth element. A zero-length magnitude array is permissible, and will
421 * result in a BigInteger value of 0, whether signum is -1, 0 or 1. The
422 * {@code magnitude} array is assumed to be unchanged for the duration of
423 * the constructor call.
424 *
425 * @param signum signum of the number (-1 for negative, 0 for zero, 1
426 * for positive).
427 * @param magnitude big-endian binary representation of the magnitude of
428 * the number.
429 * @throws NumberFormatException {@code signum} is not one of the three
430 * legal values (-1, 0, and 1), or {@code signum} is 0 and
431 * {@code magnitude} contains one or more non-zero bytes.
432 */
433 public BigInteger(int signum, byte[] magnitude) {
434 this(signum, magnitude, 0, magnitude.length);
435 }
436
437 /**
438 * A constructor for internal use that translates the sign-magnitude
439 * representation of a BigInteger into a BigInteger. It checks the
440 * arguments and copies the magnitude so this constructor would be
441 * safe for external use. The {@code magnitude} array is assumed to be
442 * unchanged for the duration of the constructor call.
443 */
444 private BigInteger(int signum, int[] magnitude) {
445 this.mag = stripLeadingZeroInts(magnitude);
446
447 if (signum < -1 || signum > 1)
448 throw(new NumberFormatException("Invalid signum value"));
449
450 if (this.mag.length == 0) {
451 this.signum = 0;
452 } else {
453 if (signum == 0)
454 throw(new NumberFormatException("signum-magnitude mismatch"));
455 this.signum = signum;
456 }
457 if (mag.length >= MAX_MAG_LENGTH) {
458 checkRange();
459 }
460 }
461
462 /**
463 * Translates the String representation of a BigInteger in the
464 * specified radix into a BigInteger. The String representation
465 * consists of an optional minus or plus sign followed by a
466 * sequence of one or more digits in the specified radix. The
467 * character-to-digit mapping is provided by {@code
468 * Character.digit}. The String may not contain any extraneous
469 * characters (whitespace, for example).
470 *
471 * @param val String representation of BigInteger.
472 * @param radix radix to be used in interpreting {@code val}.
473 * @throws NumberFormatException {@code val} is not a valid representation
474 * of a BigInteger in the specified radix, or {@code radix} is
475 * outside the range from {@link Character#MIN_RADIX} to
476 * {@link Character#MAX_RADIX}, inclusive.
477 * @see Character#digit
478 */
479 public BigInteger(String val, int radix) {
480 int cursor = 0, numDigits;
481 final int len = val.length();
482
483 if (radix < Character.MIN_RADIX || radix > Character.MAX_RADIX)
484 throw new NumberFormatException("Radix out of range");
485 if (len == 0)
486 throw new NumberFormatException("Zero length BigInteger");
487
488 // Check for at most one leading sign
489 int sign = 1;
490 int index1 = val.lastIndexOf('-');
491 int index2 = val.lastIndexOf('+');
492 if (index1 >= 0) {
493 if (index1 != 0 || index2 >= 0) {
494 throw new NumberFormatException("Illegal embedded sign character");
495 }
496 sign = -1;
497 cursor = 1;
498 } else if (index2 >= 0) {
499 if (index2 != 0) {
500 throw new NumberFormatException("Illegal embedded sign character");
501 }
502 cursor = 1;
503 }
504 if (cursor == len)
505 throw new NumberFormatException("Zero length BigInteger");
506
507 // Skip leading zeros and compute number of digits in magnitude
508 while (cursor < len &&
509 Character.digit(val.charAt(cursor), radix) == 0) {
510 cursor++;
511 }
512
513 if (cursor == len) {
514 signum = 0;
515 mag = ZERO.mag;
516 return;
517 }
518
519 numDigits = len - cursor;
520 signum = sign;
521
522 // Pre-allocate array of expected size. May be too large but can
523 // never be too small. Typically exact.
524 long numBits = ((numDigits * bitsPerDigit[radix]) >>> 10) + 1;
525 if (numBits + 31 >= (1L << 32)) {
526 reportOverflow();
527 }
528 int numWords = (int) (numBits + 31) >>> 5;
529 int[] magnitude = new int[numWords];
530
531 // Process first (potentially short) digit group
532 int firstGroupLen = numDigits % digitsPerInt[radix];
533 if (firstGroupLen == 0)
534 firstGroupLen = digitsPerInt[radix];
535 String group = val.substring(cursor, cursor += firstGroupLen);
536 magnitude[numWords - 1] = Integer.parseInt(group, radix);
537 if (magnitude[numWords - 1] < 0)
538 throw new NumberFormatException("Illegal digit");
539
540 // Process remaining digit groups
541 int superRadix = intRadix[radix];
542 int groupVal = 0;
543 while (cursor < len) {
544 group = val.substring(cursor, cursor += digitsPerInt[radix]);
545 groupVal = Integer.parseInt(group, radix);
546 if (groupVal < 0)
547 throw new NumberFormatException("Illegal digit");
548 destructiveMulAdd(magnitude, superRadix, groupVal);
549 }
550 // Required for cases where the array was overallocated.
551 mag = trustedStripLeadingZeroInts(magnitude);
552 if (mag.length >= MAX_MAG_LENGTH) {
553 checkRange();
554 }
555 }
556
557 /*
558 * Constructs a new BigInteger using a char array with radix=10.
559 * Sign is precalculated outside and not allowed in the val. The {@code val}
560 * array is assumed to be unchanged for the duration of the constructor
561 * call.
562 */
563 BigInteger(char[] val, int sign, int len) {
564 int cursor = 0, numDigits;
565
566 // Skip leading zeros and compute number of digits in magnitude
567 while (cursor < len && Character.digit(val[cursor], 10) == 0) {
568 cursor++;
569 }
570 if (cursor == len) {
571 signum = 0;
572 mag = ZERO.mag;
573 return;
574 }
575
576 numDigits = len - cursor;
577 signum = sign;
578 // Pre-allocate array of expected size
579 int numWords;
580 if (len < 10) {
581 numWords = 1;
582 } else {
583 long numBits = ((numDigits * bitsPerDigit[10]) >>> 10) + 1;
584 if (numBits + 31 >= (1L << 32)) {
585 reportOverflow();
586 }
587 numWords = (int) (numBits + 31) >>> 5;
588 }
589 int[] magnitude = new int[numWords];
590
591 // Process first (potentially short) digit group
592 int firstGroupLen = numDigits % digitsPerInt[10];
593 if (firstGroupLen == 0)
594 firstGroupLen = digitsPerInt[10];
595 magnitude[numWords - 1] = parseInt(val, cursor, cursor += firstGroupLen);
596
597 // Process remaining digit groups
598 while (cursor < len) {
599 int groupVal = parseInt(val, cursor, cursor += digitsPerInt[10]);
600 destructiveMulAdd(magnitude, intRadix[10], groupVal);
601 }
602 mag = trustedStripLeadingZeroInts(magnitude);
603 if (mag.length >= MAX_MAG_LENGTH) {
604 checkRange();
605 }
606 }
607
608 // Create an integer with the digits between the two indexes
609 // Assumes start < end. The result may be negative, but it
610 // is to be treated as an unsigned value.
611 private int parseInt(char[] source, int start, int end) {
612 int result = Character.digit(source[start++], 10);
613 if (result == -1)
614 throw new NumberFormatException(new String(source));
615
616 for (int index = start; index < end; index++) {
617 int nextVal = Character.digit(source[index], 10);
618 if (nextVal == -1)
619 throw new NumberFormatException(new String(source));
620 result = 10*result + nextVal;
621 }
622
623 return result;
624 }
625
626 // bitsPerDigit in the given radix times 1024
627 // Rounded up to avoid underallocation.
628 private static long bitsPerDigit[] = { 0, 0,
629 1024, 1624, 2048, 2378, 2648, 2875, 3072, 3247, 3402, 3543, 3672,
630 3790, 3899, 4001, 4096, 4186, 4271, 4350, 4426, 4498, 4567, 4633,
631 4696, 4756, 4814, 4870, 4923, 4975, 5025, 5074, 5120, 5166, 5210,
632 5253, 5295};
633
634 // Multiply x array times word y in place, and add word z
635 private static void destructiveMulAdd(int[] x, int y, int z) {
636 // Perform the multiplication word by word
637 long ylong = y & LONG_MASK;
638 long zlong = z & LONG_MASK;
639 int len = x.length;
640
641 long product = 0;
642 long carry = 0;
643 for (int i = len-1; i >= 0; i--) {
644 product = ylong * (x[i] & LONG_MASK) + carry;
645 x[i] = (int)product;
646 carry = product >>> 32;
647 }
648
649 // Perform the addition
650 long sum = (x[len-1] & LONG_MASK) + zlong;
651 x[len-1] = (int)sum;
652 carry = sum >>> 32;
653 for (int i = len-2; i >= 0; i--) {
654 sum = (x[i] & LONG_MASK) + carry;
655 x[i] = (int)sum;
656 carry = sum >>> 32;
657 }
658 }
659
660 /**
661 * Translates the decimal String representation of a BigInteger into a
662 * BigInteger. The String representation consists of an optional minus
663 * sign followed by a sequence of one or more decimal digits. The
664 * character-to-digit mapping is provided by {@code Character.digit}.
665 * The String may not contain any extraneous characters (whitespace, for
666 * example).
667 *
668 * @param val decimal String representation of BigInteger.
669 * @throws NumberFormatException {@code val} is not a valid representation
670 * of a BigInteger.
671 * @see Character#digit
672 */
673 public BigInteger(String val) {
674 this(val, 10);
675 }
676
677 /**
678 * Constructs a randomly generated BigInteger, uniformly distributed over
679 * the range 0 to (2<sup>{@code numBits}</sup> - 1), inclusive.
680 * The uniformity of the distribution assumes that a fair source of random
681 * bits is provided in {@code rnd}. Note that this constructor always
682 * constructs a non-negative BigInteger.
683 *
684 * @param numBits maximum bitLength of the new BigInteger.
685 * @param rnd source of randomness to be used in computing the new
686 * BigInteger.
687 * @throws IllegalArgumentException {@code numBits} is negative.
688 * @see #bitLength()
689 */
690 public BigInteger(int numBits, Random rnd) {
691 this(1, randomBits(numBits, rnd));
692 }
693
694 private static byte[] randomBits(int numBits, Random rnd) {
695 if (numBits < 0)
696 throw new IllegalArgumentException("numBits must be non-negative");
697 int numBytes = (int)(((long)numBits+7)/8); // avoid overflow
698 byte[] randomBits = new byte[numBytes];
699
700 // Generate random bytes and mask out any excess bits
701 if (numBytes > 0) {
702 rnd.nextBytes(randomBits);
703 int excessBits = 8*numBytes - numBits;
704 randomBits[0] &= (1 << (8-excessBits)) - 1;
705 }
706 return randomBits;
707 }
708
709 /**
710 * Constructs a randomly generated positive BigInteger that is probably
711 * prime, with the specified bitLength.
712 *
713 * <p>It is recommended that the {@link #probablePrime probablePrime}
714 * method be used in preference to this constructor unless there
715 * is a compelling need to specify a certainty.
716 *
717 * @param bitLength bitLength of the returned BigInteger.
718 * @param certainty a measure of the uncertainty that the caller is
719 * willing to tolerate. The probability that the new BigInteger
720 * represents a prime number will exceed
721 * (1 - 1/2<sup>{@code certainty}</sup>). The execution time of
722 * this constructor is proportional to the value of this parameter.
723 * @param rnd source of random bits used to select candidates to be
724 * tested for primality.
725 * @throws ArithmeticException {@code bitLength < 2} or {@code bitLength} is too large.
726 * @see #bitLength()
727 */
728 public BigInteger(int bitLength, int certainty, Random rnd) {
729 BigInteger prime;
730
731 if (bitLength < 2)
732 throw new ArithmeticException("bitLength < 2");
733 prime = (bitLength < SMALL_PRIME_THRESHOLD
734 ? smallPrime(bitLength, certainty, rnd)
735 : largePrime(bitLength, certainty, rnd));
736 signum = 1;
737 mag = prime.mag;
738 }
739
740 // Minimum size in bits that the requested prime number has
741 // before we use the large prime number generating algorithms.
742 // The cutoff of 95 was chosen empirically for best performance.
743 private static final int SMALL_PRIME_THRESHOLD = 95;
744
745 // Certainty required to meet the spec of probablePrime
746 private static final int DEFAULT_PRIME_CERTAINTY = 100;
747
748 /**
749 * Returns a positive BigInteger that is probably prime, with the
750 * specified bitLength. The probability that a BigInteger returned
751 * by this method is composite does not exceed 2<sup>-100</sup>.
752 *
753 * @param bitLength bitLength of the returned BigInteger.
754 * @param rnd source of random bits used to select candidates to be
755 * tested for primality.
756 * @return a BigInteger of {@code bitLength} bits that is probably prime
757 * @throws ArithmeticException {@code bitLength < 2} or {@code bitLength} is too large.
758 * @see #bitLength()
759 * @since 1.4
760 */
761 public static BigInteger probablePrime(int bitLength, Random rnd) {
762 if (bitLength < 2)
763 throw new ArithmeticException("bitLength < 2");
764
765 return (bitLength < SMALL_PRIME_THRESHOLD ?
766 smallPrime(bitLength, DEFAULT_PRIME_CERTAINTY, rnd) :
767 largePrime(bitLength, DEFAULT_PRIME_CERTAINTY, rnd));
768 }
769
770 /**
771 * Find a random number of the specified bitLength that is probably prime.
772 * This method is used for smaller primes, its performance degrades on
773 * larger bitlengths.
774 *
775 * This method assumes bitLength > 1.
776 */
777 private static BigInteger smallPrime(int bitLength, int certainty, Random rnd) {
778 int magLen = (bitLength + 31) >>> 5;
779 int temp[] = new int[magLen];
780 int highBit = 1 << ((bitLength+31) & 0x1f); // High bit of high int
781 int highMask = (highBit << 1) - 1; // Bits to keep in high int
782
783 while (true) {
784 // Construct a candidate
785 for (int i=0; i < magLen; i++)
786 temp[i] = rnd.nextInt();
787 temp[0] = (temp[0] & highMask) | highBit; // Ensure exact length
788 if (bitLength > 2)
789 temp[magLen-1] |= 1; // Make odd if bitlen > 2
790
791 BigInteger p = new BigInteger(temp, 1);
792
793 // Do cheap "pre-test" if applicable
794 if (bitLength > 6) {
795 long r = p.remainder(SMALL_PRIME_PRODUCT).longValue();
796 if ((r%3==0) || (r%5==0) || (r%7==0) || (r%11==0) ||
797 (r%13==0) || (r%17==0) || (r%19==0) || (r%23==0) ||
798 (r%29==0) || (r%31==0) || (r%37==0) || (r%41==0))
799 continue; // Candidate is composite; try another
800 }
801
802 // All candidates of bitLength 2 and 3 are prime by this point
803 if (bitLength < 4)
804 return p;
805
806 // Do expensive test if we survive pre-test (or it's inapplicable)
807 if (p.primeToCertainty(certainty, rnd))
808 return p;
809 }
810 }
811
812 private static final BigInteger SMALL_PRIME_PRODUCT
813 = valueOf(3L*5*7*11*13*17*19*23*29*31*37*41);
814
815 /**
816 * Find a random number of the specified bitLength that is probably prime.
817 * This method is more appropriate for larger bitlengths since it uses
818 * a sieve to eliminate most composites before using a more expensive
819 * test.
820 */
821 private static BigInteger largePrime(int bitLength, int certainty, Random rnd) {
822 BigInteger p;
823 p = new BigInteger(bitLength, rnd).setBit(bitLength-1);
824 p.mag[p.mag.length-1] &= 0xfffffffe;
825
826 // Use a sieve length likely to contain the next prime number
827 int searchLen = getPrimeSearchLen(bitLength);
828 BitSieve searchSieve = new BitSieve(p, searchLen);
829 BigInteger candidate = searchSieve.retrieve(p, certainty, rnd);
830
831 while ((candidate == null) || (candidate.bitLength() != bitLength)) {
832 p = p.add(BigInteger.valueOf(2*searchLen));
833 if (p.bitLength() != bitLength)
834 p = new BigInteger(bitLength, rnd).setBit(bitLength-1);
835 p.mag[p.mag.length-1] &= 0xfffffffe;
836 searchSieve = new BitSieve(p, searchLen);
837 candidate = searchSieve.retrieve(p, certainty, rnd);
838 }
839 return candidate;
840 }
841
842 /**
843 * Returns the first integer greater than this {@code BigInteger} that
844 * is probably prime. The probability that the number returned by this
845 * method is composite does not exceed 2<sup>-100</sup>. This method will
846 * never skip over a prime when searching: if it returns {@code p}, there
847 * is no prime {@code q} such that {@code this < q < p}.
848 *
849 * @return the first integer greater than this {@code BigInteger} that
850 * is probably prime.
851 * @throws ArithmeticException {@code this < 0} or {@code this} is too large.
852 * @since 1.5
853 */
854 public BigInteger nextProbablePrime() {
855 if (this.signum < 0)
856 throw new ArithmeticException("start < 0: " + this);
857
858 // Handle trivial cases
859 if ((this.signum == 0) || this.equals(ONE))
860 return TWO;
861
862 BigInteger result = this.add(ONE);
863
864 // Fastpath for small numbers
865 if (result.bitLength() < SMALL_PRIME_THRESHOLD) {
866
867 // Ensure an odd number
868 if (!result.testBit(0))
869 result = result.add(ONE);
870
871 while (true) {
872 // Do cheap "pre-test" if applicable
873 if (result.bitLength() > 6) {
874 long r = result.remainder(SMALL_PRIME_PRODUCT).longValue();
875 if ((r%3==0) || (r%5==0) || (r%7==0) || (r%11==0) ||
876 (r%13==0) || (r%17==0) || (r%19==0) || (r%23==0) ||
877 (r%29==0) || (r%31==0) || (r%37==0) || (r%41==0)) {
878 result = result.add(TWO);
879 continue; // Candidate is composite; try another
880 }
881 }
882
883 // All candidates of bitLength 2 and 3 are prime by this point
884 if (result.bitLength() < 4)
885 return result;
886
887 // The expensive test
888 if (result.primeToCertainty(DEFAULT_PRIME_CERTAINTY, null))
889 return result;
890
891 result = result.add(TWO);
892 }
893 }
894
895 // Start at previous even number
896 if (result.testBit(0))
897 result = result.subtract(ONE);
898
899 // Looking for the next large prime
900 int searchLen = getPrimeSearchLen(result.bitLength());
901
902 while (true) {
903 BitSieve searchSieve = new BitSieve(result, searchLen);
904 BigInteger candidate = searchSieve.retrieve(result,
905 DEFAULT_PRIME_CERTAINTY, null);
906 if (candidate != null)
907 return candidate;
908 result = result.add(BigInteger.valueOf(2 * searchLen));
909 }
910 }
911
912 private static int getPrimeSearchLen(int bitLength) {
913 if (bitLength > PRIME_SEARCH_BIT_LENGTH_LIMIT + 1) {
914 throw new ArithmeticException("Prime search implementation restriction on bitLength");
915 }
916 return bitLength / 20 * 64;
917 }
918
919 /**
920 * Returns {@code true} if this BigInteger is probably prime,
921 * {@code false} if it's definitely composite.
922 *
923 * This method assumes bitLength > 2.
924 *
925 * @param certainty a measure of the uncertainty that the caller is
926 * willing to tolerate: if the call returns {@code true}
927 * the probability that this BigInteger is prime exceeds
928 * {@code (1 - 1/2<sup>certainty</sup>)}. The execution time of
929 * this method is proportional to the value of this parameter.
930 * @return {@code true} if this BigInteger is probably prime,
931 * {@code false} if it's definitely composite.
932 */
933 boolean primeToCertainty(int certainty, Random random) {
934 int rounds = 0;
935 int n = (Math.min(certainty, Integer.MAX_VALUE-1)+1)/2;
936
937 // The relationship between the certainty and the number of rounds
938 // we perform is given in the draft standard ANSI X9.80, "PRIME
939 // NUMBER GENERATION, PRIMALITY TESTING, AND PRIMALITY CERTIFICATES".
940 int sizeInBits = this.bitLength();
941 if (sizeInBits < 100) {
942 rounds = 50;
943 rounds = n < rounds ? n : rounds;
944 return passesMillerRabin(rounds, random);
945 }
946
947 if (sizeInBits < 256) {
948 rounds = 27;
949 } else if (sizeInBits < 512) {
950 rounds = 15;
951 } else if (sizeInBits < 768) {
952 rounds = 8;
953 } else if (sizeInBits < 1024) {
954 rounds = 4;
955 } else {
956 rounds = 2;
957 }
958 rounds = n < rounds ? n : rounds;
959
960 return passesMillerRabin(rounds, random) && passesLucasLehmer();
961 }
962
963 /**
964 * Returns true iff this BigInteger is a Lucas-Lehmer probable prime.
965 *
966 * The following assumptions are made:
967 * This BigInteger is a positive, odd number.
968 */
969 private boolean passesLucasLehmer() {
970 BigInteger thisPlusOne = this.add(ONE);
971
972 // Step 1
973 int d = 5;
974 while (jacobiSymbol(d, this) != -1) {
975 // 5, -7, 9, -11, ...
976 d = (d < 0) ? Math.abs(d)+2 : -(d+2);
977 }
978
979 // Step 2
980 BigInteger u = lucasLehmerSequence(d, thisPlusOne, this);
981
982 // Step 3
983 return u.mod(this).equals(ZERO);
984 }
985
986 /**
987 * Computes Jacobi(p,n).
988 * Assumes n positive, odd, n>=3.
989 */
990 private static int jacobiSymbol(int p, BigInteger n) {
991 if (p == 0)
992 return 0;
993
994 // Algorithm and comments adapted from Colin Plumb's C library.
995 int j = 1;
996 int u = n.mag[n.mag.length-1];
997
998 // Make p positive
999 if (p < 0) {
1000 p = -p;
1001 int n8 = u & 7;
1002 if ((n8 == 3) || (n8 == 7))
1003 j = -j; // 3 (011) or 7 (111) mod 8
1004 }
1005
1006 // Get rid of factors of 2 in p
1007 while ((p & 3) == 0)
1008 p >>= 2;
1009 if ((p & 1) == 0) {
1010 p >>= 1;
1011 if (((u ^ (u>>1)) & 2) != 0)
1012 j = -j; // 3 (011) or 5 (101) mod 8
1013 }
1014 if (p == 1)
1015 return j;
1016 // Then, apply quadratic reciprocity
1017 if ((p & u & 2) != 0) // p = u = 3 (mod 4)?
1018 j = -j;
1019 // And reduce u mod p
1020 u = n.mod(BigInteger.valueOf(p)).intValue();
1021
1022 // Now compute Jacobi(u,p), u < p
1023 while (u != 0) {
1024 while ((u & 3) == 0)
1025 u >>= 2;
1026 if ((u & 1) == 0) {
1027 u >>= 1;
1028 if (((p ^ (p>>1)) & 2) != 0)
1029 j = -j; // 3 (011) or 5 (101) mod 8
1030 }
1031 if (u == 1)
1032 return j;
1033 // Now both u and p are odd, so use quadratic reciprocity
1034 assert (u < p);
1035 int t = u; u = p; p = t;
1036 if ((u & p & 2) != 0) // u = p = 3 (mod 4)?
1037 j = -j;
1038 // Now u >= p, so it can be reduced
1039 u %= p;
1040 }
1041 return 0;
1042 }
1043
1044 private static BigInteger lucasLehmerSequence(int z, BigInteger k, BigInteger n) {
1045 BigInteger d = BigInteger.valueOf(z);
1046 BigInteger u = ONE; BigInteger u2;
1047 BigInteger v = ONE; BigInteger v2;
1048
1049 for (int i=k.bitLength()-2; i >= 0; i--) {
1050 u2 = u.multiply(v).mod(n);
1051
1052 v2 = v.square().add(d.multiply(u.square())).mod(n);
1053 if (v2.testBit(0))
1054 v2 = v2.subtract(n);
1055
1056 v2 = v2.shiftRight(1);
1057
1058 u = u2; v = v2;
1059 if (k.testBit(i)) {
1060 u2 = u.add(v).mod(n);
1061 if (u2.testBit(0))
1062 u2 = u2.subtract(n);
1063
1064 u2 = u2.shiftRight(1);
1065 v2 = v.add(d.multiply(u)).mod(n);
1066 if (v2.testBit(0))
1067 v2 = v2.subtract(n);
1068 v2 = v2.shiftRight(1);
1069
1070 u = u2; v = v2;
1071 }
1072 }
1073 return u;
1074 }
1075
1076 /**
1077 * Returns true iff this BigInteger passes the specified number of
1078 * Miller-Rabin tests. This test is taken from the DSA spec (NIST FIPS
1079 * 186-2).
1080 *
1081 * The following assumptions are made:
1082 * This BigInteger is a positive, odd number greater than 2.
1083 * iterations<=50.
1084 */
1085 private boolean passesMillerRabin(int iterations, Random rnd) {
1086 // Find a and m such that m is odd and this == 1 + 2**a * m
1087 BigInteger thisMinusOne = this.subtract(ONE);
1088 BigInteger m = thisMinusOne;
1089 int a = m.getLowestSetBit();
1090 m = m.shiftRight(a);
1091
1092 // Do the tests
1093 if (rnd == null) {
1094 rnd = ThreadLocalRandom.current();
1095 }
1096 for (int i=0; i < iterations; i++) {
1097 // Generate a uniform random on (1, this)
1098 BigInteger b;
1099 do {
1100 b = new BigInteger(this.bitLength(), rnd);
1101 } while (b.compareTo(ONE) <= 0 || b.compareTo(this) >= 0);
1102
1103 int j = 0;
1104 BigInteger z = b.modPow(m, this);
1105 while (!((j == 0 && z.equals(ONE)) || z.equals(thisMinusOne))) {
1106 if (j > 0 && z.equals(ONE) || ++j == a)
1107 return false;
1108 z = z.modPow(TWO, this);
1109 }
1110 }
1111 return true;
1112 }
1113
1114 /**
1115 * This internal constructor differs from its public cousin
1116 * with the arguments reversed in two ways: it assumes that its
1117 * arguments are correct, and it doesn't copy the magnitude array.
1118 */
1119 BigInteger(int[] magnitude, int signum) {
1120 this.signum = (magnitude.length == 0 ? 0 : signum);
1121 this.mag = magnitude;
1122 if (mag.length >= MAX_MAG_LENGTH) {
1123 checkRange();
1124 }
1125 }
1126
1127 /**
1128 * This private constructor is for internal use and assumes that its
1129 * arguments are correct. The {@code magnitude} array is assumed to be
1130 * unchanged for the duration of the constructor call.
1131 */
1132 private BigInteger(byte[] magnitude, int signum) {
1133 this.signum = (magnitude.length == 0 ? 0 : signum);
1134 this.mag = stripLeadingZeroBytes(magnitude, 0, magnitude.length);
1135 if (mag.length >= MAX_MAG_LENGTH) {
1136 checkRange();
1137 }
1138 }
1139
1140 /**
1141 * Throws an {@code ArithmeticException} if the {@code BigInteger} would be
1142 * out of the supported range.
1143 *
1144 * @throws ArithmeticException if {@code this} exceeds the supported range.
1145 */
1146 private void checkRange() {
1147 if (mag.length > MAX_MAG_LENGTH || mag.length == MAX_MAG_LENGTH && mag[0] < 0) {
1148 reportOverflow();
1149 }
1150 }
1151
1152 private static void reportOverflow() {
1153 throw new ArithmeticException("BigInteger would overflow supported range");
1154 }
1155
1156 //Static Factory Methods
1157
1158 /**
1159 * Returns a BigInteger whose value is equal to that of the
1160 * specified {@code long}. This "static factory method" is
1161 * provided in preference to a ({@code long}) constructor
1162 * because it allows for reuse of frequently used BigIntegers.
1163 *
1164 * @param val value of the BigInteger to return.
1165 * @return a BigInteger with the specified value.
1166 */
1167 public static BigInteger valueOf(long val) {
1168 // If -MAX_CONSTANT < val < MAX_CONSTANT, return stashed constant
1169 if (val == 0)
1170 return ZERO;
1171 if (val > 0 && val <= MAX_CONSTANT)
1172 return posConst[(int) val];
1173 else if (val < 0 && val >= -MAX_CONSTANT)
1174 return negConst[(int) -val];
1175
1176 return new BigInteger(val);
1177 }
1178
1179 /**
1180 * Constructs a BigInteger with the specified value, which may not be zero.
1181 */
1182 private BigInteger(long val) {
1183 if (val < 0) {
1184 val = -val;
1185 signum = -1;
1186 } else {
1187 signum = 1;
1188 }
1189
1190 int highWord = (int)(val >>> 32);
1191 if (highWord == 0) {
1192 mag = new int[1];
1193 mag[0] = (int)val;
1194 } else {
1195 mag = new int[2];
1196 mag[0] = highWord;
1197 mag[1] = (int)val;
1198 }
1199 }
1200
1201 /**
1202 * Returns a BigInteger with the given two's complement representation.
1203 * Assumes that the input array will not be modified (the returned
1204 * BigInteger will reference the input array if feasible).
1205 */
1206 private static BigInteger valueOf(int val[]) {
1207 return (val[0] > 0 ? new BigInteger(val, 1) : new BigInteger(val));
1208 }
1209
1210 // Constants
1211
1212 /**
1213 * Initialize static constant array when class is loaded.
1214 */
1215 private static final int MAX_CONSTANT = 16;
1216 private static BigInteger posConst[] = new BigInteger[MAX_CONSTANT+1];
1217 private static BigInteger negConst[] = new BigInteger[MAX_CONSTANT+1];
1218
1219 /**
1220 * The cache of powers of each radix. This allows us to not have to
1221 * recalculate powers of radix^(2^n) more than once. This speeds
1222 * Schoenhage recursive base conversion significantly.
1223 */
1224 private static volatile BigInteger[][] powerCache;
1225
1226 /** The cache of logarithms of radices for base conversion. */
1227 private static final double[] logCache;
1228
1229 /** The natural log of 2. This is used in computing cache indices. */
1230 private static final double LOG_TWO = Math.log(2.0);
1231
1232 static {
1233 for (int i = 1; i <= MAX_CONSTANT; i++) {
1234 int[] magnitude = new int[1];
1235 magnitude[0] = i;
1236 posConst[i] = new BigInteger(magnitude, 1);
1237 negConst[i] = new BigInteger(magnitude, -1);
1238 }
1239
1240 /*
1241 * Initialize the cache of radix^(2^x) values used for base conversion
1242 * with just the very first value. Additional values will be created
1243 * on demand.
1244 */
1245 powerCache = new BigInteger[Character.MAX_RADIX+1][];
1246 logCache = new double[Character.MAX_RADIX+1];
1247
1248 for (int i=Character.MIN_RADIX; i <= Character.MAX_RADIX; i++) {
1249 powerCache[i] = new BigInteger[] { BigInteger.valueOf(i) };
1250 logCache[i] = Math.log(i);
1251 }
1252 }
1253
1254 /**
1255 * The BigInteger constant zero.
1256 *
1257 * @since 1.2
1258 */
1259 public static final BigInteger ZERO = new BigInteger(new int[0], 0);
1260
1261 /**
1262 * The BigInteger constant one.
1263 *
1264 * @since 1.2
1265 */
1266 public static final BigInteger ONE = valueOf(1);
1267
1268 /**
1269 * The BigInteger constant two. (Not exported.)
1270 */
1271 private static final BigInteger TWO = valueOf(2);
1272
1273 /**
1274 * The BigInteger constant -1. (Not exported.)
1275 */
1276 private static final BigInteger NEGATIVE_ONE = valueOf(-1);
1277
1278 /**
1279 * The BigInteger constant ten.
1280 *
1281 * @since 1.5
1282 */
1283 public static final BigInteger TEN = valueOf(10);
1284
1285 // Arithmetic Operations
1286
1287 /**
1288 * Returns a BigInteger whose value is {@code (this + val)}.
1289 *
1290 * @param val value to be added to this BigInteger.
1291 * @return {@code this + val}
1292 */
1293 public BigInteger add(BigInteger val) {
1294 if (val.signum == 0)
1295 return this;
1296 if (signum == 0)
1297 return val;
1298 if (val.signum == signum)
1299 return new BigInteger(add(mag, val.mag), signum);
1300
1301 int cmp = compareMagnitude(val);
1302 if (cmp == 0)
1303 return ZERO;
1304 int[] resultMag = (cmp > 0 ? subtract(mag, val.mag)
1305 : subtract(val.mag, mag));
1306 resultMag = trustedStripLeadingZeroInts(resultMag);
1307
1308 return new BigInteger(resultMag, cmp == signum ? 1 : -1);
1309 }
1310
1311 /**
1312 * Package private methods used by BigDecimal code to add a BigInteger
1313 * with a long. Assumes val is not equal to INFLATED.
1314 */
1315 BigInteger add(long val) {
1316 if (val == 0)
1317 return this;
1318 if (signum == 0)
1319 return valueOf(val);
1320 if (Long.signum(val) == signum)
1321 return new BigInteger(add(mag, Math.abs(val)), signum);
1322 int cmp = compareMagnitude(val);
1323 if (cmp == 0)
1324 return ZERO;
1325 int[] resultMag = (cmp > 0 ? subtract(mag, Math.abs(val)) : subtract(Math.abs(val), mag));
1326 resultMag = trustedStripLeadingZeroInts(resultMag);
1327 return new BigInteger(resultMag, cmp == signum ? 1 : -1);
1328 }
1329
1330 /**
1331 * Adds the contents of the int array x and long value val. This
1332 * method allocates a new int array to hold the answer and returns
1333 * a reference to that array. Assumes x.length > 0 and val is
1334 * non-negative
1335 */
1336 private static int[] add(int[] x, long val) {
1337 int[] y;
1338 long sum = 0;
1339 int xIndex = x.length;
1340 int[] result;
1341 int highWord = (int)(val >>> 32);
1342 if (highWord == 0) {
1343 result = new int[xIndex];
1344 sum = (x[--xIndex] & LONG_MASK) + val;
1345 result[xIndex] = (int)sum;
1346 } else {
1347 if (xIndex == 1) {
1348 result = new int[2];
1349 sum = val + (x[0] & LONG_MASK);
1350 result[1] = (int)sum;
1351 result[0] = (int)(sum >>> 32);
1352 return result;
1353 } else {
1354 result = new int[xIndex];
1355 sum = (x[--xIndex] & LONG_MASK) + (val & LONG_MASK);
1356 result[xIndex] = (int)sum;
1357 sum = (x[--xIndex] & LONG_MASK) + (highWord & LONG_MASK) + (sum >>> 32);
1358 result[xIndex] = (int)sum;
1359 }
1360 }
1361 // Copy remainder of longer number while carry propagation is required
1362 boolean carry = (sum >>> 32 != 0);
1363 while (xIndex > 0 && carry)
1364 carry = ((result[--xIndex] = x[xIndex] + 1) == 0);
1365 // Copy remainder of longer number
1366 while (xIndex > 0)
1367 result[--xIndex] = x[xIndex];
1368 // Grow result if necessary
1369 if (carry) {
1370 int bigger[] = new int[result.length + 1];
1371 System.arraycopy(result, 0, bigger, 1, result.length);
1372 bigger[0] = 0x01;
1373 return bigger;
1374 }
1375 return result;
1376 }
1377
1378 /**
1379 * Adds the contents of the int arrays x and y. This method allocates
1380 * a new int array to hold the answer and returns a reference to that
1381 * array.
1382 */
1383 private static int[] add(int[] x, int[] y) {
1384 // If x is shorter, swap the two arrays
1385 if (x.length < y.length) {
1386 int[] tmp = x;
1387 x = y;
1388 y = tmp;
1389 }
1390
1391 int xIndex = x.length;
1392 int yIndex = y.length;
1393 int result[] = new int[xIndex];
1394 long sum = 0;
1395 if (yIndex == 1) {
1396 sum = (x[--xIndex] & LONG_MASK) + (y[0] & LONG_MASK) ;
1397 result[xIndex] = (int)sum;
1398 } else {
1399 // Add common parts of both numbers
1400 while (yIndex > 0) {
1401 sum = (x[--xIndex] & LONG_MASK) +
1402 (y[--yIndex] & LONG_MASK) + (sum >>> 32);
1403 result[xIndex] = (int)sum;
1404 }
1405 }
1406 // Copy remainder of longer number while carry propagation is required
1407 boolean carry = (sum >>> 32 != 0);
1408 while (xIndex > 0 && carry)
1409 carry = ((result[--xIndex] = x[xIndex] + 1) == 0);
1410
1411 // Copy remainder of longer number
1412 while (xIndex > 0)
1413 result[--xIndex] = x[xIndex];
1414
1415 // Grow result if necessary
1416 if (carry) {
1417 int bigger[] = new int[result.length + 1];
1418 System.arraycopy(result, 0, bigger, 1, result.length);
1419 bigger[0] = 0x01;
1420 return bigger;
1421 }
1422 return result;
1423 }
1424
1425 private static int[] subtract(long val, int[] little) {
1426 int highWord = (int)(val >>> 32);
1427 if (highWord == 0) {
1428 int result[] = new int[1];
1429 result[0] = (int)(val - (little[0] & LONG_MASK));
1430 return result;
1431 } else {
1432 int result[] = new int[2];
1433 if (little.length == 1) {
1434 long difference = ((int)val & LONG_MASK) - (little[0] & LONG_MASK);
1435 result[1] = (int)difference;
1436 // Subtract remainder of longer number while borrow propagates
1437 boolean borrow = (difference >> 32 != 0);
1438 if (borrow) {
1439 result[0] = highWord - 1;
1440 } else { // Copy remainder of longer number
1441 result[0] = highWord;
1442 }
1443 return result;
1444 } else { // little.length == 2
1445 long difference = ((int)val & LONG_MASK) - (little[1] & LONG_MASK);
1446 result[1] = (int)difference;
1447 difference = (highWord & LONG_MASK) - (little[0] & LONG_MASK) + (difference >> 32);
1448 result[0] = (int)difference;
1449 return result;
1450 }
1451 }
1452 }
1453
1454 /**
1455 * Subtracts the contents of the second argument (val) from the
1456 * first (big). The first int array (big) must represent a larger number
1457 * than the second. This method allocates the space necessary to hold the
1458 * answer.
1459 * assumes val >= 0
1460 */
1461 private static int[] subtract(int[] big, long val) {
1462 int highWord = (int)(val >>> 32);
1463 int bigIndex = big.length;
1464 int result[] = new int[bigIndex];
1465 long difference = 0;
1466
1467 if (highWord == 0) {
1468 difference = (big[--bigIndex] & LONG_MASK) - val;
1469 result[bigIndex] = (int)difference;
1470 } else {
1471 difference = (big[--bigIndex] & LONG_MASK) - (val & LONG_MASK);
1472 result[bigIndex] = (int)difference;
1473 difference = (big[--bigIndex] & LONG_MASK) - (highWord & LONG_MASK) + (difference >> 32);
1474 result[bigIndex] = (int)difference;
1475 }
1476
1477 // Subtract remainder of longer number while borrow propagates
1478 boolean borrow = (difference >> 32 != 0);
1479 while (bigIndex > 0 && borrow)
1480 borrow = ((result[--bigIndex] = big[bigIndex] - 1) == -1);
1481
1482 // Copy remainder of longer number
1483 while (bigIndex > 0)
1484 result[--bigIndex] = big[bigIndex];
1485
1486 return result;
1487 }
1488
1489 /**
1490 * Returns a BigInteger whose value is {@code (this - val)}.
1491 *
1492 * @param val value to be subtracted from this BigInteger.
1493 * @return {@code this - val}
1494 */
1495 public BigInteger subtract(BigInteger val) {
1496 if (val.signum == 0)
1497 return this;
1498 if (signum == 0)
1499 return val.negate();
1500 if (val.signum != signum)
1501 return new BigInteger(add(mag, val.mag), signum);
1502
1503 int cmp = compareMagnitude(val);
1504 if (cmp == 0)
1505 return ZERO;
1506 int[] resultMag = (cmp > 0 ? subtract(mag, val.mag)
1507 : subtract(val.mag, mag));
1508 resultMag = trustedStripLeadingZeroInts(resultMag);
1509 return new BigInteger(resultMag, cmp == signum ? 1 : -1);
1510 }
1511
1512 /**
1513 * Subtracts the contents of the second int arrays (little) from the
1514 * first (big). The first int array (big) must represent a larger number
1515 * than the second. This method allocates the space necessary to hold the
1516 * answer.
1517 */
1518 private static int[] subtract(int[] big, int[] little) {
1519 int bigIndex = big.length;
1520 int result[] = new int[bigIndex];
1521 int littleIndex = little.length;
1522 long difference = 0;
1523
1524 // Subtract common parts of both numbers
1525 while (littleIndex > 0) {
1526 difference = (big[--bigIndex] & LONG_MASK) -
1527 (little[--littleIndex] & LONG_MASK) +
1528 (difference >> 32);
1529 result[bigIndex] = (int)difference;
1530 }
1531
1532 // Subtract remainder of longer number while borrow propagates
1533 boolean borrow = (difference >> 32 != 0);
1534 while (bigIndex > 0 && borrow)
1535 borrow = ((result[--bigIndex] = big[bigIndex] - 1) == -1);
1536
1537 // Copy remainder of longer number
1538 while (bigIndex > 0)
1539 result[--bigIndex] = big[bigIndex];
1540
1541 return result;
1542 }
1543
1544 /**
1545 * Returns a BigInteger whose value is {@code (this * val)}.
1546 *
1547 * @implNote An implementation may offer better algorithmic
1548 * performance when {@code val == this}.
1549 *
1550 * @param val value to be multiplied by this BigInteger.
1551 * @return {@code this * val}
1552 */
1553 public BigInteger multiply(BigInteger val) {
1554 if (val.signum == 0 || signum == 0)
1555 return ZERO;
1556
1557 int xlen = mag.length;
1558
1559 if (val == this && xlen > MULTIPLY_SQUARE_THRESHOLD) {
1560 return square();
1561 }
1562
1563 int ylen = val.mag.length;
1564
1565 if ((xlen < KARATSUBA_THRESHOLD) || (ylen < KARATSUBA_THRESHOLD)) {
1566 int resultSign = signum == val.signum ? 1 : -1;
1567 if (val.mag.length == 1) {
1568 return multiplyByInt(mag,val.mag[0], resultSign);
1569 }
1570 if (mag.length == 1) {
1571 return multiplyByInt(val.mag,mag[0], resultSign);
1572 }
1573 int[] result = multiplyToLen(mag, xlen,
1574 val.mag, ylen, null);
1575 result = trustedStripLeadingZeroInts(result);
1576 return new BigInteger(result, resultSign);
1577 } else {
1578 if ((xlen < TOOM_COOK_THRESHOLD) && (ylen < TOOM_COOK_THRESHOLD)) {
1579 return multiplyKaratsuba(this, val);
1580 } else {
1581 return multiplyToomCook3(this, val);
1582 }
1583 }
1584 }
1585
1586 private static BigInteger multiplyByInt(int[] x, int y, int sign) {
1587 if (Integer.bitCount(y) == 1) {
1588 return new BigInteger(shiftLeft(x,Integer.numberOfTrailingZeros(y)), sign);
1589 }
1590 int xlen = x.length;
1591 int[] rmag = new int[xlen + 1];
1592 long carry = 0;
1593 long yl = y & LONG_MASK;
1594 int rstart = rmag.length - 1;
1595 for (int i = xlen - 1; i >= 0; i--) {
1596 long product = (x[i] & LONG_MASK) * yl + carry;
1597 rmag[rstart--] = (int)product;
1598 carry = product >>> 32;
1599 }
1600 if (carry == 0L) {
1601 rmag = java.util.Arrays.copyOfRange(rmag, 1, rmag.length);
1602 } else {
1603 rmag[rstart] = (int)carry;
1604 }
1605 return new BigInteger(rmag, sign);
1606 }
1607
1608 /**
1609 * Package private methods used by BigDecimal code to multiply a BigInteger
1610 * with a long. Assumes v is not equal to INFLATED.
1611 */
1612 BigInteger multiply(long v) {
1613 if (v == 0 || signum == 0)
1614 return ZERO;
1615 if (v == BigDecimal.INFLATED)
1616 return multiply(BigInteger.valueOf(v));
1617 int rsign = (v > 0 ? signum : -signum);
1618 if (v < 0)
1619 v = -v;
1620 long dh = v >>> 32; // higher order bits
1621 long dl = v & LONG_MASK; // lower order bits
1622
1623 int xlen = mag.length;
1624 int[] value = mag;
1625 int[] rmag = (dh == 0L) ? (new int[xlen + 1]) : (new int[xlen + 2]);
1626 long carry = 0;
1627 int rstart = rmag.length - 1;
1628 for (int i = xlen - 1; i >= 0; i--) {
1629 long product = (value[i] & LONG_MASK) * dl + carry;
1630 rmag[rstart--] = (int)product;
1631 carry = product >>> 32;
1632 }
1633 rmag[rstart] = (int)carry;
1634 if (dh != 0L) {
1635 carry = 0;
1636 rstart = rmag.length - 2;
1637 for (int i = xlen - 1; i >= 0; i--) {
1638 long product = (value[i] & LONG_MASK) * dh +
1639 (rmag[rstart] & LONG_MASK) + carry;
1640 rmag[rstart--] = (int)product;
1641 carry = product >>> 32;
1642 }
1643 rmag[0] = (int)carry;
1644 }
1645 if (carry == 0L)
1646 rmag = java.util.Arrays.copyOfRange(rmag, 1, rmag.length);
1647 return new BigInteger(rmag, rsign);
1648 }
1649
1650 /**
1651 * Multiplies int arrays x and y to the specified lengths and places
1652 * the result into z. There will be no leading zeros in the resultant array.
1653 */
1654 private static int[] multiplyToLen(int[] x, int xlen, int[] y, int ylen, int[] z) {
1655 multiplyToLenCheck(x, xlen);
1656 multiplyToLenCheck(y, ylen);
1657 return implMultiplyToLen(x, xlen, y, ylen, z);
1658 }
1659
1660 @HotSpotIntrinsicCandidate
1661 private static int[] implMultiplyToLen(int[] x, int xlen, int[] y, int ylen, int[] z) {
1662 int xstart = xlen - 1;
1663 int ystart = ylen - 1;
1664
1665 if (z == null || z.length < (xlen+ ylen))
1666 z = new int[xlen+ylen];
1667
1668 long carry = 0;
1669 for (int j=ystart, k=ystart+1+xstart; j >= 0; j--, k--) {
1670 long product = (y[j] & LONG_MASK) *
1671 (x[xstart] & LONG_MASK) + carry;
1672 z[k] = (int)product;
1673 carry = product >>> 32;
1674 }
1675 z[xstart] = (int)carry;
1676
1677 for (int i = xstart-1; i >= 0; i--) {
1678 carry = 0;
1679 for (int j=ystart, k=ystart+1+i; j >= 0; j--, k--) {
1680 long product = (y[j] & LONG_MASK) *
1681 (x[i] & LONG_MASK) +
1682 (z[k] & LONG_MASK) + carry;
1683 z[k] = (int)product;
1684 carry = product >>> 32;
1685 }
1686 z[i] = (int)carry;
1687 }
1688 return z;
1689 }
1690
1691 private static void multiplyToLenCheck(int[] array, int length) {
1692 if (length <= 0) {
1693 return; // not an error because multiplyToLen won't execute if len <= 0
1694 }
1695
1696 Objects.requireNonNull(array);
1697
1698 if (length > array.length) {
1699 throw new ArrayIndexOutOfBoundsException(length - 1);
1700 }
1701 }
1702
1703 /**
1704 * Multiplies two BigIntegers using the Karatsuba multiplication
1705 * algorithm. This is a recursive divide-and-conquer algorithm which is
1706 * more efficient for large numbers than what is commonly called the
1707 * "grade-school" algorithm used in multiplyToLen. If the numbers to be
1708 * multiplied have length n, the "grade-school" algorithm has an
1709 * asymptotic complexity of O(n^2). In contrast, the Karatsuba algorithm
1710 * has complexity of O(n^(log2(3))), or O(n^1.585). It achieves this
1711 * increased performance by doing 3 multiplies instead of 4 when
1712 * evaluating the product. As it has some overhead, should be used when
1713 * both numbers are larger than a certain threshold (found
1714 * experimentally).
1715 *
1716 * See: http://en.wikipedia.org/wiki/Karatsuba_algorithm
1717 */
1718 private static BigInteger multiplyKaratsuba(BigInteger x, BigInteger y) {
1719 int xlen = x.mag.length;
1720 int ylen = y.mag.length;
1721
1722 // The number of ints in each half of the number.
1723 int half = (Math.max(xlen, ylen)+1) / 2;
1724
1725 // xl and yl are the lower halves of x and y respectively,
1726 // xh and yh are the upper halves.
1727 BigInteger xl = x.getLower(half);
1728 BigInteger xh = x.getUpper(half);
1729 BigInteger yl = y.getLower(half);
1730 BigInteger yh = y.getUpper(half);
1731
1732 BigInteger p1 = xh.multiply(yh); // p1 = xh*yh
1733 BigInteger p2 = xl.multiply(yl); // p2 = xl*yl
1734
1735 // p3=(xh+xl)*(yh+yl)
1736 BigInteger p3 = xh.add(xl).multiply(yh.add(yl));
1737
1738 // result = p1 * 2^(32*2*half) + (p3 - p1 - p2) * 2^(32*half) + p2
1739 BigInteger result = p1.shiftLeft(32*half).add(p3.subtract(p1).subtract(p2)).shiftLeft(32*half).add(p2);
1740
1741 if (x.signum != y.signum) {
1742 return result.negate();
1743 } else {
1744 return result;
1745 }
1746 }
1747
1748 /**
1749 * Multiplies two BigIntegers using a 3-way Toom-Cook multiplication
1750 * algorithm. This is a recursive divide-and-conquer algorithm which is
1751 * more efficient for large numbers than what is commonly called the
1752 * "grade-school" algorithm used in multiplyToLen. If the numbers to be
1753 * multiplied have length n, the "grade-school" algorithm has an
1754 * asymptotic complexity of O(n^2). In contrast, 3-way Toom-Cook has a
1755 * complexity of about O(n^1.465). It achieves this increased asymptotic
1756 * performance by breaking each number into three parts and by doing 5
1757 * multiplies instead of 9 when evaluating the product. Due to overhead
1758 * (additions, shifts, and one division) in the Toom-Cook algorithm, it
1759 * should only be used when both numbers are larger than a certain
1760 * threshold (found experimentally). This threshold is generally larger
1761 * than that for Karatsuba multiplication, so this algorithm is generally
1762 * only used when numbers become significantly larger.
1763 *
1764 * The algorithm used is the "optimal" 3-way Toom-Cook algorithm outlined
1765 * by Marco Bodrato.
1766 *
1767 * See: http://bodrato.it/toom-cook/
1768 * http://bodrato.it/papers/#WAIFI2007
1769 *
1770 * "Towards Optimal Toom-Cook Multiplication for Univariate and
1771 * Multivariate Polynomials in Characteristic 2 and 0." by Marco BODRATO;
1772 * In C.Carlet and B.Sunar, Eds., "WAIFI'07 proceedings", p. 116-133,
1773 * LNCS #4547. Springer, Madrid, Spain, June 21-22, 2007.
1774 *
1775 */
1776 private static BigInteger multiplyToomCook3(BigInteger a, BigInteger b) {
1777 int alen = a.mag.length;
1778 int blen = b.mag.length;
1779
1780 int largest = Math.max(alen, blen);
1781
1782 // k is the size (in ints) of the lower-order slices.
1783 int k = (largest+2)/3; // Equal to ceil(largest/3)
1784
1785 // r is the size (in ints) of the highest-order slice.
1786 int r = largest - 2*k;
1787
1788 // Obtain slices of the numbers. a2 and b2 are the most significant
1789 // bits of the numbers a and b, and a0 and b0 the least significant.
1790 BigInteger a0, a1, a2, b0, b1, b2;
1791 a2 = a.getToomSlice(k, r, 0, largest);
1792 a1 = a.getToomSlice(k, r, 1, largest);
1793 a0 = a.getToomSlice(k, r, 2, largest);
1794 b2 = b.getToomSlice(k, r, 0, largest);
1795 b1 = b.getToomSlice(k, r, 1, largest);
1796 b0 = b.getToomSlice(k, r, 2, largest);
1797
1798 BigInteger v0, v1, v2, vm1, vinf, t1, t2, tm1, da1, db1;
1799
1800 v0 = a0.multiply(b0);
1801 da1 = a2.add(a0);
1802 db1 = b2.add(b0);
1803 vm1 = da1.subtract(a1).multiply(db1.subtract(b1));
1804 da1 = da1.add(a1);
1805 db1 = db1.add(b1);
1806 v1 = da1.multiply(db1);
1807 v2 = da1.add(a2).shiftLeft(1).subtract(a0).multiply(
1808 db1.add(b2).shiftLeft(1).subtract(b0));
1809 vinf = a2.multiply(b2);
1810
1811 // The algorithm requires two divisions by 2 and one by 3.
1812 // All divisions are known to be exact, that is, they do not produce
1813 // remainders, and all results are positive. The divisions by 2 are
1814 // implemented as right shifts which are relatively efficient, leaving
1815 // only an exact division by 3, which is done by a specialized
1816 // linear-time algorithm.
1817 t2 = v2.subtract(vm1).exactDivideBy3();
1818 tm1 = v1.subtract(vm1).shiftRight(1);
1819 t1 = v1.subtract(v0);
1820 t2 = t2.subtract(t1).shiftRight(1);
1821 t1 = t1.subtract(tm1).subtract(vinf);
1822 t2 = t2.subtract(vinf.shiftLeft(1));
1823 tm1 = tm1.subtract(t2);
1824
1825 // Number of bits to shift left.
1826 int ss = k*32;
1827
1828 BigInteger result = vinf.shiftLeft(ss).add(t2).shiftLeft(ss).add(t1).shiftLeft(ss).add(tm1).shiftLeft(ss).add(v0);
1829
1830 if (a.signum != b.signum) {
1831 return result.negate();
1832 } else {
1833 return result;
1834 }
1835 }
1836
1837
1838 /**
1839 * Returns a slice of a BigInteger for use in Toom-Cook multiplication.
1840 *
1841 * @param lowerSize The size of the lower-order bit slices.
1842 * @param upperSize The size of the higher-order bit slices.
1843 * @param slice The index of which slice is requested, which must be a
1844 * number from 0 to size-1. Slice 0 is the highest-order bits, and slice
1845 * size-1 are the lowest-order bits. Slice 0 may be of different size than
1846 * the other slices.
1847 * @param fullsize The size of the larger integer array, used to align
1848 * slices to the appropriate position when multiplying different-sized
1849 * numbers.
1850 */
1851 private BigInteger getToomSlice(int lowerSize, int upperSize, int slice,
1852 int fullsize) {
1853 int start, end, sliceSize, len, offset;
1854
1855 len = mag.length;
1856 offset = fullsize - len;
1857
1858 if (slice == 0) {
1859 start = 0 - offset;
1860 end = upperSize - 1 - offset;
1861 } else {
1862 start = upperSize + (slice-1)*lowerSize - offset;
1863 end = start + lowerSize - 1;
1864 }
1865
1866 if (start < 0) {
1867 start = 0;
1868 }
1869 if (end < 0) {
1870 return ZERO;
1871 }
1872
1873 sliceSize = (end-start) + 1;
1874
1875 if (sliceSize <= 0) {
1876 return ZERO;
1877 }
1878
1879 // While performing Toom-Cook, all slices are positive and
1880 // the sign is adjusted when the final number is composed.
1881 if (start == 0 && sliceSize >= len) {
1882 return this.abs();
1883 }
1884
1885 int intSlice[] = new int[sliceSize];
1886 System.arraycopy(mag, start, intSlice, 0, sliceSize);
1887
1888 return new BigInteger(trustedStripLeadingZeroInts(intSlice), 1);
1889 }
1890
1891 /**
1892 * Does an exact division (that is, the remainder is known to be zero)
1893 * of the specified number by 3. This is used in Toom-Cook
1894 * multiplication. This is an efficient algorithm that runs in linear
1895 * time. If the argument is not exactly divisible by 3, results are
1896 * undefined. Note that this is expected to be called with positive
1897 * arguments only.
1898 */
1899 private BigInteger exactDivideBy3() {
1900 int len = mag.length;
1901 int[] result = new int[len];
1902 long x, w, q, borrow;
1903 borrow = 0L;
1904 for (int i=len-1; i >= 0; i--) {
1905 x = (mag[i] & LONG_MASK);
1906 w = x - borrow;
1907 if (borrow > x) { // Did we make the number go negative?
1908 borrow = 1L;
1909 } else {
1910 borrow = 0L;
1911 }
1912
1913 // 0xAAAAAAAB is the modular inverse of 3 (mod 2^32). Thus,
1914 // the effect of this is to divide by 3 (mod 2^32).
1915 // This is much faster than division on most architectures.
1916 q = (w * 0xAAAAAAABL) & LONG_MASK;
1917 result[i] = (int) q;
1918
1919 // Now check the borrow. The second check can of course be
1920 // eliminated if the first fails.
1921 if (q >= 0x55555556L) {
1922 borrow++;
1923 if (q >= 0xAAAAAAABL)
1924 borrow++;
1925 }
1926 }
1927 result = trustedStripLeadingZeroInts(result);
1928 return new BigInteger(result, signum);
1929 }
1930
1931 /**
1932 * Returns a new BigInteger representing n lower ints of the number.
1933 * This is used by Karatsuba multiplication and Karatsuba squaring.
1934 */
1935 private BigInteger getLower(int n) {
1936 int len = mag.length;
1937
1938 if (len <= n) {
1939 return abs();
1940 }
1941
1942 int lowerInts[] = new int[n];
1943 System.arraycopy(mag, len-n, lowerInts, 0, n);
1944
1945 return new BigInteger(trustedStripLeadingZeroInts(lowerInts), 1);
1946 }
1947
1948 /**
1949 * Returns a new BigInteger representing mag.length-n upper
1950 * ints of the number. This is used by Karatsuba multiplication and
1951 * Karatsuba squaring.
1952 */
1953 private BigInteger getUpper(int n) {
1954 int len = mag.length;
1955
1956 if (len <= n) {
1957 return ZERO;
1958 }
1959
1960 int upperLen = len - n;
1961 int upperInts[] = new int[upperLen];
1962 System.arraycopy(mag, 0, upperInts, 0, upperLen);
1963
1964 return new BigInteger(trustedStripLeadingZeroInts(upperInts), 1);
1965 }
1966
1967 // Squaring
1968
1969 /**
1970 * Returns a BigInteger whose value is {@code (this<sup>2</sup>)}.
1971 *
1972 * @return {@code this<sup>2</sup>}
1973 */
1974 private BigInteger square() {
1975 if (signum == 0) {
1976 return ZERO;
1977 }
1978 int len = mag.length;
1979
1980 if (len < KARATSUBA_SQUARE_THRESHOLD) {
1981 int[] z = squareToLen(mag, len, null);
1982 return new BigInteger(trustedStripLeadingZeroInts(z), 1);
1983 } else {
1984 if (len < TOOM_COOK_SQUARE_THRESHOLD) {
1985 return squareKaratsuba();
1986 } else {
1987 return squareToomCook3();
1988 }
1989 }
1990 }
1991
1992 /**
1993 * Squares the contents of the int array x. The result is placed into the
1994 * int array z. The contents of x are not changed.
1995 */
1996 private static final int[] squareToLen(int[] x, int len, int[] z) {
1997 int zlen = len << 1;
1998 if (z == null || z.length < zlen)
1999 z = new int[zlen];
2000
2001 // Execute checks before calling intrinsified method.
2002 implSquareToLenChecks(x, len, z, zlen);
2003 return implSquareToLen(x, len, z, zlen);
2004 }
2005
2006 /**
2007 * Parameters validation.
2008 */
2009 private static void implSquareToLenChecks(int[] x, int len, int[] z, int zlen) throws RuntimeException {
2010 if (len < 1) {
2011 throw new IllegalArgumentException("invalid input length: " + len);
2012 }
2013 if (len > x.length) {
2014 throw new IllegalArgumentException("input length out of bound: " +
2015 len + " > " + x.length);
2016 }
2017 if (len * 2 > z.length) {
2018 throw new IllegalArgumentException("input length out of bound: " +
2019 (len * 2) + " > " + z.length);
2020 }
2021 if (zlen < 1) {
2022 throw new IllegalArgumentException("invalid input length: " + zlen);
2023 }
2024 if (zlen > z.length) {
2025 throw new IllegalArgumentException("input length out of bound: " +
2026 len + " > " + z.length);
2027 }
2028 }
2029
2030 /**
2031 * Java Runtime may use intrinsic for this method.
2032 */
2033 @HotSpotIntrinsicCandidate
2034 private static final int[] implSquareToLen(int[] x, int len, int[] z, int zlen) {
2035 /*
2036 * The algorithm used here is adapted from Colin Plumb's C library.
2037 * Technique: Consider the partial products in the multiplication
2038 * of "abcde" by itself:
2039 *
2040 * a b c d e
2041 * * a b c d e
2042 * ==================
2043 * ae be ce de ee
2044 * ad bd cd dd de
2045 * ac bc cc cd ce
2046 * ab bb bc bd be
2047 * aa ab ac ad ae
2048 *
2049 * Note that everything above the main diagonal:
2050 * ae be ce de = (abcd) * e
2051 * ad bd cd = (abc) * d
2052 * ac bc = (ab) * c
2053 * ab = (a) * b
2054 *
2055 * is a copy of everything below the main diagonal:
2056 * de
2057 * cd ce
2058 * bc bd be
2059 * ab ac ad ae
2060 *
2061 * Thus, the sum is 2 * (off the diagonal) + diagonal.
2062 *
2063 * This is accumulated beginning with the diagonal (which
2064 * consist of the squares of the digits of the input), which is then
2065 * divided by two, the off-diagonal added, and multiplied by two
2066 * again. The low bit is simply a copy of the low bit of the
2067 * input, so it doesn't need special care.
2068 */
2069
2070 // Store the squares, right shifted one bit (i.e., divided by 2)
2071 int lastProductLowWord = 0;
2072 for (int j=0, i=0; j < len; j++) {
2073 long piece = (x[j] & LONG_MASK);
2074 long product = piece * piece;
2075 z[i++] = (lastProductLowWord << 31) | (int)(product >>> 33);
2076 z[i++] = (int)(product >>> 1);
2077 lastProductLowWord = (int)product;
2078 }
2079
2080 // Add in off-diagonal sums
2081 for (int i=len, offset=1; i > 0; i--, offset+=2) {
2082 int t = x[i-1];
2083 t = mulAdd(z, x, offset, i-1, t);
2084 addOne(z, offset-1, i, t);
2085 }
2086
2087 // Shift back up and set low bit
2088 primitiveLeftShift(z, zlen, 1);
2089 z[zlen-1] |= x[len-1] & 1;
2090
2091 return z;
2092 }
2093
2094 /**
2095 * Squares a BigInteger using the Karatsuba squaring algorithm. It should
2096 * be used when both numbers are larger than a certain threshold (found
2097 * experimentally). It is a recursive divide-and-conquer algorithm that
2098 * has better asymptotic performance than the algorithm used in
2099 * squareToLen.
2100 */
2101 private BigInteger squareKaratsuba() {
2102 int half = (mag.length+1) / 2;
2103
2104 BigInteger xl = getLower(half);
2105 BigInteger xh = getUpper(half);
2106
2107 BigInteger xhs = xh.square(); // xhs = xh^2
2108 BigInteger xls = xl.square(); // xls = xl^2
2109
2110 // xh^2 << 64 + (((xl+xh)^2 - (xh^2 + xl^2)) << 32) + xl^2
2111 return xhs.shiftLeft(half*32).add(xl.add(xh).square().subtract(xhs.add(xls))).shiftLeft(half*32).add(xls);
2112 }
2113
2114 /**
2115 * Squares a BigInteger using the 3-way Toom-Cook squaring algorithm. It
2116 * should be used when both numbers are larger than a certain threshold
2117 * (found experimentally). It is a recursive divide-and-conquer algorithm
2118 * that has better asymptotic performance than the algorithm used in
2119 * squareToLen or squareKaratsuba.
2120 */
2121 private BigInteger squareToomCook3() {
2122 int len = mag.length;
2123
2124 // k is the size (in ints) of the lower-order slices.
2125 int k = (len+2)/3; // Equal to ceil(largest/3)
2126
2127 // r is the size (in ints) of the highest-order slice.
2128 int r = len - 2*k;
2129
2130 // Obtain slices of the numbers. a2 is the most significant
2131 // bits of the number, and a0 the least significant.
2132 BigInteger a0, a1, a2;
2133 a2 = getToomSlice(k, r, 0, len);
2134 a1 = getToomSlice(k, r, 1, len);
2135 a0 = getToomSlice(k, r, 2, len);
2136 BigInteger v0, v1, v2, vm1, vinf, t1, t2, tm1, da1;
2137
2138 v0 = a0.square();
2139 da1 = a2.add(a0);
2140 vm1 = da1.subtract(a1).square();
2141 da1 = da1.add(a1);
2142 v1 = da1.square();
2143 vinf = a2.square();
2144 v2 = da1.add(a2).shiftLeft(1).subtract(a0).square();
2145
2146 // The algorithm requires two divisions by 2 and one by 3.
2147 // All divisions are known to be exact, that is, they do not produce
2148 // remainders, and all results are positive. The divisions by 2 are
2149 // implemented as right shifts which are relatively efficient, leaving
2150 // only a division by 3.
2151 // The division by 3 is done by an optimized algorithm for this case.
2152 t2 = v2.subtract(vm1).exactDivideBy3();
2153 tm1 = v1.subtract(vm1).shiftRight(1);
2154 t1 = v1.subtract(v0);
2155 t2 = t2.subtract(t1).shiftRight(1);
2156 t1 = t1.subtract(tm1).subtract(vinf);
2157 t2 = t2.subtract(vinf.shiftLeft(1));
2158 tm1 = tm1.subtract(t2);
2159
2160 // Number of bits to shift left.
2161 int ss = k*32;
2162
2163 return vinf.shiftLeft(ss).add(t2).shiftLeft(ss).add(t1).shiftLeft(ss).add(tm1).shiftLeft(ss).add(v0);
2164 }
2165
2166 // Division
2167
2168 /**
2169 * Returns a BigInteger whose value is {@code (this / val)}.
2170 *
2171 * @param val value by which this BigInteger is to be divided.
2172 * @return {@code this / val}
2173 * @throws ArithmeticException if {@code val} is zero.
2174 */
2175 public BigInteger divide(BigInteger val) {
2176 if (val.mag.length < BURNIKEL_ZIEGLER_THRESHOLD ||
2177 mag.length - val.mag.length < BURNIKEL_ZIEGLER_OFFSET) {
2178 return divideKnuth(val);
2179 } else {
2180 return divideBurnikelZiegler(val);
2181 }
2182 }
2183
2184 /**
2185 * Returns a BigInteger whose value is {@code (this / val)} using an O(n^2) algorithm from Knuth.
2186 *
2187 * @param val value by which this BigInteger is to be divided.
2188 * @return {@code this / val}
2189 * @throws ArithmeticException if {@code val} is zero.
2190 * @see MutableBigInteger#divideKnuth(MutableBigInteger, MutableBigInteger, boolean)
2191 */
2192 private BigInteger divideKnuth(BigInteger val) {
2193 MutableBigInteger q = new MutableBigInteger(),
2194 a = new MutableBigInteger(this.mag),
2195 b = new MutableBigInteger(val.mag);
2196
2197 a.divideKnuth(b, q, false);
2198 return q.toBigInteger(this.signum * val.signum);
2199 }
2200
2201 /**
2202 * Returns an array of two BigIntegers containing {@code (this / val)}
2203 * followed by {@code (this % val)}.
2204 *
2205 * @param val value by which this BigInteger is to be divided, and the
2206 * remainder computed.
2207 * @return an array of two BigIntegers: the quotient {@code (this / val)}
2208 * is the initial element, and the remainder {@code (this % val)}
2209 * is the final element.
2210 * @throws ArithmeticException if {@code val} is zero.
2211 */
2212 public BigInteger[] divideAndRemainder(BigInteger val) {
2213 if (val.mag.length < BURNIKEL_ZIEGLER_THRESHOLD ||
2214 mag.length - val.mag.length < BURNIKEL_ZIEGLER_OFFSET) {
2215 return divideAndRemainderKnuth(val);
2216 } else {
2217 return divideAndRemainderBurnikelZiegler(val);
2218 }
2219 }
2220
2221 /** Long division */
2222 private BigInteger[] divideAndRemainderKnuth(BigInteger val) {
2223 BigInteger[] result = new BigInteger[2];
2224 MutableBigInteger q = new MutableBigInteger(),
2225 a = new MutableBigInteger(this.mag),
2226 b = new MutableBigInteger(val.mag);
2227 MutableBigInteger r = a.divideKnuth(b, q);
2228 result[0] = q.toBigInteger(this.signum == val.signum ? 1 : -1);
2229 result[1] = r.toBigInteger(this.signum);
2230 return result;
2231 }
2232
2233 /**
2234 * Returns a BigInteger whose value is {@code (this % val)}.
2235 *
2236 * @param val value by which this BigInteger is to be divided, and the
2237 * remainder computed.
2238 * @return {@code this % val}
2239 * @throws ArithmeticException if {@code val} is zero.
2240 */
2241 public BigInteger remainder(BigInteger val) {
2242 if (val.mag.length < BURNIKEL_ZIEGLER_THRESHOLD ||
2243 mag.length - val.mag.length < BURNIKEL_ZIEGLER_OFFSET) {
2244 return remainderKnuth(val);
2245 } else {
2246 return remainderBurnikelZiegler(val);
2247 }
2248 }
2249
2250 /** Long division */
2251 private BigInteger remainderKnuth(BigInteger val) {
2252 MutableBigInteger q = new MutableBigInteger(),
2253 a = new MutableBigInteger(this.mag),
2254 b = new MutableBigInteger(val.mag);
2255
2256 return a.divideKnuth(b, q).toBigInteger(this.signum);
2257 }
2258
2259 /**
2260 * Calculates {@code this / val} using the Burnikel-Ziegler algorithm.
2261 * @param val the divisor
2262 * @return {@code this / val}
2263 */
2264 private BigInteger divideBurnikelZiegler(BigInteger val) {
2265 return divideAndRemainderBurnikelZiegler(val)[0];
2266 }
2267
2268 /**
2269 * Calculates {@code this % val} using the Burnikel-Ziegler algorithm.
2270 * @param val the divisor
2271 * @return {@code this % val}
2272 */
2273 private BigInteger remainderBurnikelZiegler(BigInteger val) {
2274 return divideAndRemainderBurnikelZiegler(val)[1];
2275 }
2276
2277 /**
2278 * Computes {@code this / val} and {@code this % val} using the
2279 * Burnikel-Ziegler algorithm.
2280 * @param val the divisor
2281 * @return an array containing the quotient and remainder
2282 */
2283 private BigInteger[] divideAndRemainderBurnikelZiegler(BigInteger val) {
2284 MutableBigInteger q = new MutableBigInteger();
2285 MutableBigInteger r = new MutableBigInteger(this).divideAndRemainderBurnikelZiegler(new MutableBigInteger(val), q);
2286 BigInteger qBigInt = q.isZero() ? ZERO : q.toBigInteger(signum*val.signum);
2287 BigInteger rBigInt = r.isZero() ? ZERO : r.toBigInteger(signum);
2288 return new BigInteger[] {qBigInt, rBigInt};
2289 }
2290
2291 /**
2292 * Returns a BigInteger whose value is <code>(this<sup>exponent</sup>)</code>.
2293 * Note that {@code exponent} is an integer rather than a BigInteger.
2294 *
2295 * @param exponent exponent to which this BigInteger is to be raised.
2296 * @return <code>this<sup>exponent</sup></code>
2297 * @throws ArithmeticException {@code exponent} is negative. (This would
2298 * cause the operation to yield a non-integer value.)
2299 */
2300 public BigInteger pow(int exponent) {
2301 if (exponent < 0) {
2302 throw new ArithmeticException("Negative exponent");
2303 }
2304 if (signum == 0) {
2305 return (exponent == 0 ? ONE : this);
2306 }
2307
2308 BigInteger partToSquare = this.abs();
2309
2310 // Factor out powers of two from the base, as the exponentiation of
2311 // these can be done by left shifts only.
2312 // The remaining part can then be exponentiated faster. The
2313 // powers of two will be multiplied back at the end.
2314 int powersOfTwo = partToSquare.getLowestSetBit();
2315 long bitsToShift = (long)powersOfTwo * exponent;
2316 if (bitsToShift > Integer.MAX_VALUE) {
2317 reportOverflow();
2318 }
2319
2320 int remainingBits;
2321
2322 // Factor the powers of two out quickly by shifting right, if needed.
2323 if (powersOfTwo > 0) {
2324 partToSquare = partToSquare.shiftRight(powersOfTwo);
2325 remainingBits = partToSquare.bitLength();
2326 if (remainingBits == 1) { // Nothing left but +/- 1?
2327 if (signum < 0 && (exponent&1) == 1) {
2328 return NEGATIVE_ONE.shiftLeft(powersOfTwo*exponent);
2329 } else {
2330 return ONE.shiftLeft(powersOfTwo*exponent);
2331 }
2332 }
2333 } else {
2334 remainingBits = partToSquare.bitLength();
2335 if (remainingBits == 1) { // Nothing left but +/- 1?
2336 if (signum < 0 && (exponent&1) == 1) {
2337 return NEGATIVE_ONE;
2338 } else {
2339 return ONE;
2340 }
2341 }
2342 }
2343
2344 // This is a quick way to approximate the size of the result,
2345 // similar to doing log2[n] * exponent. This will give an upper bound
2346 // of how big the result can be, and which algorithm to use.
2347 long scaleFactor = (long)remainingBits * exponent;
2348
2349 // Use slightly different algorithms for small and large operands.
2350 // See if the result will safely fit into a long. (Largest 2^63-1)
2351 if (partToSquare.mag.length == 1 && scaleFactor <= 62) {
2352 // Small number algorithm. Everything fits into a long.
2353 int newSign = (signum <0 && (exponent&1) == 1 ? -1 : 1);
2354 long result = 1;
2355 long baseToPow2 = partToSquare.mag[0] & LONG_MASK;
2356
2357 int workingExponent = exponent;
2358
2359 // Perform exponentiation using repeated squaring trick
2360 while (workingExponent != 0) {
2361 if ((workingExponent & 1) == 1) {
2362 result = result * baseToPow2;
2363 }
2364
2365 if ((workingExponent >>>= 1) != 0) {
2366 baseToPow2 = baseToPow2 * baseToPow2;
2367 }
2368 }
2369
2370 // Multiply back the powers of two (quickly, by shifting left)
2371 if (powersOfTwo > 0) {
2372 if (bitsToShift + scaleFactor <= 62) { // Fits in long?
2373 return valueOf((result << bitsToShift) * newSign);
2374 } else {
2375 return valueOf(result*newSign).shiftLeft((int) bitsToShift);
2376 }
2377 }
2378 else {
2379 return valueOf(result*newSign);
2380 }
2381 } else {
2382 // Large number algorithm. This is basically identical to
2383 // the algorithm above, but calls multiply() and square()
2384 // which may use more efficient algorithms for large numbers.
2385 BigInteger answer = ONE;
2386
2387 int workingExponent = exponent;
2388 // Perform exponentiation using repeated squaring trick
2389 while (workingExponent != 0) {
2390 if ((workingExponent & 1) == 1) {
2391 answer = answer.multiply(partToSquare);
2392 }
2393
2394 if ((workingExponent >>>= 1) != 0) {
2395 partToSquare = partToSquare.square();
2396 }
2397 }
2398 // Multiply back the (exponentiated) powers of two (quickly,
2399 // by shifting left)
2400 if (powersOfTwo > 0) {
2401 answer = answer.shiftLeft(powersOfTwo*exponent);
2402 }
2403
2404 if (signum < 0 && (exponent&1) == 1) {
2405 return answer.negate();
2406 } else {
2407 return answer;
2408 }
2409 }
2410 }
2411
2412 /**
2413 * Returns the integer square root of this BigInteger. The integer square
2414 * root of the corresponding mathematical integer {@code n} is the largest
2415 * mathematical integer {@code s} such that {@code s*s <= n}. It is equal
2416 * to the value of {@code floor(sqrt(n))}, where {@code sqrt(n)} denotes the
2417 * real square root of {@code n} treated as a real. Note that the integer
2418 * square root will be less than the real square root if the latter is not
2419 * representable as an integral value.
2420 *
2421 * @return the integer square root of {@code this}
2422 * @throws ArithmeticException if {@code this} is negative. (The square
2423 * root of a negative integer {@code val} is
2424 * {@code (i * sqrt(-val))} where <i>i</i> is the
2425 * <i>imaginary unit</i> and is equal to
2426 * {@code sqrt(-1)}.)
2427 * @since 1.9
2428 */
2429 public BigInteger sqrt() {
2430 if (this.signum < 0) {
2431 throw new ArithmeticException("Negative BigInteger");
2432 }
2433
2434 return new MutableBigInteger(this.mag).sqrt().toBigInteger();
2435 }
2436
2437 /**
2438 * Returns an array of two BigIntegers containing the integer square root
2439 * {@code s} of {@code this} and its remainder {@code this - s*s},
2440 * respectively.
2441 *
2442 * @return an array of two BigIntegers with the integer square root at
2443 * offset 0 and the remainder at offset 1
2444 * @throws ArithmeticException if {@code this} is negative. (The square
2445 * root of a negative integer {@code val} is
2446 * {@code (i * sqrt(-val))} where <i>i</i> is the
2447 * <i>imaginary unit</i> and is equal to
2448 * {@code sqrt(-1)}.)
2449 * @see #sqrt()
2450 * @since 1.9
2451 */
2452 public BigInteger[] sqrtAndRemainder() {
2453 BigInteger s = sqrt();
2454 BigInteger r = this.subtract(s.square());
2455 assert r.compareTo(BigInteger.ZERO) >= 0;
2456 return new BigInteger[] {s, r};
2457 }
2458
2459 /**
2460 * Returns a BigInteger whose value is the greatest common divisor of
2461 * {@code abs(this)} and {@code abs(val)}. Returns 0 if
2462 * {@code this == 0 && val == 0}.
2463 *
2464 * @param val value with which the GCD is to be computed.
2465 * @return {@code GCD(abs(this), abs(val))}
2466 */
2467 public BigInteger gcd(BigInteger val) {
2468 if (val.signum == 0)
2469 return this.abs();
2470 else if (this.signum == 0)
2471 return val.abs();
2472
2473 MutableBigInteger a = new MutableBigInteger(this);
2474 MutableBigInteger b = new MutableBigInteger(val);
2475
2476 MutableBigInteger result = a.hybridGCD(b);
2477
2478 return result.toBigInteger(1);
2479 }
2480
2481 /**
2482 * Package private method to return bit length for an integer.
2483 */
2484 static int bitLengthForInt(int n) {
2485 return 32 - Integer.numberOfLeadingZeros(n);
2486 }
2487
2488 /**
2489 * Left shift int array a up to len by n bits. Returns the array that
2490 * results from the shift since space may have to be reallocated.
2491 */
2492 private static int[] leftShift(int[] a, int len, int n) {
2493 int nInts = n >>> 5;
2494 int nBits = n&0x1F;
2495 int bitsInHighWord = bitLengthForInt(a[0]);
2496
2497 // If shift can be done without recopy, do so
2498 if (n <= (32-bitsInHighWord)) {
2499 primitiveLeftShift(a, len, nBits);
2500 return a;
2501 } else { // Array must be resized
2502 if (nBits <= (32-bitsInHighWord)) {
2503 int result[] = new int[nInts+len];
2504 System.arraycopy(a, 0, result, 0, len);
2505 primitiveLeftShift(result, result.length, nBits);
2506 return result;
2507 } else {
2508 int result[] = new int[nInts+len+1];
2509 System.arraycopy(a, 0, result, 0, len);
2510 primitiveRightShift(result, result.length, 32 - nBits);
2511 return result;
2512 }
2513 }
2514 }
2515
2516 // shifts a up to len right n bits assumes no leading zeros, 0<n<32
2517 static void primitiveRightShift(int[] a, int len, int n) {
2518 int n2 = 32 - n;
2519 for (int i=len-1, c=a[i]; i > 0; i--) {
2520 int b = c;
2521 c = a[i-1];
2522 a[i] = (c << n2) | (b >>> n);
2523 }
2524 a[0] >>>= n;
2525 }
2526
2527 // shifts a up to len left n bits assumes no leading zeros, 0<=n<32
2528 static void primitiveLeftShift(int[] a, int len, int n) {
2529 if (len == 0 || n == 0)
2530 return;
2531
2532 int n2 = 32 - n;
2533 for (int i=0, c=a[i], m=i+len-1; i < m; i++) {
2534 int b = c;
2535 c = a[i+1];
2536 a[i] = (b << n) | (c >>> n2);
2537 }
2538 a[len-1] <<= n;
2539 }
2540
2541 /**
2542 * Calculate bitlength of contents of the first len elements an int array,
2543 * assuming there are no leading zero ints.
2544 */
2545 private static int bitLength(int[] val, int len) {
2546 if (len == 0)
2547 return 0;
2548 return ((len - 1) << 5) + bitLengthForInt(val[0]);
2549 }
2550
2551 /**
2552 * Returns a BigInteger whose value is the absolute value of this
2553 * BigInteger.
2554 *
2555 * @return {@code abs(this)}
2556 */
2557 public BigInteger abs() {
2558 return (signum >= 0 ? this : this.negate());
2559 }
2560
2561 /**
2562 * Returns a BigInteger whose value is {@code (-this)}.
2563 *
2564 * @return {@code -this}
2565 */
2566 public BigInteger negate() {
2567 return new BigInteger(this.mag, -this.signum);
2568 }
2569
2570 /**
2571 * Returns the signum function of this BigInteger.
2572 *
2573 * @return -1, 0 or 1 as the value of this BigInteger is negative, zero or
2574 * positive.
2575 */
2576 public int signum() {
2577 return this.signum;
2578 }
2579
2580 // Modular Arithmetic Operations
2581
2582 /**
2583 * Returns a BigInteger whose value is {@code (this mod m}). This method
2584 * differs from {@code remainder} in that it always returns a
2585 * <i>non-negative</i> BigInteger.
2586 *
2587 * @param m the modulus.
2588 * @return {@code this mod m}
2589 * @throws ArithmeticException {@code m} ≤ 0
2590 * @see #remainder
2591 */
2592 public BigInteger mod(BigInteger m) {
2593 if (m.signum <= 0)
2594 throw new ArithmeticException("BigInteger: modulus not positive");
2595
2596 BigInteger result = this.remainder(m);
2597 return (result.signum >= 0 ? result : result.add(m));
2598 }
2599
2600 /**
2601 * Returns a BigInteger whose value is
2602 * <code>(this<sup>exponent</sup> mod m)</code>. (Unlike {@code pow}, this
2603 * method permits negative exponents.)
2604 *
2605 * @param exponent the exponent.
2606 * @param m the modulus.
2607 * @return <code>this<sup>exponent</sup> mod m</code>
2608 * @throws ArithmeticException {@code m} ≤ 0 or the exponent is
2609 * negative and this BigInteger is not <i>relatively
2610 * prime</i> to {@code m}.
2611 * @see #modInverse
2612 */
2613 public BigInteger modPow(BigInteger exponent, BigInteger m) {
2614 if (m.signum <= 0)
2615 throw new ArithmeticException("BigInteger: modulus not positive");
2616
2617 // Trivial cases
2618 if (exponent.signum == 0)
2619 return (m.equals(ONE) ? ZERO : ONE);
2620
2621 if (this.equals(ONE))
2622 return (m.equals(ONE) ? ZERO : ONE);
2623
2624 if (this.equals(ZERO) && exponent.signum >= 0)
2625 return ZERO;
2626
2627 if (this.equals(negConst[1]) && (!exponent.testBit(0)))
2628 return (m.equals(ONE) ? ZERO : ONE);
2629
2630 boolean invertResult;
2631 if ((invertResult = (exponent.signum < 0)))
2632 exponent = exponent.negate();
2633
2634 BigInteger base = (this.signum < 0 || this.compareTo(m) >= 0
2635 ? this.mod(m) : this);
2636 BigInteger result;
2637 if (m.testBit(0)) { // odd modulus
2638 result = base.oddModPow(exponent, m);
2639 } else {
2640 /*
2641 * Even modulus. Tear it into an "odd part" (m1) and power of two
2642 * (m2), exponentiate mod m1, manually exponentiate mod m2, and
2643 * use Chinese Remainder Theorem to combine results.
2644 */
2645
2646 // Tear m apart into odd part (m1) and power of 2 (m2)
2647 int p = m.getLowestSetBit(); // Max pow of 2 that divides m
2648
2649 BigInteger m1 = m.shiftRight(p); // m/2**p
2650 BigInteger m2 = ONE.shiftLeft(p); // 2**p
2651
2652 // Calculate new base from m1
2653 BigInteger base2 = (this.signum < 0 || this.compareTo(m1) >= 0
2654 ? this.mod(m1) : this);
2655
2656 // Caculate (base ** exponent) mod m1.
2657 BigInteger a1 = (m1.equals(ONE) ? ZERO :
2658 base2.oddModPow(exponent, m1));
2659
2660 // Calculate (this ** exponent) mod m2
2661 BigInteger a2 = base.modPow2(exponent, p);
2662
2663 // Combine results using Chinese Remainder Theorem
2664 BigInteger y1 = m2.modInverse(m1);
2665 BigInteger y2 = m1.modInverse(m2);
2666
2667 if (m.mag.length < MAX_MAG_LENGTH / 2) {
2668 result = a1.multiply(m2).multiply(y1).add(a2.multiply(m1).multiply(y2)).mod(m);
2669 } else {
2670 MutableBigInteger t1 = new MutableBigInteger();
2671 new MutableBigInteger(a1.multiply(m2)).multiply(new MutableBigInteger(y1), t1);
2672 MutableBigInteger t2 = new MutableBigInteger();
2673 new MutableBigInteger(a2.multiply(m1)).multiply(new MutableBigInteger(y2), t2);
2674 t1.add(t2);
2675 MutableBigInteger q = new MutableBigInteger();
2676 result = t1.divide(new MutableBigInteger(m), q).toBigInteger();
2677 }
2678 }
2679
2680 return (invertResult ? result.modInverse(m) : result);
2681 }
2682
2683 // Montgomery multiplication. These are wrappers for
2684 // implMontgomeryXX routines which are expected to be replaced by
2685 // virtual machine intrinsics. We don't use the intrinsics for
2686 // very large operands: MONTGOMERY_INTRINSIC_THRESHOLD should be
2687 // larger than any reasonable crypto key.
2688 private static int[] montgomeryMultiply(int[] a, int[] b, int[] n, int len, long inv,
2689 int[] product) {
2690 implMontgomeryMultiplyChecks(a, b, n, len, product);
2691 if (len > MONTGOMERY_INTRINSIC_THRESHOLD) {
2692 // Very long argument: do not use an intrinsic
2693 product = multiplyToLen(a, len, b, len, product);
2694 return montReduce(product, n, len, (int)inv);
2695 } else {
2696 return implMontgomeryMultiply(a, b, n, len, inv, materialize(product, len));
2697 }
2698 }
2699 private static int[] montgomerySquare(int[] a, int[] n, int len, long inv,
2700 int[] product) {
2701 implMontgomeryMultiplyChecks(a, a, n, len, product);
2702 if (len > MONTGOMERY_INTRINSIC_THRESHOLD) {
2703 // Very long argument: do not use an intrinsic
2704 product = squareToLen(a, len, product);
2705 return montReduce(product, n, len, (int)inv);
2706 } else {
2707 return implMontgomerySquare(a, n, len, inv, materialize(product, len));
2708 }
2709 }
2710
2711 // Range-check everything.
2712 private static void implMontgomeryMultiplyChecks
2713 (int[] a, int[] b, int[] n, int len, int[] product) throws RuntimeException {
2714 if (len % 2 != 0) {
2715 throw new IllegalArgumentException("input array length must be even: " + len);
2716 }
2717
2718 if (len < 1) {
2719 throw new IllegalArgumentException("invalid input length: " + len);
2720 }
2721
2722 if (len > a.length ||
2723 len > b.length ||
2724 len > n.length ||
2725 (product != null && len > product.length)) {
2726 throw new IllegalArgumentException("input array length out of bound: " + len);
2727 }
2728 }
2729
2730 // Make sure that the int array z (which is expected to contain
2731 // the result of a Montgomery multiplication) is present and
2732 // sufficiently large.
2733 private static int[] materialize(int[] z, int len) {
2734 if (z == null || z.length < len)
2735 z = new int[len];
2736 return z;
2737 }
2738
2739 // These methods are intended to be be replaced by virtual machine
2740 // intrinsics.
2741 @HotSpotIntrinsicCandidate
2742 private static int[] implMontgomeryMultiply(int[] a, int[] b, int[] n, int len,
2743 long inv, int[] product) {
2744 product = multiplyToLen(a, len, b, len, product);
2745 return montReduce(product, n, len, (int)inv);
2746 }
2747 @HotSpotIntrinsicCandidate
2748 private static int[] implMontgomerySquare(int[] a, int[] n, int len,
2749 long inv, int[] product) {
2750 product = squareToLen(a, len, product);
2751 return montReduce(product, n, len, (int)inv);
2752 }
2753
2754 static int[] bnExpModThreshTable = {7, 25, 81, 241, 673, 1793,
2755 Integer.MAX_VALUE}; // Sentinel
2756
2757 /**
2758 * Returns a BigInteger whose value is x to the power of y mod z.
2759 * Assumes: z is odd && x < z.
2760 */
2761 private BigInteger oddModPow(BigInteger y, BigInteger z) {
2762 /*
2763 * The algorithm is adapted from Colin Plumb's C library.
2764 *
2765 * The window algorithm:
2766 * The idea is to keep a running product of b1 = n^(high-order bits of exp)
2767 * and then keep appending exponent bits to it. The following patterns
2768 * apply to a 3-bit window (k = 3):
2769 * To append 0: square
2770 * To append 1: square, multiply by n^1
2771 * To append 10: square, multiply by n^1, square
2772 * To append 11: square, square, multiply by n^3
2773 * To append 100: square, multiply by n^1, square, square
2774 * To append 101: square, square, square, multiply by n^5
2775 * To append 110: square, square, multiply by n^3, square
2776 * To append 111: square, square, square, multiply by n^7
2777 *
2778 * Since each pattern involves only one multiply, the longer the pattern
2779 * the better, except that a 0 (no multiplies) can be appended directly.
2780 * We precompute a table of odd powers of n, up to 2^k, and can then
2781 * multiply k bits of exponent at a time. Actually, assuming random
2782 * exponents, there is on average one zero bit between needs to
2783 * multiply (1/2 of the time there's none, 1/4 of the time there's 1,
2784 * 1/8 of the time, there's 2, 1/32 of the time, there's 3, etc.), so
2785 * you have to do one multiply per k+1 bits of exponent.
2786 *
2787 * The loop walks down the exponent, squaring the result buffer as
2788 * it goes. There is a wbits+1 bit lookahead buffer, buf, that is
2789 * filled with the upcoming exponent bits. (What is read after the
2790 * end of the exponent is unimportant, but it is filled with zero here.)
2791 * When the most-significant bit of this buffer becomes set, i.e.
2792 * (buf & tblmask) != 0, we have to decide what pattern to multiply
2793 * by, and when to do it. We decide, remember to do it in future
2794 * after a suitable number of squarings have passed (e.g. a pattern
2795 * of "100" in the buffer requires that we multiply by n^1 immediately;
2796 * a pattern of "110" calls for multiplying by n^3 after one more
2797 * squaring), clear the buffer, and continue.
2798 *
2799 * When we start, there is one more optimization: the result buffer
2800 * is implcitly one, so squaring it or multiplying by it can be
2801 * optimized away. Further, if we start with a pattern like "100"
2802 * in the lookahead window, rather than placing n into the buffer
2803 * and then starting to square it, we have already computed n^2
2804 * to compute the odd-powers table, so we can place that into
2805 * the buffer and save a squaring.
2806 *
2807 * This means that if you have a k-bit window, to compute n^z,
2808 * where z is the high k bits of the exponent, 1/2 of the time
2809 * it requires no squarings. 1/4 of the time, it requires 1
2810 * squaring, ... 1/2^(k-1) of the time, it reqires k-2 squarings.
2811 * And the remaining 1/2^(k-1) of the time, the top k bits are a
2812 * 1 followed by k-1 0 bits, so it again only requires k-2
2813 * squarings, not k-1. The average of these is 1. Add that
2814 * to the one squaring we have to do to compute the table,
2815 * and you'll see that a k-bit window saves k-2 squarings
2816 * as well as reducing the multiplies. (It actually doesn't
2817 * hurt in the case k = 1, either.)
2818 */
2819 // Special case for exponent of one
2820 if (y.equals(ONE))
2821 return this;
2822
2823 // Special case for base of zero
2824 if (signum == 0)
2825 return ZERO;
2826
2827 int[] base = mag.clone();
2828 int[] exp = y.mag;
2829 int[] mod = z.mag;
2830 int modLen = mod.length;
2831
2832 // Make modLen even. It is conventional to use a cryptographic
2833 // modulus that is 512, 768, 1024, or 2048 bits, so this code
2834 // will not normally be executed. However, it is necessary for
2835 // the correct functioning of the HotSpot intrinsics.
2836 if ((modLen & 1) != 0) {
2837 int[] x = new int[modLen + 1];
2838 System.arraycopy(mod, 0, x, 1, modLen);
2839 mod = x;
2840 modLen++;
2841 }
2842
2843 // Select an appropriate window size
2844 int wbits = 0;
2845 int ebits = bitLength(exp, exp.length);
2846 // if exponent is 65537 (0x10001), use minimum window size
2847 if ((ebits != 17) || (exp[0] != 65537)) {
2848 while (ebits > bnExpModThreshTable[wbits]) {
2849 wbits++;
2850 }
2851 }
2852
2853 // Calculate appropriate table size
2854 int tblmask = 1 << wbits;
2855
2856 // Allocate table for precomputed odd powers of base in Montgomery form
2857 int[][] table = new int[tblmask][];
2858 for (int i=0; i < tblmask; i++)
2859 table[i] = new int[modLen];
2860
2861 // Compute the modular inverse of the least significant 64-bit
2862 // digit of the modulus
2863 long n0 = (mod[modLen-1] & LONG_MASK) + ((mod[modLen-2] & LONG_MASK) << 32);
2864 long inv = -MutableBigInteger.inverseMod64(n0);
2865
2866 // Convert base to Montgomery form
2867 int[] a = leftShift(base, base.length, modLen << 5);
2868
2869 MutableBigInteger q = new MutableBigInteger(),
2870 a2 = new MutableBigInteger(a),
2871 b2 = new MutableBigInteger(mod);
2872 b2.normalize(); // MutableBigInteger.divide() assumes that its
2873 // divisor is in normal form.
2874
2875 MutableBigInteger r= a2.divide(b2, q);
2876 table[0] = r.toIntArray();
2877
2878 // Pad table[0] with leading zeros so its length is at least modLen
2879 if (table[0].length < modLen) {
2880 int offset = modLen - table[0].length;
2881 int[] t2 = new int[modLen];
2882 System.arraycopy(table[0], 0, t2, offset, table[0].length);
2883 table[0] = t2;
2884 }
2885
2886 // Set b to the square of the base
2887 int[] b = montgomerySquare(table[0], mod, modLen, inv, null);
2888
2889 // Set t to high half of b
2890 int[] t = Arrays.copyOf(b, modLen);
2891
2892 // Fill in the table with odd powers of the base
2893 for (int i=1; i < tblmask; i++) {
2894 table[i] = montgomeryMultiply(t, table[i-1], mod, modLen, inv, null);
2895 }
2896
2897 // Pre load the window that slides over the exponent
2898 int bitpos = 1 << ((ebits-1) & (32-1));
2899
2900 int buf = 0;
2901 int elen = exp.length;
2902 int eIndex = 0;
2903 for (int i = 0; i <= wbits; i++) {
2904 buf = (buf << 1) | (((exp[eIndex] & bitpos) != 0)?1:0);
2905 bitpos >>>= 1;
2906 if (bitpos == 0) {
2907 eIndex++;
2908 bitpos = 1 << (32-1);
2909 elen--;
2910 }
2911 }
2912
2913 int multpos = ebits;
2914
2915 // The first iteration, which is hoisted out of the main loop
2916 ebits--;
2917 boolean isone = true;
2918
2919 multpos = ebits - wbits;
2920 while ((buf & 1) == 0) {
2921 buf >>>= 1;
2922 multpos++;
2923 }
2924
2925 int[] mult = table[buf >>> 1];
2926
2927 buf = 0;
2928 if (multpos == ebits)
2929 isone = false;
2930
2931 // The main loop
2932 while (true) {
2933 ebits--;
2934 // Advance the window
2935 buf <<= 1;
2936
2937 if (elen != 0) {
2938 buf |= ((exp[eIndex] & bitpos) != 0) ? 1 : 0;
2939 bitpos >>>= 1;
2940 if (bitpos == 0) {
2941 eIndex++;
2942 bitpos = 1 << (32-1);
2943 elen--;
2944 }
2945 }
2946
2947 // Examine the window for pending multiplies
2948 if ((buf & tblmask) != 0) {
2949 multpos = ebits - wbits;
2950 while ((buf & 1) == 0) {
2951 buf >>>= 1;
2952 multpos++;
2953 }
2954 mult = table[buf >>> 1];
2955 buf = 0;
2956 }
2957
2958 // Perform multiply
2959 if (ebits == multpos) {
2960 if (isone) {
2961 b = mult.clone();
2962 isone = false;
2963 } else {
2964 t = b;
2965 a = montgomeryMultiply(t, mult, mod, modLen, inv, a);
2966 t = a; a = b; b = t;
2967 }
2968 }
2969
2970 // Check if done
2971 if (ebits == 0)
2972 break;
2973
2974 // Square the input
2975 if (!isone) {
2976 t = b;
2977 a = montgomerySquare(t, mod, modLen, inv, a);
2978 t = a; a = b; b = t;
2979 }
2980 }
2981
2982 // Convert result out of Montgomery form and return
2983 int[] t2 = new int[2*modLen];
2984 System.arraycopy(b, 0, t2, modLen, modLen);
2985
2986 b = montReduce(t2, mod, modLen, (int)inv);
2987
2988 t2 = Arrays.copyOf(b, modLen);
2989
2990 return new BigInteger(1, t2);
2991 }
2992
2993 /**
2994 * Montgomery reduce n, modulo mod. This reduces modulo mod and divides
2995 * by 2^(32*mlen). Adapted from Colin Plumb's C library.
2996 */
2997 private static int[] montReduce(int[] n, int[] mod, int mlen, int inv) {
2998 int c=0;
2999 int len = mlen;
3000 int offset=0;
3001
3002 do {
3003 int nEnd = n[n.length-1-offset];
3004 int carry = mulAdd(n, mod, offset, mlen, inv * nEnd);
3005 c += addOne(n, offset, mlen, carry);
3006 offset++;
3007 } while (--len > 0);
3008
3009 while (c > 0)
3010 c += subN(n, mod, mlen);
3011
3012 while (intArrayCmpToLen(n, mod, mlen) >= 0)
3013 subN(n, mod, mlen);
3014
3015 return n;
3016 }
3017
3018
3019 /*
3020 * Returns -1, 0 or +1 as big-endian unsigned int array arg1 is less than,
3021 * equal to, or greater than arg2 up to length len.
3022 */
3023 private static int intArrayCmpToLen(int[] arg1, int[] arg2, int len) {
3024 for (int i=0; i < len; i++) {
3025 long b1 = arg1[i] & LONG_MASK;
3026 long b2 = arg2[i] & LONG_MASK;
3027 if (b1 < b2)
3028 return -1;
3029 if (b1 > b2)
3030 return 1;
3031 }
3032 return 0;
3033 }
3034
3035 /**
3036 * Subtracts two numbers of same length, returning borrow.
3037 */
3038 private static int subN(int[] a, int[] b, int len) {
3039 long sum = 0;
3040
3041 while (--len >= 0) {
3042 sum = (a[len] & LONG_MASK) -
3043 (b[len] & LONG_MASK) + (sum >> 32);
3044 a[len] = (int)sum;
3045 }
3046
3047 return (int)(sum >> 32);
3048 }
3049
3050 /**
3051 * Multiply an array by one word k and add to result, return the carry
3052 */
3053 static int mulAdd(int[] out, int[] in, int offset, int len, int k) {
3054 implMulAddCheck(out, in, offset, len, k);
3055 return implMulAdd(out, in, offset, len, k);
3056 }
3057
3058 /**
3059 * Parameters validation.
3060 */
3061 private static void implMulAddCheck(int[] out, int[] in, int offset, int len, int k) {
3062 if (len > in.length) {
3063 throw new IllegalArgumentException("input length is out of bound: " + len + " > " + in.length);
3064 }
3065 if (offset < 0) {
3066 throw new IllegalArgumentException("input offset is invalid: " + offset);
3067 }
3068 if (offset > (out.length - 1)) {
3069 throw new IllegalArgumentException("input offset is out of bound: " + offset + " > " + (out.length - 1));
3070 }
3071 if (len > (out.length - offset)) {
3072 throw new IllegalArgumentException("input len is out of bound: " + len + " > " + (out.length - offset));
3073 }
3074 }
3075
3076 /**
3077 * Java Runtime may use intrinsic for this method.
3078 */
3079 @HotSpotIntrinsicCandidate
3080 private static int implMulAdd(int[] out, int[] in, int offset, int len, int k) {
3081 long kLong = k & LONG_MASK;
3082 long carry = 0;
3083
3084 offset = out.length-offset - 1;
3085 for (int j=len-1; j >= 0; j--) {
3086 long product = (in[j] & LONG_MASK) * kLong +
3087 (out[offset] & LONG_MASK) + carry;
3088 out[offset--] = (int)product;
3089 carry = product >>> 32;
3090 }
3091 return (int)carry;
3092 }
3093
3094 /**
3095 * Add one word to the number a mlen words into a. Return the resulting
3096 * carry.
3097 */
3098 static int addOne(int[] a, int offset, int mlen, int carry) {
3099 offset = a.length-1-mlen-offset;
3100 long t = (a[offset] & LONG_MASK) + (carry & LONG_MASK);
3101
3102 a[offset] = (int)t;
3103 if ((t >>> 32) == 0)
3104 return 0;
3105 while (--mlen >= 0) {
3106 if (--offset < 0) { // Carry out of number
3107 return 1;
3108 } else {
3109 a[offset]++;
3110 if (a[offset] != 0)
3111 return 0;
3112 }
3113 }
3114 return 1;
3115 }
3116
3117 /**
3118 * Returns a BigInteger whose value is (this ** exponent) mod (2**p)
3119 */
3120 private BigInteger modPow2(BigInteger exponent, int p) {
3121 /*
3122 * Perform exponentiation using repeated squaring trick, chopping off
3123 * high order bits as indicated by modulus.
3124 */
3125 BigInteger result = ONE;
3126 BigInteger baseToPow2 = this.mod2(p);
3127 int expOffset = 0;
3128
3129 int limit = exponent.bitLength();
3130
3131 if (this.testBit(0))
3132 limit = (p-1) < limit ? (p-1) : limit;
3133
3134 while (expOffset < limit) {
3135 if (exponent.testBit(expOffset))
3136 result = result.multiply(baseToPow2).mod2(p);
3137 expOffset++;
3138 if (expOffset < limit)
3139 baseToPow2 = baseToPow2.square().mod2(p);
3140 }
3141
3142 return result;
3143 }
3144
3145 /**
3146 * Returns a BigInteger whose value is this mod(2**p).
3147 * Assumes that this {@code BigInteger >= 0} and {@code p > 0}.
3148 */
3149 private BigInteger mod2(int p) {
3150 if (bitLength() <= p)
3151 return this;
3152
3153 // Copy remaining ints of mag
3154 int numInts = (p + 31) >>> 5;
3155 int[] mag = new int[numInts];
3156 System.arraycopy(this.mag, (this.mag.length - numInts), mag, 0, numInts);
3157
3158 // Mask out any excess bits
3159 int excessBits = (numInts << 5) - p;
3160 mag[0] &= (1L << (32-excessBits)) - 1;
3161
3162 return (mag[0] == 0 ? new BigInteger(1, mag) : new BigInteger(mag, 1));
3163 }
3164
3165 /**
3166 * Returns a BigInteger whose value is {@code (this}<sup>-1</sup> {@code mod m)}.
3167 *
3168 * @param m the modulus.
3169 * @return {@code this}<sup>-1</sup> {@code mod m}.
3170 * @throws ArithmeticException {@code m} ≤ 0, or this BigInteger
3171 * has no multiplicative inverse mod m (that is, this BigInteger
3172 * is not <i>relatively prime</i> to m).
3173 */
3174 public BigInteger modInverse(BigInteger m) {
3175 if (m.signum != 1)
3176 throw new ArithmeticException("BigInteger: modulus not positive");
3177
3178 if (m.equals(ONE))
3179 return ZERO;
3180
3181 // Calculate (this mod m)
3182 BigInteger modVal = this;
3183 if (signum < 0 || (this.compareMagnitude(m) >= 0))
3184 modVal = this.mod(m);
3185
3186 if (modVal.equals(ONE))
3187 return ONE;
3188
3189 MutableBigInteger a = new MutableBigInteger(modVal);
3190 MutableBigInteger b = new MutableBigInteger(m);
3191
3192 MutableBigInteger result = a.mutableModInverse(b);
3193 return result.toBigInteger(1);
3194 }
3195
3196 // Shift Operations
3197
3198 /**
3199 * Returns a BigInteger whose value is {@code (this << n)}.
3200 * The shift distance, {@code n}, may be negative, in which case
3201 * this method performs a right shift.
3202 * (Computes <code>floor(this * 2<sup>n</sup>)</code>.)
3203 *
3204 * @param n shift distance, in bits.
3205 * @return {@code this << n}
3206 * @see #shiftRight
3207 */
3208 public BigInteger shiftLeft(int n) {
3209 if (signum == 0)
3210 return ZERO;
3211 if (n > 0) {
3212 return new BigInteger(shiftLeft(mag, n), signum);
3213 } else if (n == 0) {
3214 return this;
3215 } else {
3216 // Possible int overflow in (-n) is not a trouble,
3217 // because shiftRightImpl considers its argument unsigned
3218 return shiftRightImpl(-n);
3219 }
3220 }
3221
3222 /**
3223 * Returns a magnitude array whose value is {@code (mag << n)}.
3224 * The shift distance, {@code n}, is considered unnsigned.
3225 * (Computes <code>this * 2<sup>n</sup></code>.)
3226 *
3227 * @param mag magnitude, the most-significant int ({@code mag[0]}) must be non-zero.
3228 * @param n unsigned shift distance, in bits.
3229 * @return {@code mag << n}
3230 */
3231 private static int[] shiftLeft(int[] mag, int n) {
3232 int nInts = n >>> 5;
3233 int nBits = n & 0x1f;
3234 int magLen = mag.length;
3235 int newMag[] = null;
3236
3237 if (nBits == 0) {
3238 newMag = new int[magLen + nInts];
3239 System.arraycopy(mag, 0, newMag, 0, magLen);
3240 } else {
3241 int i = 0;
3242 int nBits2 = 32 - nBits;
3243 int highBits = mag[0] >>> nBits2;
3244 if (highBits != 0) {
3245 newMag = new int[magLen + nInts + 1];
3246 newMag[i++] = highBits;
3247 } else {
3248 newMag = new int[magLen + nInts];
3249 }
3250 int j=0;
3251 while (j < magLen-1)
3252 newMag[i++] = mag[j++] << nBits | mag[j] >>> nBits2;
3253 newMag[i] = mag[j] << nBits;
3254 }
3255 return newMag;
3256 }
3257
3258 /**
3259 * Returns a BigInteger whose value is {@code (this >> n)}. Sign
3260 * extension is performed. The shift distance, {@code n}, may be
3261 * negative, in which case this method performs a left shift.
3262 * (Computes <code>floor(this / 2<sup>n</sup>)</code>.)
3263 *
3264 * @param n shift distance, in bits.
3265 * @return {@code this >> n}
3266 * @see #shiftLeft
3267 */
3268 public BigInteger shiftRight(int n) {
3269 if (signum == 0)
3270 return ZERO;
3271 if (n > 0) {
3272 return shiftRightImpl(n);
3273 } else if (n == 0) {
3274 return this;
3275 } else {
3276 // Possible int overflow in {@code -n} is not a trouble,
3277 // because shiftLeft considers its argument unsigned
3278 return new BigInteger(shiftLeft(mag, -n), signum);
3279 }
3280 }
3281
3282 /**
3283 * Returns a BigInteger whose value is {@code (this >> n)}. The shift
3284 * distance, {@code n}, is considered unsigned.
3285 * (Computes <code>floor(this * 2<sup>-n</sup>)</code>.)
3286 *
3287 * @param n unsigned shift distance, in bits.
3288 * @return {@code this >> n}
3289 */
3290 private BigInteger shiftRightImpl(int n) {
3291 int nInts = n >>> 5;
3292 int nBits = n & 0x1f;
3293 int magLen = mag.length;
3294 int newMag[] = null;
3295
3296 // Special case: entire contents shifted off the end
3297 if (nInts >= magLen)
3298 return (signum >= 0 ? ZERO : negConst[1]);
3299
3300 if (nBits == 0) {
3301 int newMagLen = magLen - nInts;
3302 newMag = Arrays.copyOf(mag, newMagLen);
3303 } else {
3304 int i = 0;
3305 int highBits = mag[0] >>> nBits;
3306 if (highBits != 0) {
3307 newMag = new int[magLen - nInts];
3308 newMag[i++] = highBits;
3309 } else {
3310 newMag = new int[magLen - nInts -1];
3311 }
3312
3313 int nBits2 = 32 - nBits;
3314 int j=0;
3315 while (j < magLen - nInts - 1)
3316 newMag[i++] = (mag[j++] << nBits2) | (mag[j] >>> nBits);
3317 }
3318
3319 if (signum < 0) {
3320 // Find out whether any one-bits were shifted off the end.
3321 boolean onesLost = false;
3322 for (int i=magLen-1, j=magLen-nInts; i >= j && !onesLost; i--)
3323 onesLost = (mag[i] != 0);
3324 if (!onesLost && nBits != 0)
3325 onesLost = (mag[magLen - nInts - 1] << (32 - nBits) != 0);
3326
3327 if (onesLost)
3328 newMag = javaIncrement(newMag);
3329 }
3330
3331 return new BigInteger(newMag, signum);
3332 }
3333
3334 int[] javaIncrement(int[] val) {
3335 int lastSum = 0;
3336 for (int i=val.length-1; i >= 0 && lastSum == 0; i--)
3337 lastSum = (val[i] += 1);
3338 if (lastSum == 0) {
3339 val = new int[val.length+1];
3340 val[0] = 1;
3341 }
3342 return val;
3343 }
3344
3345 // Bitwise Operations
3346
3347 /**
3348 * Returns a BigInteger whose value is {@code (this & val)}. (This
3349 * method returns a negative BigInteger if and only if this and val are
3350 * both negative.)
3351 *
3352 * @param val value to be AND'ed with this BigInteger.
3353 * @return {@code this & val}
3354 */
3355 public BigInteger and(BigInteger val) {
3356 int[] result = new int[Math.max(intLength(), val.intLength())];
3357 for (int i=0; i < result.length; i++)
3358 result[i] = (getInt(result.length-i-1)
3359 & val.getInt(result.length-i-1));
3360
3361 return valueOf(result);
3362 }
3363
3364 /**
3365 * Returns a BigInteger whose value is {@code (this | val)}. (This method
3366 * returns a negative BigInteger if and only if either this or val is
3367 * negative.)
3368 *
3369 * @param val value to be OR'ed with this BigInteger.
3370 * @return {@code this | val}
3371 */
3372 public BigInteger or(BigInteger val) {
3373 int[] result = new int[Math.max(intLength(), val.intLength())];
3374 for (int i=0; i < result.length; i++)
3375 result[i] = (getInt(result.length-i-1)
3376 | val.getInt(result.length-i-1));
3377
3378 return valueOf(result);
3379 }
3380
3381 /**
3382 * Returns a BigInteger whose value is {@code (this ^ val)}. (This method
3383 * returns a negative BigInteger if and only if exactly one of this and
3384 * val are negative.)
3385 *
3386 * @param val value to be XOR'ed with this BigInteger.
3387 * @return {@code this ^ val}
3388 */
3389 public BigInteger xor(BigInteger val) {
3390 int[] result = new int[Math.max(intLength(), val.intLength())];
3391 for (int i=0; i < result.length; i++)
3392 result[i] = (getInt(result.length-i-1)
3393 ^ val.getInt(result.length-i-1));
3394
3395 return valueOf(result);
3396 }
3397
3398 /**
3399 * Returns a BigInteger whose value is {@code (~this)}. (This method
3400 * returns a negative value if and only if this BigInteger is
3401 * non-negative.)
3402 *
3403 * @return {@code ~this}
3404 */
3405 public BigInteger not() {
3406 int[] result = new int[intLength()];
3407 for (int i=0; i < result.length; i++)
3408 result[i] = ~getInt(result.length-i-1);
3409
3410 return valueOf(result);
3411 }
3412
3413 /**
3414 * Returns a BigInteger whose value is {@code (this & ~val)}. This
3415 * method, which is equivalent to {@code and(val.not())}, is provided as
3416 * a convenience for masking operations. (This method returns a negative
3417 * BigInteger if and only if {@code this} is negative and {@code val} is
3418 * positive.)
3419 *
3420 * @param val value to be complemented and AND'ed with this BigInteger.
3421 * @return {@code this & ~val}
3422 */
3423 public BigInteger andNot(BigInteger val) {
3424 int[] result = new int[Math.max(intLength(), val.intLength())];
3425 for (int i=0; i < result.length; i++)
3426 result[i] = (getInt(result.length-i-1)
3427 & ~val.getInt(result.length-i-1));
3428
3429 return valueOf(result);
3430 }
3431
3432
3433 // Single Bit Operations
3434
3435 /**
3436 * Returns {@code true} if and only if the designated bit is set.
3437 * (Computes {@code ((this & (1<<n)) != 0)}.)
3438 *
3439 * @param n index of bit to test.
3440 * @return {@code true} if and only if the designated bit is set.
3441 * @throws ArithmeticException {@code n} is negative.
3442 */
3443 public boolean testBit(int n) {
3444 if (n < 0)
3445 throw new ArithmeticException("Negative bit address");
3446
3447 return (getInt(n >>> 5) & (1 << (n & 31))) != 0;
3448 }
3449
3450 /**
3451 * Returns a BigInteger whose value is equivalent to this BigInteger
3452 * with the designated bit set. (Computes {@code (this | (1<<n))}.)
3453 *
3454 * @param n index of bit to set.
3455 * @return {@code this | (1<<n)}
3456 * @throws ArithmeticException {@code n} is negative.
3457 */
3458 public BigInteger setBit(int n) {
3459 if (n < 0)
3460 throw new ArithmeticException("Negative bit address");
3461
3462 int intNum = n >>> 5;
3463 int[] result = new int[Math.max(intLength(), intNum+2)];
3464
3465 for (int i=0; i < result.length; i++)
3466 result[result.length-i-1] = getInt(i);
3467
3468 result[result.length-intNum-1] |= (1 << (n & 31));
3469
3470 return valueOf(result);
3471 }
3472
3473 /**
3474 * Returns a BigInteger whose value is equivalent to this BigInteger
3475 * with the designated bit cleared.
3476 * (Computes {@code (this & ~(1<<n))}.)
3477 *
3478 * @param n index of bit to clear.
3479 * @return {@code this & ~(1<<n)}
3480 * @throws ArithmeticException {@code n} is negative.
3481 */
3482 public BigInteger clearBit(int n) {
3483 if (n < 0)
3484 throw new ArithmeticException("Negative bit address");
3485
3486 int intNum = n >>> 5;
3487 int[] result = new int[Math.max(intLength(), ((n + 1) >>> 5) + 1)];
3488
3489 for (int i=0; i < result.length; i++)
3490 result[result.length-i-1] = getInt(i);
3491
3492 result[result.length-intNum-1] &= ~(1 << (n & 31));
3493
3494 return valueOf(result);
3495 }
3496
3497 /**
3498 * Returns a BigInteger whose value is equivalent to this BigInteger
3499 * with the designated bit flipped.
3500 * (Computes {@code (this ^ (1<<n))}.)
3501 *
3502 * @param n index of bit to flip.
3503 * @return {@code this ^ (1<<n)}
3504 * @throws ArithmeticException {@code n} is negative.
3505 */
3506 public BigInteger flipBit(int n) {
3507 if (n < 0)
3508 throw new ArithmeticException("Negative bit address");
3509
3510 int intNum = n >>> 5;
3511 int[] result = new int[Math.max(intLength(), intNum+2)];
3512
3513 for (int i=0; i < result.length; i++)
3514 result[result.length-i-1] = getInt(i);
3515
3516 result[result.length-intNum-1] ^= (1 << (n & 31));
3517
3518 return valueOf(result);
3519 }
3520
3521 /**
3522 * Returns the index of the rightmost (lowest-order) one bit in this
3523 * BigInteger (the number of zero bits to the right of the rightmost
3524 * one bit). Returns -1 if this BigInteger contains no one bits.
3525 * (Computes {@code (this == 0? -1 : log2(this & -this))}.)
3526 *
3527 * @return index of the rightmost one bit in this BigInteger.
3528 */
3529 public int getLowestSetBit() {
3530 int lsb = lowestSetBitPlusTwo - 2;
3531 if (lsb == -2) { // lowestSetBit not initialized yet
3532 lsb = 0;
3533 if (signum == 0) {
3534 lsb -= 1;
3535 } else {
3536 // Search for lowest order nonzero int
3537 int i,b;
3538 for (i=0; (b = getInt(i)) == 0; i++)
3539 ;
3540 lsb += (i << 5) + Integer.numberOfTrailingZeros(b);
3541 }
3542 lowestSetBitPlusTwo = lsb + 2;
3543 }
3544 return lsb;
3545 }
3546
3547
3548 // Miscellaneous Bit Operations
3549
3550 /**
3551 * Returns the number of bits in the minimal two's-complement
3552 * representation of this BigInteger, <i>excluding</i> a sign bit.
3553 * For positive BigIntegers, this is equivalent to the number of bits in
3554 * the ordinary binary representation. (Computes
3555 * {@code (ceil(log2(this < 0 ? -this : this+1)))}.)
3556 *
3557 * @return number of bits in the minimal two's-complement
3558 * representation of this BigInteger, <i>excluding</i> a sign bit.
3559 */
3560 public int bitLength() {
3561 int n = bitLengthPlusOne - 1;
3562 if (n == -1) { // bitLength not initialized yet
3563 int[] m = mag;
3564 int len = m.length;
3565 if (len == 0) {
3566 n = 0; // offset by one to initialize
3567 } else {
3568 // Calculate the bit length of the magnitude
3569 int magBitLength = ((len - 1) << 5) + bitLengthForInt(mag[0]);
3570 if (signum < 0) {
3571 // Check if magnitude is a power of two
3572 boolean pow2 = (Integer.bitCount(mag[0]) == 1);
3573 for (int i=1; i< len && pow2; i++)
3574 pow2 = (mag[i] == 0);
3575
3576 n = (pow2 ? magBitLength -1 : magBitLength);
3577 } else {
3578 n = magBitLength;
3579 }
3580 }
3581 bitLengthPlusOne = n + 1;
3582 }
3583 return n;
3584 }
3585
3586 /**
3587 * Returns the number of bits in the two's complement representation
3588 * of this BigInteger that differ from its sign bit. This method is
3589 * useful when implementing bit-vector style sets atop BigIntegers.
3590 *
3591 * @return number of bits in the two's complement representation
3592 * of this BigInteger that differ from its sign bit.
3593 */
3594 public int bitCount() {
3595 int bc = bitCountPlusOne - 1;
3596 if (bc == -1) { // bitCount not initialized yet
3597 bc = 0; // offset by one to initialize
3598 // Count the bits in the magnitude
3599 for (int i=0; i < mag.length; i++)
3600 bc += Integer.bitCount(mag[i]);
3601 if (signum < 0) {
3602 // Count the trailing zeros in the magnitude
3603 int magTrailingZeroCount = 0, j;
3604 for (j=mag.length-1; mag[j] == 0; j--)
3605 magTrailingZeroCount += 32;
3606 magTrailingZeroCount += Integer.numberOfTrailingZeros(mag[j]);
3607 bc += magTrailingZeroCount - 1;
3608 }
3609 bitCountPlusOne = bc + 1;
3610 }
3611 return bc;
3612 }
3613
3614 // Primality Testing
3615
3616 /**
3617 * Returns {@code true} if this BigInteger is probably prime,
3618 * {@code false} if it's definitely composite. If
3619 * {@code certainty} is ≤ 0, {@code true} is
3620 * returned.
3621 *
3622 * @param certainty a measure of the uncertainty that the caller is
3623 * willing to tolerate: if the call returns {@code true}
3624 * the probability that this BigInteger is prime exceeds
3625 * (1 - 1/2<sup>{@code certainty}</sup>). The execution time of
3626 * this method is proportional to the value of this parameter.
3627 * @return {@code true} if this BigInteger is probably prime,
3628 * {@code false} if it's definitely composite.
3629 */
3630 public boolean isProbablePrime(int certainty) {
3631 if (certainty <= 0)
3632 return true;
3633 BigInteger w = this.abs();
3634 if (w.equals(TWO))
3635 return true;
3636 if (!w.testBit(0) || w.equals(ONE))
3637 return false;
3638
3639 return w.primeToCertainty(certainty, null);
3640 }
3641
3642 // Comparison Operations
3643
3644 /**
3645 * Compares this BigInteger with the specified BigInteger. This
3646 * method is provided in preference to individual methods for each
3647 * of the six boolean comparison operators ({@literal <}, ==,
3648 * {@literal >}, {@literal >=}, !=, {@literal <=}). The suggested
3649 * idiom for performing these comparisons is: {@code
3650 * (x.compareTo(y)} <<i>op</i>> {@code 0)}, where
3651 * <<i>op</i>> is one of the six comparison operators.
3652 *
3653 * @param val BigInteger to which this BigInteger is to be compared.
3654 * @return -1, 0 or 1 as this BigInteger is numerically less than, equal
3655 * to, or greater than {@code val}.
3656 */
3657 public int compareTo(BigInteger val) {
3658 if (signum == val.signum) {
3659 switch (signum) {
3660 case 1:
3661 return compareMagnitude(val);
3662 case -1:
3663 return val.compareMagnitude(this);
3664 default:
3665 return 0;
3666 }
3667 }
3668 return signum > val.signum ? 1 : -1;
3669 }
3670
3671 /**
3672 * Compares the magnitude array of this BigInteger with the specified
3673 * BigInteger's. This is the version of compareTo ignoring sign.
3674 *
3675 * @param val BigInteger whose magnitude array to be compared.
3676 * @return -1, 0 or 1 as this magnitude array is less than, equal to or
3677 * greater than the magnitude aray for the specified BigInteger's.
3678 */
3679 final int compareMagnitude(BigInteger val) {
3680 int[] m1 = mag;
3681 int len1 = m1.length;
3682 int[] m2 = val.mag;
3683 int len2 = m2.length;
3684 if (len1 < len2)
3685 return -1;
3686 if (len1 > len2)
3687 return 1;
3688 for (int i = 0; i < len1; i++) {
3689 int a = m1[i];
3690 int b = m2[i];
3691 if (a != b)
3692 return ((a & LONG_MASK) < (b & LONG_MASK)) ? -1 : 1;
3693 }
3694 return 0;
3695 }
3696
3697 /**
3698 * Version of compareMagnitude that compares magnitude with long value.
3699 * val can't be Long.MIN_VALUE.
3700 */
3701 final int compareMagnitude(long val) {
3702 assert val != Long.MIN_VALUE;
3703 int[] m1 = mag;
3704 int len = m1.length;
3705 if (len > 2) {
3706 return 1;
3707 }
3708 if (val < 0) {
3709 val = -val;
3710 }
3711 int highWord = (int)(val >>> 32);
3712 if (highWord == 0) {
3713 if (len < 1)
3714 return -1;
3715 if (len > 1)
3716 return 1;
3717 int a = m1[0];
3718 int b = (int)val;
3719 if (a != b) {
3720 return ((a & LONG_MASK) < (b & LONG_MASK))? -1 : 1;
3721 }
3722 return 0;
3723 } else {
3724 if (len < 2)
3725 return -1;
3726 int a = m1[0];
3727 int b = highWord;
3728 if (a != b) {
3729 return ((a & LONG_MASK) < (b & LONG_MASK))? -1 : 1;
3730 }
3731 a = m1[1];
3732 b = (int)val;
3733 if (a != b) {
3734 return ((a & LONG_MASK) < (b & LONG_MASK))? -1 : 1;
3735 }
3736 return 0;
3737 }
3738 }
3739
3740 /**
3741 * Compares this BigInteger with the specified Object for equality.
3742 *
3743 * @param x Object to which this BigInteger is to be compared.
3744 * @return {@code true} if and only if the specified Object is a
3745 * BigInteger whose value is numerically equal to this BigInteger.
3746 */
3747 public boolean equals(Object x) {
3748 // This test is just an optimization, which may or may not help
3749 if (x == this)
3750 return true;
3751
3752 if (!(x instanceof BigInteger))
3753 return false;
3754
3755 BigInteger xInt = (BigInteger) x;
3756 if (xInt.signum != signum)
3757 return false;
3758
3759 int[] m = mag;
3760 int len = m.length;
3761 int[] xm = xInt.mag;
3762 if (len != xm.length)
3763 return false;
3764
3765 for (int i = 0; i < len; i++)
3766 if (xm[i] != m[i])
3767 return false;
3768
3769 return true;
3770 }
3771
3772 /**
3773 * Returns the minimum of this BigInteger and {@code val}.
3774 *
3775 * @param val value with which the minimum is to be computed.
3776 * @return the BigInteger whose value is the lesser of this BigInteger and
3777 * {@code val}. If they are equal, either may be returned.
3778 */
3779 public BigInteger min(BigInteger val) {
3780 return (compareTo(val) < 0 ? this : val);
3781 }
3782
3783 /**
3784 * Returns the maximum of this BigInteger and {@code val}.
3785 *
3786 * @param val value with which the maximum is to be computed.
3787 * @return the BigInteger whose value is the greater of this and
3788 * {@code val}. If they are equal, either may be returned.
3789 */
3790 public BigInteger max(BigInteger val) {
3791 return (compareTo(val) > 0 ? this : val);
3792 }
3793
3794
3795 // Hash Function
3796
3797 /**
3798 * Returns the hash code for this BigInteger.
3799 *
3800 * @return hash code for this BigInteger.
3801 */
3802 public int hashCode() {
3803 int hashCode = 0;
3804
3805 for (int i=0; i < mag.length; i++)
3806 hashCode = (int)(31*hashCode + (mag[i] & LONG_MASK));
3807
3808 return hashCode * signum;
3809 }
3810
3811 /**
3812 * Returns the String representation of this BigInteger in the
3813 * given radix. If the radix is outside the range from {@link
3814 * Character#MIN_RADIX} to {@link Character#MAX_RADIX} inclusive,
3815 * it will default to 10 (as is the case for
3816 * {@code Integer.toString}). The digit-to-character mapping
3817 * provided by {@code Character.forDigit} is used, and a minus
3818 * sign is prepended if appropriate. (This representation is
3819 * compatible with the {@link #BigInteger(String, int) (String,
3820 * int)} constructor.)
3821 *
3822 * @param radix radix of the String representation.
3823 * @return String representation of this BigInteger in the given radix.
3824 * @see Integer#toString
3825 * @see Character#forDigit
3826 * @see #BigInteger(java.lang.String, int)
3827 */
3828 public String toString(int radix) {
3829 if (signum == 0)
3830 return "0";
3831 if (radix < Character.MIN_RADIX || radix > Character.MAX_RADIX)
3832 radix = 10;
3833
3834 // If it's small enough, use smallToString.
3835 if (mag.length <= SCHOENHAGE_BASE_CONVERSION_THRESHOLD)
3836 return smallToString(radix);
3837
3838 // Otherwise use recursive toString, which requires positive arguments.
3839 // The results will be concatenated into this StringBuilder
3840 StringBuilder sb = new StringBuilder();
3841 if (signum < 0) {
3842 toString(this.negate(), sb, radix, 0);
3843 sb.insert(0, '-');
3844 }
3845 else
3846 toString(this, sb, radix, 0);
3847
3848 return sb.toString();
3849 }
3850
3851 /** This method is used to perform toString when arguments are small. */
3852 private String smallToString(int radix) {
3853 if (signum == 0) {
3854 return "0";
3855 }
3856
3857 // Compute upper bound on number of digit groups and allocate space
3858 int maxNumDigitGroups = (4*mag.length + 6)/7;
3859 String digitGroup[] = new String[maxNumDigitGroups];
3860
3861 // Translate number to string, a digit group at a time
3862 BigInteger tmp = this.abs();
3863 int numGroups = 0;
3864 while (tmp.signum != 0) {
3865 BigInteger d = longRadix[radix];
3866
3867 MutableBigInteger q = new MutableBigInteger(),
3868 a = new MutableBigInteger(tmp.mag),
3869 b = new MutableBigInteger(d.mag);
3870 MutableBigInteger r = a.divide(b, q);
3871 BigInteger q2 = q.toBigInteger(tmp.signum * d.signum);
3872 BigInteger r2 = r.toBigInteger(tmp.signum * d.signum);
3873
3874 digitGroup[numGroups++] = Long.toString(r2.longValue(), radix);
3875 tmp = q2;
3876 }
3877
3878 // Put sign (if any) and first digit group into result buffer
3879 StringBuilder buf = new StringBuilder(numGroups*digitsPerLong[radix]+1);
3880 if (signum < 0) {
3881 buf.append('-');
3882 }
3883 buf.append(digitGroup[numGroups-1]);
3884
3885 // Append remaining digit groups padded with leading zeros
3886 for (int i=numGroups-2; i >= 0; i--) {
3887 // Prepend (any) leading zeros for this digit group
3888 int numLeadingZeros = digitsPerLong[radix]-digitGroup[i].length();
3889 if (numLeadingZeros != 0) {
3890 buf.append(zeros[numLeadingZeros]);
3891 }
3892 buf.append(digitGroup[i]);
3893 }
3894 return buf.toString();
3895 }
3896
3897 /**
3898 * Converts the specified BigInteger to a string and appends to
3899 * {@code sb}. This implements the recursive Schoenhage algorithm
3900 * for base conversions.
3901 * <p>
3902 * See Knuth, Donald, _The Art of Computer Programming_, Vol. 2,
3903 * Answers to Exercises (4.4) Question 14.
3904 *
3905 * @param u The number to convert to a string.
3906 * @param sb The StringBuilder that will be appended to in place.
3907 * @param radix The base to convert to.
3908 * @param digits The minimum number of digits to pad to.
3909 */
3910 private static void toString(BigInteger u, StringBuilder sb, int radix,
3911 int digits) {
3912 // If we're smaller than a certain threshold, use the smallToString
3913 // method, padding with leading zeroes when necessary.
3914 if (u.mag.length <= SCHOENHAGE_BASE_CONVERSION_THRESHOLD) {
3915 String s = u.smallToString(radix);
3916
3917 // Pad with internal zeros if necessary.
3918 // Don't pad if we're at the beginning of the string.
3919 if ((s.length() < digits) && (sb.length() > 0)) {
3920 for (int i=s.length(); i < digits; i++) {
3921 sb.append('0');
3922 }
3923 }
3924
3925 sb.append(s);
3926 return;
3927 }
3928
3929 int b, n;
3930 b = u.bitLength();
3931
3932 // Calculate a value for n in the equation radix^(2^n) = u
3933 // and subtract 1 from that value. This is used to find the
3934 // cache index that contains the best value to divide u.
3935 n = (int) Math.round(Math.log(b * LOG_TWO / logCache[radix]) / LOG_TWO - 1.0);
3936 BigInteger v = getRadixConversionCache(radix, n);
3937 BigInteger[] results;
3938 results = u.divideAndRemainder(v);
3939
3940 int expectedDigits = 1 << n;
3941
3942 // Now recursively build the two halves of each number.
3943 toString(results[0], sb, radix, digits-expectedDigits);
3944 toString(results[1], sb, radix, expectedDigits);
3945 }
3946
3947 /**
3948 * Returns the value radix^(2^exponent) from the cache.
3949 * If this value doesn't already exist in the cache, it is added.
3950 * <p>
3951 * This could be changed to a more complicated caching method using
3952 * {@code Future}.
3953 */
3954 private static BigInteger getRadixConversionCache(int radix, int exponent) {
3955 BigInteger[] cacheLine = powerCache[radix]; // volatile read
3956 if (exponent < cacheLine.length) {
3957 return cacheLine[exponent];
3958 }
3959
3960 int oldLength = cacheLine.length;
3961 cacheLine = Arrays.copyOf(cacheLine, exponent + 1);
3962 for (int i = oldLength; i <= exponent; i++) {
3963 cacheLine[i] = cacheLine[i - 1].pow(2);
3964 }
3965
3966 BigInteger[][] pc = powerCache; // volatile read again
3967 if (exponent >= pc[radix].length) {
3968 pc = pc.clone();
3969 pc[radix] = cacheLine;
3970 powerCache = pc; // volatile write, publish
3971 }
3972 return cacheLine[exponent];
3973 }
3974
3975 /* zero[i] is a string of i consecutive zeros. */
3976 private static String zeros[] = new String[64];
3977 static {
3978 zeros[63] =
3979 "000000000000000000000000000000000000000000000000000000000000000";
3980 for (int i=0; i < 63; i++)
3981 zeros[i] = zeros[63].substring(0, i);
3982 }
3983
3984 /**
3985 * Returns the decimal String representation of this BigInteger.
3986 * The digit-to-character mapping provided by
3987 * {@code Character.forDigit} is used, and a minus sign is
3988 * prepended if appropriate. (This representation is compatible
3989 * with the {@link #BigInteger(String) (String)} constructor, and
3990 * allows for String concatenation with Java's + operator.)
3991 *
3992 * @return decimal String representation of this BigInteger.
3993 * @see Character#forDigit
3994 * @see #BigInteger(java.lang.String)
3995 */
3996 public String toString() {
3997 return toString(10);
3998 }
3999
4000 /**
4001 * Returns a byte array containing the two's-complement
4002 * representation of this BigInteger. The byte array will be in
4003 * <i>big-endian</i> byte-order: the most significant byte is in
4004 * the zeroth element. The array will contain the minimum number
4005 * of bytes required to represent this BigInteger, including at
4006 * least one sign bit, which is {@code (ceil((this.bitLength() +
4007 * 1)/8))}. (This representation is compatible with the
4008 * {@link #BigInteger(byte[]) (byte[])} constructor.)
4009 *
4010 * @return a byte array containing the two's-complement representation of
4011 * this BigInteger.
4012 * @see #BigInteger(byte[])
4013 */
4014 public byte[] toByteArray() {
4015 int byteLen = bitLength()/8 + 1;
4016 byte[] byteArray = new byte[byteLen];
4017
4018 for (int i=byteLen-1, bytesCopied=4, nextInt=0, intIndex=0; i >= 0; i--) {
4019 if (bytesCopied == 4) {
4020 nextInt = getInt(intIndex++);
4021 bytesCopied = 1;
4022 } else {
4023 nextInt >>>= 8;
4024 bytesCopied++;
4025 }
4026 byteArray[i] = (byte)nextInt;
4027 }
4028 return byteArray;
4029 }
4030
4031 /**
4032 * Converts this BigInteger to an {@code int}. This
4033 * conversion is analogous to a
4034 * <i>narrowing primitive conversion</i> from {@code long} to
4035 * {@code int} as defined in section 5.1.3 of
4036 * <cite>The Java™ Language Specification</cite>:
4037 * if this BigInteger is too big to fit in an
4038 * {@code int}, only the low-order 32 bits are returned.
4039 * Note that this conversion can lose information about the
4040 * overall magnitude of the BigInteger value as well as return a
4041 * result with the opposite sign.
4042 *
4043 * @return this BigInteger converted to an {@code int}.
4044 * @see #intValueExact()
4045 */
4046 public int intValue() {
4047 int result = 0;
4048 result = getInt(0);
4049 return result;
4050 }
4051
4052 /**
4053 * Converts this BigInteger to a {@code long}. This
4054 * conversion is analogous to a
4055 * <i>narrowing primitive conversion</i> from {@code long} to
4056 * {@code int} as defined in section 5.1.3 of
4057 * <cite>The Java™ Language Specification</cite>:
4058 * if this BigInteger is too big to fit in a
4059 * {@code long}, only the low-order 64 bits are returned.
4060 * Note that this conversion can lose information about the
4061 * overall magnitude of the BigInteger value as well as return a
4062 * result with the opposite sign.
4063 *
4064 * @return this BigInteger converted to a {@code long}.
4065 * @see #longValueExact()
4066 */
4067 public long longValue() {
4068 long result = 0;
4069
4070 for (int i=1; i >= 0; i--)
4071 result = (result << 32) + (getInt(i) & LONG_MASK);
4072 return result;
4073 }
4074
4075 /**
4076 * Converts this BigInteger to a {@code float}. This
4077 * conversion is similar to the
4078 * <i>narrowing primitive conversion</i> from {@code double} to
4079 * {@code float} as defined in section 5.1.3 of
4080 * <cite>The Java™ Language Specification</cite>:
4081 * if this BigInteger has too great a magnitude
4082 * to represent as a {@code float}, it will be converted to
4083 * {@link Float#NEGATIVE_INFINITY} or {@link
4084 * Float#POSITIVE_INFINITY} as appropriate. Note that even when
4085 * the return value is finite, this conversion can lose
4086 * information about the precision of the BigInteger value.
4087 *
4088 * @return this BigInteger converted to a {@code float}.
4089 */
4090 public float floatValue() {
4091 if (signum == 0) {
4092 return 0.0f;
4093 }
4094
4095 int exponent = ((mag.length - 1) << 5) + bitLengthForInt(mag[0]) - 1;
4096
4097 // exponent == floor(log2(abs(this)))
4098 if (exponent < Long.SIZE - 1) {
4099 return longValue();
4100 } else if (exponent > Float.MAX_EXPONENT) {
4101 return signum > 0 ? Float.POSITIVE_INFINITY : Float.NEGATIVE_INFINITY;
4102 }
4103
4104 /*
4105 * We need the top SIGNIFICAND_WIDTH bits, including the "implicit"
4106 * one bit. To make rounding easier, we pick out the top
4107 * SIGNIFICAND_WIDTH + 1 bits, so we have one to help us round up or
4108 * down. twiceSignifFloor will contain the top SIGNIFICAND_WIDTH + 1
4109 * bits, and signifFloor the top SIGNIFICAND_WIDTH.
4110 *
4111 * It helps to consider the real number signif = abs(this) *
4112 * 2^(SIGNIFICAND_WIDTH - 1 - exponent).
4113 */
4114 int shift = exponent - FloatConsts.SIGNIFICAND_WIDTH;
4115
4116 int twiceSignifFloor;
4117 // twiceSignifFloor will be == abs().shiftRight(shift).intValue()
4118 // We do the shift into an int directly to improve performance.
4119
4120 int nBits = shift & 0x1f;
4121 int nBits2 = 32 - nBits;
4122
4123 if (nBits == 0) {
4124 twiceSignifFloor = mag[0];
4125 } else {
4126 twiceSignifFloor = mag[0] >>> nBits;
4127 if (twiceSignifFloor == 0) {
4128 twiceSignifFloor = (mag[0] << nBits2) | (mag[1] >>> nBits);
4129 }
4130 }
4131
4132 int signifFloor = twiceSignifFloor >> 1;
4133 signifFloor &= FloatConsts.SIGNIF_BIT_MASK; // remove the implied bit
4134
4135 /*
4136 * We round up if either the fractional part of signif is strictly
4137 * greater than 0.5 (which is true if the 0.5 bit is set and any lower
4138 * bit is set), or if the fractional part of signif is >= 0.5 and
4139 * signifFloor is odd (which is true if both the 0.5 bit and the 1 bit
4140 * are set). This is equivalent to the desired HALF_EVEN rounding.
4141 */
4142 boolean increment = (twiceSignifFloor & 1) != 0
4143 && ((signifFloor & 1) != 0 || abs().getLowestSetBit() < shift);
4144 int signifRounded = increment ? signifFloor + 1 : signifFloor;
4145 int bits = ((exponent + FloatConsts.EXP_BIAS))
4146 << (FloatConsts.SIGNIFICAND_WIDTH - 1);
4147 bits += signifRounded;
4148 /*
4149 * If signifRounded == 2^24, we'd need to set all of the significand
4150 * bits to zero and add 1 to the exponent. This is exactly the behavior
4151 * we get from just adding signifRounded to bits directly. If the
4152 * exponent is Float.MAX_EXPONENT, we round up (correctly) to
4153 * Float.POSITIVE_INFINITY.
4154 */
4155 bits |= signum & FloatConsts.SIGN_BIT_MASK;
4156 return Float.intBitsToFloat(bits);
4157 }
4158
4159 /**
4160 * Converts this BigInteger to a {@code double}. This
4161 * conversion is similar to the
4162 * <i>narrowing primitive conversion</i> from {@code double} to
4163 * {@code float} as defined in section 5.1.3 of
4164 * <cite>The Java™ Language Specification</cite>:
4165 * if this BigInteger has too great a magnitude
4166 * to represent as a {@code double}, it will be converted to
4167 * {@link Double#NEGATIVE_INFINITY} or {@link
4168 * Double#POSITIVE_INFINITY} as appropriate. Note that even when
4169 * the return value is finite, this conversion can lose
4170 * information about the precision of the BigInteger value.
4171 *
4172 * @return this BigInteger converted to a {@code double}.
4173 */
4174 public double doubleValue() {
4175 if (signum == 0) {
4176 return 0.0;
4177 }
4178
4179 int exponent = ((mag.length - 1) << 5) + bitLengthForInt(mag[0]) - 1;
4180
4181 // exponent == floor(log2(abs(this))Double)
4182 if (exponent < Long.SIZE - 1) {
4183 return longValue();
4184 } else if (exponent > Double.MAX_EXPONENT) {
4185 return signum > 0 ? Double.POSITIVE_INFINITY : Double.NEGATIVE_INFINITY;
4186 }
4187
4188 /*
4189 * We need the top SIGNIFICAND_WIDTH bits, including the "implicit"
4190 * one bit. To make rounding easier, we pick out the top
4191 * SIGNIFICAND_WIDTH + 1 bits, so we have one to help us round up or
4192 * down. twiceSignifFloor will contain the top SIGNIFICAND_WIDTH + 1
4193 * bits, and signifFloor the top SIGNIFICAND_WIDTH.
4194 *
4195 * It helps to consider the real number signif = abs(this) *
4196 * 2^(SIGNIFICAND_WIDTH - 1 - exponent).
4197 */
4198 int shift = exponent - DoubleConsts.SIGNIFICAND_WIDTH;
4199
4200 long twiceSignifFloor;
4201 // twiceSignifFloor will be == abs().shiftRight(shift).longValue()
4202 // We do the shift into a long directly to improve performance.
4203
4204 int nBits = shift & 0x1f;
4205 int nBits2 = 32 - nBits;
4206
4207 int highBits;
4208 int lowBits;
4209 if (nBits == 0) {
4210 highBits = mag[0];
4211 lowBits = mag[1];
4212 } else {
4213 highBits = mag[0] >>> nBits;
4214 lowBits = (mag[0] << nBits2) | (mag[1] >>> nBits);
4215 if (highBits == 0) {
4216 highBits = lowBits;
4217 lowBits = (mag[1] << nBits2) | (mag[2] >>> nBits);
4218 }
4219 }
4220
4221 twiceSignifFloor = ((highBits & LONG_MASK) << 32)
4222 | (lowBits & LONG_MASK);
4223
4224 long signifFloor = twiceSignifFloor >> 1;
4225 signifFloor &= DoubleConsts.SIGNIF_BIT_MASK; // remove the implied bit
4226
4227 /*
4228 * We round up if either the fractional part of signif is strictly
4229 * greater than 0.5 (which is true if the 0.5 bit is set and any lower
4230 * bit is set), or if the fractional part of signif is >= 0.5 and
4231 * signifFloor is odd (which is true if both the 0.5 bit and the 1 bit
4232 * are set). This is equivalent to the desired HALF_EVEN rounding.
4233 */
4234 boolean increment = (twiceSignifFloor & 1) != 0
4235 && ((signifFloor & 1) != 0 || abs().getLowestSetBit() < shift);
4236 long signifRounded = increment ? signifFloor + 1 : signifFloor;
4237 long bits = (long) ((exponent + DoubleConsts.EXP_BIAS))
4238 << (DoubleConsts.SIGNIFICAND_WIDTH - 1);
4239 bits += signifRounded;
4240 /*
4241 * If signifRounded == 2^53, we'd need to set all of the significand
4242 * bits to zero and add 1 to the exponent. This is exactly the behavior
4243 * we get from just adding signifRounded to bits directly. If the
4244 * exponent is Double.MAX_EXPONENT, we round up (correctly) to
4245 * Double.POSITIVE_INFINITY.
4246 */
4247 bits |= signum & DoubleConsts.SIGN_BIT_MASK;
4248 return Double.longBitsToDouble(bits);
4249 }
4250
4251 /**
4252 * Returns a copy of the input array stripped of any leading zero bytes.
4253 */
4254 private static int[] stripLeadingZeroInts(int val[]) {
4255 int vlen = val.length;
4256 int keep;
4257
4258 // Find first nonzero byte
4259 for (keep = 0; keep < vlen && val[keep] == 0; keep++)
4260 ;
4261 return java.util.Arrays.copyOfRange(val, keep, vlen);
4262 }
4263
4264 /**
4265 * Returns the input array stripped of any leading zero bytes.
4266 * Since the source is trusted the copying may be skipped.
4267 */
4268 private static int[] trustedStripLeadingZeroInts(int val[]) {
4269 int vlen = val.length;
4270 int keep;
4271
4272 // Find first nonzero byte
4273 for (keep = 0; keep < vlen && val[keep] == 0; keep++)
4274 ;
4275 return keep == 0 ? val : java.util.Arrays.copyOfRange(val, keep, vlen);
4276 }
4277
4278 /**
4279 * Returns a copy of the input array stripped of any leading zero bytes.
4280 */
4281 private static int[] stripLeadingZeroBytes(byte a[], int off, int len) {
4282 int indexBound = off + len;
4283 int keep;
4284
4285 // Find first nonzero byte
4286 for (keep = off; keep < indexBound && a[keep] == 0; keep++)
4287 ;
4288
4289 // Allocate new array and copy relevant part of input array
4290 int intLength = ((indexBound - keep) + 3) >>> 2;
4291 int[] result = new int[intLength];
4292 int b = indexBound - 1;
4293 for (int i = intLength-1; i >= 0; i--) {
4294 result[i] = a[b--] & 0xff;
4295 int bytesRemaining = b - keep + 1;
4296 int bytesToTransfer = Math.min(3, bytesRemaining);
4297 for (int j=8; j <= (bytesToTransfer << 3); j += 8)
4298 result[i] |= ((a[b--] & 0xff) << j);
4299 }
4300 return result;
4301 }
4302
4303 /**
4304 * Takes an array a representing a negative 2's-complement number and
4305 * returns the minimal (no leading zero bytes) unsigned whose value is -a.
4306 */
4307 private static int[] makePositive(byte a[], int off, int len) {
4308 int keep, k;
4309 int indexBound = off + len;
4310
4311 // Find first non-sign (0xff) byte of input
4312 for (keep=off; keep < indexBound && a[keep] == -1; keep++)
4313 ;
4314
4315
4316 /* Allocate output array. If all non-sign bytes are 0x00, we must
4317 * allocate space for one extra output byte. */
4318 for (k=keep; k < indexBound && a[k] == 0; k++)
4319 ;
4320
4321 int extraByte = (k == indexBound) ? 1 : 0;
4322 int intLength = ((indexBound - keep + extraByte) + 3) >>> 2;
4323 int result[] = new int[intLength];
4324
4325 /* Copy one's complement of input into output, leaving extra
4326 * byte (if it exists) == 0x00 */
4327 int b = indexBound - 1;
4328 for (int i = intLength-1; i >= 0; i--) {
4329 result[i] = a[b--] & 0xff;
4330 int numBytesToTransfer = Math.min(3, b-keep+1);
4331 if (numBytesToTransfer < 0)
4332 numBytesToTransfer = 0;
4333 for (int j=8; j <= 8*numBytesToTransfer; j += 8)
4334 result[i] |= ((a[b--] & 0xff) << j);
4335
4336 // Mask indicates which bits must be complemented
4337 int mask = -1 >>> (8*(3-numBytesToTransfer));
4338 result[i] = ~result[i] & mask;
4339 }
4340
4341 // Add one to one's complement to generate two's complement
4342 for (int i=result.length-1; i >= 0; i--) {
4343 result[i] = (int)((result[i] & LONG_MASK) + 1);
4344 if (result[i] != 0)
4345 break;
4346 }
4347
4348 return result;
4349 }
4350
4351 /**
4352 * Takes an array a representing a negative 2's-complement number and
4353 * returns the minimal (no leading zero ints) unsigned whose value is -a.
4354 */
4355 private static int[] makePositive(int a[]) {
4356 int keep, j;
4357
4358 // Find first non-sign (0xffffffff) int of input
4359 for (keep=0; keep < a.length && a[keep] == -1; keep++)
4360 ;
4361
4362 /* Allocate output array. If all non-sign ints are 0x00, we must
4363 * allocate space for one extra output int. */
4364 for (j=keep; j < a.length && a[j] == 0; j++)
4365 ;
4366 int extraInt = (j == a.length ? 1 : 0);
4367 int result[] = new int[a.length - keep + extraInt];
4368
4369 /* Copy one's complement of input into output, leaving extra
4370 * int (if it exists) == 0x00 */
4371 for (int i = keep; i < a.length; i++)
4372 result[i - keep + extraInt] = ~a[i];
4373
4374 // Add one to one's complement to generate two's complement
4375 for (int i=result.length-1; ++result[i] == 0; i--)
4376 ;
4377
4378 return result;
4379 }
4380
4381 /*
4382 * The following two arrays are used for fast String conversions. Both
4383 * are indexed by radix. The first is the number of digits of the given
4384 * radix that can fit in a Java long without "going negative", i.e., the
4385 * highest integer n such that radix**n < 2**63. The second is the
4386 * "long radix" that tears each number into "long digits", each of which
4387 * consists of the number of digits in the corresponding element in
4388 * digitsPerLong (longRadix[i] = i**digitPerLong[i]). Both arrays have
4389 * nonsense values in their 0 and 1 elements, as radixes 0 and 1 are not
4390 * used.
4391 */
4392 private static int digitsPerLong[] = {0, 0,
4393 62, 39, 31, 27, 24, 22, 20, 19, 18, 18, 17, 17, 16, 16, 15, 15, 15, 14,
4394 14, 14, 14, 13, 13, 13, 13, 13, 13, 12, 12, 12, 12, 12, 12, 12, 12};
4395
4396 private static BigInteger longRadix[] = {null, null,
4397 valueOf(0x4000000000000000L), valueOf(0x383d9170b85ff80bL),
4398 valueOf(0x4000000000000000L), valueOf(0x6765c793fa10079dL),
4399 valueOf(0x41c21cb8e1000000L), valueOf(0x3642798750226111L),
4400 valueOf(0x1000000000000000L), valueOf(0x12bf307ae81ffd59L),
4401 valueOf( 0xde0b6b3a7640000L), valueOf(0x4d28cb56c33fa539L),
4402 valueOf(0x1eca170c00000000L), valueOf(0x780c7372621bd74dL),
4403 valueOf(0x1e39a5057d810000L), valueOf(0x5b27ac993df97701L),
4404 valueOf(0x1000000000000000L), valueOf(0x27b95e997e21d9f1L),
4405 valueOf(0x5da0e1e53c5c8000L), valueOf( 0xb16a458ef403f19L),
4406 valueOf(0x16bcc41e90000000L), valueOf(0x2d04b7fdd9c0ef49L),
4407 valueOf(0x5658597bcaa24000L), valueOf( 0x6feb266931a75b7L),
4408 valueOf( 0xc29e98000000000L), valueOf(0x14adf4b7320334b9L),
4409 valueOf(0x226ed36478bfa000L), valueOf(0x383d9170b85ff80bL),
4410 valueOf(0x5a3c23e39c000000L), valueOf( 0x4e900abb53e6b71L),
4411 valueOf( 0x7600ec618141000L), valueOf( 0xaee5720ee830681L),
4412 valueOf(0x1000000000000000L), valueOf(0x172588ad4f5f0981L),
4413 valueOf(0x211e44f7d02c1000L), valueOf(0x2ee56725f06e5c71L),
4414 valueOf(0x41c21cb8e1000000L)};
4415
4416 /*
4417 * These two arrays are the integer analogue of above.
4418 */
4419 private static int digitsPerInt[] = {0, 0, 30, 19, 15, 13, 11,
4420 11, 10, 9, 9, 8, 8, 8, 8, 7, 7, 7, 7, 7, 7, 7, 6, 6, 6, 6,
4421 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 5};
4422
4423 private static int intRadix[] = {0, 0,
4424 0x40000000, 0x4546b3db, 0x40000000, 0x48c27395, 0x159fd800,
4425 0x75db9c97, 0x40000000, 0x17179149, 0x3b9aca00, 0xcc6db61,
4426 0x19a10000, 0x309f1021, 0x57f6c100, 0xa2f1b6f, 0x10000000,
4427 0x18754571, 0x247dbc80, 0x3547667b, 0x4c4b4000, 0x6b5a6e1d,
4428 0x6c20a40, 0x8d2d931, 0xb640000, 0xe8d4a51, 0x1269ae40,
4429 0x17179149, 0x1cb91000, 0x23744899, 0x2b73a840, 0x34e63b41,
4430 0x40000000, 0x4cfa3cc1, 0x5c13d840, 0x6d91b519, 0x39aa400
4431 };
4432
4433 /**
4434 * These routines provide access to the two's complement representation
4435 * of BigIntegers.
4436 */
4437
4438 /**
4439 * Returns the length of the two's complement representation in ints,
4440 * including space for at least one sign bit.
4441 */
4442 private int intLength() {
4443 return (bitLength() >>> 5) + 1;
4444 }
4445
4446 /* Returns sign bit */
4447 private int signBit() {
4448 return signum < 0 ? 1 : 0;
4449 }
4450
4451 /* Returns an int of sign bits */
4452 private int signInt() {
4453 return signum < 0 ? -1 : 0;
4454 }
4455
4456 /**
4457 * Returns the specified int of the little-endian two's complement
4458 * representation (int 0 is the least significant). The int number can
4459 * be arbitrarily high (values are logically preceded by infinitely many
4460 * sign ints).
4461 */
4462 private int getInt(int n) {
4463 if (n < 0)
4464 return 0;
4465 if (n >= mag.length)
4466 return signInt();
4467
4468 int magInt = mag[mag.length-n-1];
4469
4470 return (signum >= 0 ? magInt :
4471 (n <= firstNonzeroIntNum() ? -magInt : ~magInt));
4472 }
4473
4474 /**
4475 * Returns the index of the int that contains the first nonzero int in the
4476 * little-endian binary representation of the magnitude (int 0 is the
4477 * least significant). If the magnitude is zero, return value is undefined.
4478 *
4479 * <p>Note: never used for a BigInteger with a magnitude of zero.
4480 * @see #getInt.
4481 */
4482 private int firstNonzeroIntNum() {
4483 int fn = firstNonzeroIntNumPlusTwo - 2;
4484 if (fn == -2) { // firstNonzeroIntNum not initialized yet
4485 // Search for the first nonzero int
4486 int i;
4487 int mlen = mag.length;
4488 for (i = mlen - 1; i >= 0 && mag[i] == 0; i--)
4489 ;
4490 fn = mlen - i - 1;
4491 firstNonzeroIntNumPlusTwo = fn + 2; // offset by two to initialize
4492 }
4493 return fn;
4494 }
4495
4496 /** use serialVersionUID from JDK 1.1. for interoperability */
4497 private static final long serialVersionUID = -8287574255936472291L;
4498
4499 /**
4500 * Serializable fields for BigInteger.
4501 *
4502 * @serialField signum int
4503 * signum of this BigInteger
4504 * @serialField magnitude byte[]
4505 * magnitude array of this BigInteger
4506 * @serialField bitCount int
4507 * appears in the serialized form for backward compatibility
4508 * @serialField bitLength int
4509 * appears in the serialized form for backward compatibility
4510 * @serialField firstNonzeroByteNum int
4511 * appears in the serialized form for backward compatibility
4512 * @serialField lowestSetBit int
4513 * appears in the serialized form for backward compatibility
4514 */
4515 private static final ObjectStreamField[] serialPersistentFields = {
4516 new ObjectStreamField("signum", Integer.TYPE),
4517 new ObjectStreamField("magnitude", byte[].class),
4518 new ObjectStreamField("bitCount", Integer.TYPE),
4519 new ObjectStreamField("bitLength", Integer.TYPE),
4520 new ObjectStreamField("firstNonzeroByteNum", Integer.TYPE),
4521 new ObjectStreamField("lowestSetBit", Integer.TYPE)
4522 };
4523
4524 /**
4525 * Reconstitute the {@code BigInteger} instance from a stream (that is,
4526 * deserialize it). The magnitude is read in as an array of bytes
4527 * for historical reasons, but it is converted to an array of ints
4528 * and the byte array is discarded.
4529 * Note:
4530 * The current convention is to initialize the cache fields, bitCountPlusOne,
4531 * bitLengthPlusOne and lowestSetBitPlusTwo, to 0 rather than some other
4532 * marker value. Therefore, no explicit action to set these fields needs to
4533 * be taken in readObject because those fields already have a 0 value by
4534 * default since defaultReadObject is not being used.
4535 */
4536 private void readObject(java.io.ObjectInputStream s)
4537 throws java.io.IOException, ClassNotFoundException {
4538 // prepare to read the alternate persistent fields
4539 ObjectInputStream.GetField fields = s.readFields();
4540
4541 // Read the alternate persistent fields that we care about
4542 int sign = fields.get("signum", -2);
4543 byte[] magnitude = (byte[])fields.get("magnitude", null);
4544
4545 // Validate signum
4546 if (sign < -1 || sign > 1) {
4547 String message = "BigInteger: Invalid signum value";
4548 if (fields.defaulted("signum"))
4549 message = "BigInteger: Signum not present in stream";
4550 throw new java.io.StreamCorruptedException(message);
4551 }
4552 int[] mag = stripLeadingZeroBytes(magnitude, 0, magnitude.length);
4553 if ((mag.length == 0) != (sign == 0)) {
4554 String message = "BigInteger: signum-magnitude mismatch";
4555 if (fields.defaulted("magnitude"))
4556 message = "BigInteger: Magnitude not present in stream";
4557 throw new java.io.StreamCorruptedException(message);
4558 }
4559
4560 // Commit final fields via Unsafe
4561 UnsafeHolder.putSign(this, sign);
4562
4563 // Calculate mag field from magnitude and discard magnitude
4564 UnsafeHolder.putMag(this, mag);
4565 if (mag.length >= MAX_MAG_LENGTH) {
4566 try {
4567 checkRange();
4568 } catch (ArithmeticException e) {
4569 throw new java.io.StreamCorruptedException("BigInteger: Out of the supported range");
4570 }
4571 }
4572 }
4573
4574 // Support for resetting final fields while deserializing
4575 private static class UnsafeHolder {
4576 private static final jdk.internal.misc.Unsafe unsafe;
4577 private static final long signumOffset;
4578 private static final long magOffset;
4579 static {
4580 try {
4581 unsafe = jdk.internal.misc.Unsafe.getUnsafe();
4582 signumOffset = unsafe.objectFieldOffset
4583 (BigInteger.class.getDeclaredField("signum"));
4584 magOffset = unsafe.objectFieldOffset
4585 (BigInteger.class.getDeclaredField("mag"));
4586 } catch (Exception ex) {
4587 throw new ExceptionInInitializerError(ex);
4588 }
4589 }
4590
4591 static void putSign(BigInteger bi, int sign) {
4592 unsafe.putInt(bi, signumOffset, sign);
4593 }
4594
4595 static void putMag(BigInteger bi, int[] magnitude) {
4596 unsafe.putObject(bi, magOffset, magnitude);
4597 }
4598 }
4599
4600 /**
4601 * Save the {@code BigInteger} instance to a stream. The magnitude of a
4602 * {@code BigInteger} is serialized as a byte array for historical reasons.
4603 * To maintain compatibility with older implementations, the integers
4604 * -1, -1, -2, and -2 are written as the values of the obsolete fields
4605 * {@code bitCount}, {@code bitLength}, {@code lowestSetBit}, and
4606 * {@code firstNonzeroByteNum}, respectively. These values are compatible
4607 * with older implementations, but will be ignored by current
4608 * implementations.
4609 */
4610 private void writeObject(ObjectOutputStream s) throws IOException {
4611 // set the values of the Serializable fields
4612 ObjectOutputStream.PutField fields = s.putFields();
4613 fields.put("signum", signum);
4614 fields.put("magnitude", magSerializedForm());
4615 // The values written for cached fields are compatible with older
4616 // versions, but are ignored in readObject so don't otherwise matter.
4617 fields.put("bitCount", -1);
4618 fields.put("bitLength", -1);
4619 fields.put("lowestSetBit", -2);
4620 fields.put("firstNonzeroByteNum", -2);
4621
4622 // save them
4623 s.writeFields();
4624 }
4625
4626 /**
4627 * Returns the mag array as an array of bytes.
4628 */
4629 private byte[] magSerializedForm() {
4630 int len = mag.length;
4631
4632 int bitLen = (len == 0 ? 0 : ((len - 1) << 5) + bitLengthForInt(mag[0]));
4633 int byteLen = (bitLen + 7) >>> 3;
4634 byte[] result = new byte[byteLen];
4635
4636 for (int i = byteLen - 1, bytesCopied = 4, intIndex = len - 1, nextInt = 0;
4637 i >= 0; i--) {
4638 if (bytesCopied == 4) {
4639 nextInt = mag[intIndex--];
4640 bytesCopied = 1;
4641 } else {
4642 nextInt >>>= 8;
4643 bytesCopied++;
4644 }
4645 result[i] = (byte)nextInt;
4646 }
4647 return result;
4648 }
4649
4650 /**
4651 * Converts this {@code BigInteger} to a {@code long}, checking
4652 * for lost information. If the value of this {@code BigInteger}
4653 * is out of the range of the {@code long} type, then an
4654 * {@code ArithmeticException} is thrown.
4655 *
4656 * @return this {@code BigInteger} converted to a {@code long}.
4657 * @throws ArithmeticException if the value of {@code this} will
4658 * not exactly fit in a {@code long}.
4659 * @see BigInteger#longValue
4660 * @since 1.8
4661 */
4662 public long longValueExact() {
4663 if (mag.length <= 2 && bitLength() <= 63)
4664 return longValue();
4665 else
4666 throw new ArithmeticException("BigInteger out of long range");
4667 }
4668
4669 /**
4670 * Converts this {@code BigInteger} to an {@code int}, checking
4671 * for lost information. If the value of this {@code BigInteger}
4672 * is out of the range of the {@code int} type, then an
4673 * {@code ArithmeticException} is thrown.
4674 *
4675 * @return this {@code BigInteger} converted to an {@code int}.
4676 * @throws ArithmeticException if the value of {@code this} will
4677 * not exactly fit in a {@code int}.
4678 * @see BigInteger#intValue
4679 * @since 1.8
4680 */
4681 public int intValueExact() {
4682 if (mag.length <= 1 && bitLength() <= 31)
4683 return intValue();
4684 else
4685 throw new ArithmeticException("BigInteger out of int range");
4686 }
4687
4688 /**
4689 * Converts this {@code BigInteger} to a {@code short}, checking
4690 * for lost information. If the value of this {@code BigInteger}
4691 * is out of the range of the {@code short} type, then an
4692 * {@code ArithmeticException} is thrown.
4693 *
4694 * @return this {@code BigInteger} converted to a {@code short}.
4695 * @throws ArithmeticException if the value of {@code this} will
4696 * not exactly fit in a {@code short}.
4697 * @see BigInteger#shortValue
4698 * @since 1.8
4699 */
4700 public short shortValueExact() {
4701 if (mag.length <= 1 && bitLength() <= 31) {
4702 int value = intValue();
4703 if (value >= Short.MIN_VALUE && value <= Short.MAX_VALUE)
4704 return shortValue();
4705 }
4706 throw new ArithmeticException("BigInteger out of short range");
4707 }
4708
4709 /**
4710 * Converts this {@code BigInteger} to a {@code byte}, checking
4711 * for lost information. If the value of this {@code BigInteger}
4712 * is out of the range of the {@code byte} type, then an
4713 * {@code ArithmeticException} is thrown.
4714 *
4715 * @return this {@code BigInteger} converted to a {@code byte}.
4716 * @throws ArithmeticException if the value of {@code this} will
4717 * not exactly fit in a {@code byte}.
4718 * @see BigInteger#byteValue
4719 * @since 1.8
4720 */
4721 public byte byteValueExact() {
4722 if (mag.length <= 1 && bitLength() <= 31) {
4723 int value = intValue();
4724 if (value >= Byte.MIN_VALUE && value <= Byte.MAX_VALUE)
4725 return byteValue();
4726 }
4727 throw new ArithmeticException("BigInteger out of byte range");
4728 }
4729 }