1 /*
2 * Copyright (c) 1996, 2014, 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.Random;
38 import java.util.concurrent.ThreadLocalRandom;
39 import sun.misc.DoubleConsts;
40 import sun.misc.FloatConsts;
41
42 /**
43 * Immutable arbitrary-precision integers. All operations behave as if
44 * BigIntegers were represented in two's-complement notation (like Java's
45 * primitive integer types). BigInteger provides analogues to all of Java's
46 * primitive integer operators, and all relevant methods from java.lang.Math.
47 * Additionally, BigInteger provides operations for modular arithmetic, GCD
48 * calculation, primality testing, prime generation, bit manipulation,
49 * and a few other miscellaneous operations.
50 *
51 * <p>Semantics of arithmetic operations exactly mimic those of Java's integer
52 * arithmetic operators, as defined in <i>The Java Language Specification</i>.
53 * For example, division by zero throws an {@code ArithmeticException}, and
54 * division of a negative by a positive yields a negative (or zero) remainder.
55 * All of the details in the Spec concerning overflow are ignored, as
56 * BigIntegers are made as large as necessary to accommodate the results of an
57 * operation.
58 *
59 * <p>Semantics of shift operations extend those of Java's shift operators
60 * to allow for negative shift distances. A right-shift with a negative
61 * shift distance results in a left shift, and vice-versa. The unsigned
62 * right shift operator ({@code >>>}) is omitted, as this operation makes
63 * little sense in combination with the "infinite word size" abstraction
64 * provided by this class.
65 *
66 * <p>Semantics of bitwise logical operations exactly mimic those of Java's
67 * bitwise integer operators. The binary operators ({@code and},
68 * {@code or}, {@code xor}) implicitly perform sign extension on the shorter
69 * of the two operands prior to performing the operation.
70 *
71 * <p>Comparison operations perform signed integer comparisons, analogous to
72 * those performed by Java's relational and equality operators.
73 *
74 * <p>Modular arithmetic operations are provided to compute residues, perform
75 * exponentiation, and compute multiplicative inverses. These methods always
76 * return a non-negative result, between {@code 0} and {@code (modulus - 1)},
77 * inclusive.
78 *
79 * <p>Bit operations operate on a single bit of the two's-complement
80 * representation of their operand. If necessary, the operand is sign-
81 * extended so that it contains the designated bit. None of the single-bit
82 * operations can produce a BigInteger with a different sign from the
83 * BigInteger being operated on, as they affect only a single bit, and the
84 * "infinite word size" abstraction provided by this class ensures that there
85 * are infinitely many "virtual sign bits" preceding each BigInteger.
86 *
87 * <p>For the sake of brevity and clarity, pseudo-code is used throughout the
88 * descriptions of BigInteger methods. The pseudo-code expression
89 * {@code (i + j)} is shorthand for "a BigInteger whose value is
90 * that of the BigInteger {@code i} plus that of the BigInteger {@code j}."
91 * The pseudo-code expression {@code (i == j)} is shorthand for
92 * "{@code true} if and only if the BigInteger {@code i} represents the same
93 * value as the BigInteger {@code j}." Other pseudo-code expressions are
94 * interpreted similarly.
95 *
96 * <p>All methods and constructors in this class throw
97 * {@code NullPointerException} when passed
98 * a null object reference for any input parameter.
99 *
100 * BigInteger must support values in the range
101 * -2<sup>{@code Integer.MAX_VALUE}</sup> (exclusive) to
102 * +2<sup>{@code Integer.MAX_VALUE}</sup> (exclusive)
103 * and may support values outside of that range.
104 *
105 * The range of probable prime values is limited and may be less than
106 * the full supported positive range of {@code BigInteger}.
107 * The range must be at least 1 to 2<sup>500000000</sup>.
108 *
109 * @implNote
110 * BigInteger constructors and operations throw {@code ArithmeticException} when
111 * the result is out of the supported range of
112 * -2<sup>{@code Integer.MAX_VALUE}</sup> (exclusive) to
113 * +2<sup>{@code Integer.MAX_VALUE}</sup> (exclusive).
114 *
115 * @see BigDecimal
116 * @author Josh Bloch
117 * @author Michael McCloskey
118 * @author Alan Eliasen
119 * @author Timothy Buktu
120 * @since 1.1
121 */
122
123 public class BigInteger extends Number implements Comparable<BigInteger> {
124 /**
125 * The signum of this BigInteger: -1 for negative, 0 for zero, or
126 * 1 for positive. Note that the BigInteger zero <i>must</i> have
127 * a signum of 0. This is necessary to ensures that there is exactly one
128 * representation for each BigInteger value.
129 */
130 final int signum;
131
132 /**
133 * The magnitude of this BigInteger, in <i>big-endian</i> order: the
134 * zeroth element of this array is the most-significant int of the
135 * magnitude. The magnitude must be "minimal" in that the most-significant
136 * int ({@code mag[0]}) must be non-zero. This is necessary to
137 * ensure that there is exactly one representation for each BigInteger
138 * value. Note that this implies that the BigInteger zero has a
139 * zero-length mag array.
140 */
141 final int[] mag;
142
143 // The following fields are stable variables. A stable variable's value
144 // changes at most once from the default zero value to a non-zero stable
145 // value. A stable value is calculated lazily on demand.
146
147 /**
148 * One plus the bitCount of this BigInteger. This is a stable variable.
149 *
150 * @see #bitCount
151 */
152 private int bitCountPlusOne;
153
154 /**
155 * One plus the bitLength of this BigInteger. This is a stable variable.
156 * (either value is acceptable).
157 *
158 * @see #bitLength()
159 */
160 private int bitLengthPlusOne;
161
162 /**
163 * Two plus the lowest set bit of this BigInteger. This is a stable variable.
164 *
165 * @see #getLowestSetBit
166 */
167 private int lowestSetBitPlusTwo;
168
169 /**
170 * Two plus the index of the lowest-order int in the magnitude of this
171 * BigInteger that contains a nonzero int. This is a stable variable. The
172 * least significant int has int-number 0, the next int in order of
173 * increasing significance has int-number 1, and so forth.
174 *
175 * <p>Note: never used for a BigInteger with a magnitude of zero.
176 *
177 * @see #firstNonzeroIntNum()
178 */
179 private int firstNonzeroIntNumPlusTwo;
180
181 /**
182 * This mask is used to obtain the value of an int as if it were unsigned.
183 */
184 final static long LONG_MASK = 0xffffffffL;
185
186 /**
187 * This constant limits {@code mag.length} of BigIntegers to the supported
188 * range.
189 */
190 private static final int MAX_MAG_LENGTH = Integer.MAX_VALUE / Integer.SIZE + 1; // (1 << 26)
191
192 /**
193 * Bit lengths larger than this constant can cause overflow in searchLen
194 * calculation and in BitSieve.singleSearch method.
195 */
196 private static final int PRIME_SEARCH_BIT_LENGTH_LIMIT = 500000000;
197
198 /**
199 * The threshold value for using Karatsuba multiplication. If the number
200 * of ints in both mag arrays are greater than this number, then
201 * Karatsuba multiplication will be used. This value is found
202 * experimentally to work well.
203 */
204 private static final int KARATSUBA_THRESHOLD = 80;
205
206 /**
207 * The threshold value for using 3-way Toom-Cook multiplication.
208 * If the number of ints in each mag array is greater than the
209 * Karatsuba threshold, and the number of ints in at least one of
210 * the mag arrays is greater than this threshold, then Toom-Cook
211 * multiplication will be used.
212 */
213 private static final int TOOM_COOK_THRESHOLD = 240;
214
215 /**
216 * The threshold value for using Karatsuba squaring. If the number
217 * of ints in the number are larger than this value,
218 * Karatsuba squaring will be used. This value is found
219 * experimentally to work well.
220 */
221 private static final int KARATSUBA_SQUARE_THRESHOLD = 128;
222
223 /**
224 * The threshold value for using Toom-Cook squaring. If the number
225 * of ints in the number are larger than this value,
226 * Toom-Cook squaring will be used. This value is found
227 * experimentally to work well.
228 */
229 private static final int TOOM_COOK_SQUARE_THRESHOLD = 216;
230
231 /**
232 * The threshold value for using Burnikel-Ziegler division. If the number
233 * of ints in the divisor are larger than this value, Burnikel-Ziegler
234 * division may be used. This value is found experimentally to work well.
235 */
236 static final int BURNIKEL_ZIEGLER_THRESHOLD = 80;
237
238 /**
239 * The offset value for using Burnikel-Ziegler division. If the number
240 * of ints in the divisor exceeds the Burnikel-Ziegler threshold, and the
241 * number of ints in the dividend is greater than the number of ints in the
242 * divisor plus this value, Burnikel-Ziegler division will be used. This
243 * value is found experimentally to work well.
244 */
245 static final int BURNIKEL_ZIEGLER_OFFSET = 40;
246
247 /**
248 * The threshold value for using Schoenhage recursive base conversion. If
249 * the number of ints in the number are larger than this value,
250 * the Schoenhage algorithm will be used. In practice, it appears that the
251 * Schoenhage routine is faster for any threshold down to 2, and is
252 * relatively flat for thresholds between 2-25, so this choice may be
253 * varied within this range for very small effect.
254 */
255 private static final int SCHOENHAGE_BASE_CONVERSION_THRESHOLD = 20;
256
257 /**
258 * The threshold value for using squaring code to perform multiplication
259 * of a {@code BigInteger} instance by itself. If the number of ints in
260 * the number are larger than this value, {@code multiply(this)} will
261 * return {@code square()}.
262 */
263 private static final int MULTIPLY_SQUARE_THRESHOLD = 20;
264
265 // Constructors
266
267 /**
268 * Translates a byte sub-array containing the two's-complement binary
269 * representation of a BigInteger into a BigInteger. The sub-array is
270 * specified via an offset into the array and a length. The sub-array is
271 * assumed to be in <i>big-endian</i> byte-order: the most significant
272 * byte is the element at index {@code off}. The {@code val} array is
273 * assumed to be unchanged for the duration of the constructor call.
274 *
275 * An {@code IndexOutOfBoundsException} is thrown if the length of the array
276 * {@code val} is non-zero and either {@code off} is negative, {@code len}
277 * is negative, or {@code off+len} is greater than the length of
278 * {@code val}.
279 *
280 * @param val byte array containing a sub-array which is the big-endian
281 * two's-complement binary representation of a BigInteger.
282 * @param off the start offset of the binary representation.
283 * @param len the number of bytes to use.
284 * @throws NumberFormatException {@code val} is zero bytes long.
285 * @throws IndexOutOfBoundsException if the provided array offset and
286 * length would cause an index into the byte array to be
287 * negative or greater than or equal to the array length.
288 * @since 1.9
289 */
290 public BigInteger(byte[] val, int off, int len) {
291 if (val.length == 0) {
292 throw new NumberFormatException("Zero length BigInteger");
293 } else if ((off < 0) || (off >= val.length) || (len < 0) ||
294 (len > val.length - off)) { // 0 <= off < val.length
295 throw new IndexOutOfBoundsException();
296 }
297
298 if (val[off] < 0) {
299 mag = makePositive(val, off, len);
300 signum = -1;
301 } else {
302 mag = stripLeadingZeroBytes(val, off, len);
303 signum = (mag.length == 0 ? 0 : 1);
304 }
305 if (mag.length >= MAX_MAG_LENGTH) {
306 checkRange();
307 }
308 }
309
310 /**
311 * Translates a byte array containing the two's-complement binary
312 * representation of a BigInteger into a BigInteger. The input array is
313 * assumed to be in <i>big-endian</i> byte-order: the most significant
314 * byte is in the zeroth element. The {@code val} array is assumed to be
315 * unchanged for the duration of the constructor call.
316 *
317 * @param val big-endian two's-complement binary representation of a
318 * BigInteger.
319 * @throws NumberFormatException {@code val} is zero bytes long.
320 */
321 public BigInteger(byte[] val) {
322 this(val, 0, val.length);
323 }
324
325 /**
326 * This private constructor translates an int array containing the
327 * two's-complement binary representation of a BigInteger into a
328 * BigInteger. The input array is assumed to be in <i>big-endian</i>
329 * int-order: the most significant int is in the zeroth element. The
330 * {@code val} array is assumed to be unchanged for the duration of
331 * the constructor call.
332 */
333 private BigInteger(int[] val) {
334 if (val.length == 0)
335 throw new NumberFormatException("Zero length BigInteger");
336
337 if (val[0] < 0) {
338 mag = makePositive(val);
339 signum = -1;
340 } else {
341 mag = trustedStripLeadingZeroInts(val);
342 signum = (mag.length == 0 ? 0 : 1);
343 }
344 if (mag.length >= MAX_MAG_LENGTH) {
345 checkRange();
346 }
347 }
348
349 /**
350 * Translates the sign-magnitude representation of a BigInteger into a
351 * BigInteger. The sign is represented as an integer signum value: -1 for
352 * negative, 0 for zero, or 1 for positive. The magnitude is a sub-array of
353 * a byte array in <i>big-endian</i> byte-order: the most significant byte
354 * is the element at index {@code off}. A zero value of the length
355 * {@code len} is permissible, and will result in a BigInteger value of 0,
356 * whether signum is -1, 0 or 1. The {@code magnitude} array is assumed to
357 * be unchanged for the duration of the constructor call.
358 *
359 * An {@code IndexOutOfBoundsException} is thrown if the length of the array
360 * {@code magnitude} is non-zero and either {@code off} is negative,
361 * {@code len} is negative, or {@code off+len} is greater than the length of
362 * {@code magnitude}.
363 *
364 * @param signum signum of the number (-1 for negative, 0 for zero, 1
365 * for positive).
366 * @param magnitude big-endian binary representation of the magnitude of
367 * the number.
368 * @param off the start offset of the binary representation.
369 * @param len the number of bytes to use.
370 * @throws NumberFormatException {@code signum} is not one of the three
371 * legal values (-1, 0, and 1), or {@code signum} is 0 and
372 * {@code magnitude} contains one or more non-zero bytes.
373 * @throws IndexOutOfBoundsException if the provided array offset and
374 * length would cause an index into the byte array to be
375 * negative or greater than or equal to the array length.
376 * @since 1.9
377 */
378 public BigInteger(int signum, byte[] magnitude, int off, int len) {
379 if (signum < -1 || signum > 1) {
380 throw(new NumberFormatException("Invalid signum value"));
381 } else if ((off < 0) || (len < 0) ||
382 (len > 0 &&
383 ((off >= magnitude.length) ||
384 (len > magnitude.length - off)))) { // 0 <= off < magnitude.length
385 throw new IndexOutOfBoundsException();
386 }
387
388 // stripLeadingZeroBytes() returns a zero length array if len == 0
389 this.mag = stripLeadingZeroBytes(magnitude, off, len);
390
391 if (this.mag.length == 0) {
392 this.signum = 0;
393 } else {
394 if (signum == 0)
395 throw(new NumberFormatException("signum-magnitude mismatch"));
396 this.signum = signum;
397 }
398 if (mag.length >= MAX_MAG_LENGTH) {
399 checkRange();
400 }
401 }
402
403 /**
404 * Translates the sign-magnitude representation of a BigInteger into a
405 * BigInteger. The sign is represented as an integer signum value: -1 for
406 * negative, 0 for zero, or 1 for positive. The magnitude is a byte array
407 * in <i>big-endian</i> byte-order: the most significant byte is the
408 * zeroth element. A zero-length magnitude array is permissible, and will
409 * result in a BigInteger value of 0, whether signum is -1, 0 or 1. The
410 * {@code magnitude} array is assumed to be unchanged for the duration of
411 * the constructor call.
412 *
413 * @param signum signum of the number (-1 for negative, 0 for zero, 1
414 * for positive).
415 * @param magnitude big-endian binary representation of the magnitude of
416 * the number.
417 * @throws NumberFormatException {@code signum} is not one of the three
418 * legal values (-1, 0, and 1), or {@code signum} is 0 and
419 * {@code magnitude} contains one or more non-zero bytes.
420 */
421 public BigInteger(int signum, byte[] magnitude) {
422 this(signum, magnitude, 0, magnitude.length);
423 }
424
425 /**
426 * A constructor for internal use that translates the sign-magnitude
427 * representation of a BigInteger into a BigInteger. It checks the
428 * arguments and copies the magnitude so this constructor would be
429 * safe for external use. The {@code magnitude} array is assumed to be
430 * unchanged for the duration of the constructor call.
431 */
432 private BigInteger(int signum, int[] magnitude) {
433 this.mag = stripLeadingZeroInts(magnitude);
434
435 if (signum < -1 || signum > 1)
436 throw(new NumberFormatException("Invalid signum value"));
437
438 if (this.mag.length == 0) {
439 this.signum = 0;
440 } else {
441 if (signum == 0)
442 throw(new NumberFormatException("signum-magnitude mismatch"));
443 this.signum = signum;
444 }
445 if (mag.length >= MAX_MAG_LENGTH) {
446 checkRange();
447 }
448 }
449
450 /**
451 * Translates the String representation of a BigInteger in the
452 * specified radix into a BigInteger. The String representation
453 * consists of an optional minus or plus sign followed by a
454 * sequence of one or more digits in the specified radix. The
455 * character-to-digit mapping is provided by {@code
456 * Character.digit}. The String may not contain any extraneous
457 * characters (whitespace, for example).
458 *
459 * @param val String representation of BigInteger.
460 * @param radix radix to be used in interpreting {@code val}.
461 * @throws NumberFormatException {@code val} is not a valid representation
462 * of a BigInteger in the specified radix, or {@code radix} is
463 * outside the range from {@link Character#MIN_RADIX} to
464 * {@link Character#MAX_RADIX}, inclusive.
465 * @see Character#digit
466 */
467 public BigInteger(String val, int radix) {
468 int cursor = 0, numDigits;
469 final int len = val.length();
470
471 if (radix < Character.MIN_RADIX || radix > Character.MAX_RADIX)
472 throw new NumberFormatException("Radix out of range");
473 if (len == 0)
474 throw new NumberFormatException("Zero length BigInteger");
475
476 // Check for at most one leading sign
477 int sign = 1;
478 int index1 = val.lastIndexOf('-');
479 int index2 = val.lastIndexOf('+');
480 if (index1 >= 0) {
481 if (index1 != 0 || index2 >= 0) {
482 throw new NumberFormatException("Illegal embedded sign character");
483 }
484 sign = -1;
485 cursor = 1;
486 } else if (index2 >= 0) {
487 if (index2 != 0) {
488 throw new NumberFormatException("Illegal embedded sign character");
489 }
490 cursor = 1;
491 }
492 if (cursor == len)
493 throw new NumberFormatException("Zero length BigInteger");
494
495 // Skip leading zeros and compute number of digits in magnitude
496 while (cursor < len &&
497 Character.digit(val.charAt(cursor), radix) == 0) {
498 cursor++;
499 }
500
501 if (cursor == len) {
502 signum = 0;
503 mag = ZERO.mag;
504 return;
505 }
506
507 numDigits = len - cursor;
508 signum = sign;
509
510 // Pre-allocate array of expected size. May be too large but can
511 // never be too small. Typically exact.
512 long numBits = ((numDigits * bitsPerDigit[radix]) >>> 10) + 1;
513 if (numBits + 31 >= (1L << 32)) {
514 reportOverflow();
515 }
516 int numWords = (int) (numBits + 31) >>> 5;
517 int[] magnitude = new int[numWords];
518
519 // Process first (potentially short) digit group
520 int firstGroupLen = numDigits % digitsPerInt[radix];
521 if (firstGroupLen == 0)
522 firstGroupLen = digitsPerInt[radix];
523 String group = val.substring(cursor, cursor += firstGroupLen);
524 magnitude[numWords - 1] = Integer.parseInt(group, radix);
525 if (magnitude[numWords - 1] < 0)
526 throw new NumberFormatException("Illegal digit");
527
528 // Process remaining digit groups
529 int superRadix = intRadix[radix];
530 int groupVal = 0;
531 while (cursor < len) {
532 group = val.substring(cursor, cursor += digitsPerInt[radix]);
533 groupVal = Integer.parseInt(group, radix);
534 if (groupVal < 0)
535 throw new NumberFormatException("Illegal digit");
536 destructiveMulAdd(magnitude, superRadix, groupVal);
537 }
538 // Required for cases where the array was overallocated.
539 mag = trustedStripLeadingZeroInts(magnitude);
540 if (mag.length >= MAX_MAG_LENGTH) {
541 checkRange();
542 }
543 }
544
545 /*
546 * Constructs a new BigInteger using a char array with radix=10.
547 * Sign is precalculated outside and not allowed in the val. The {@code val}
548 * array is assumed to be unchanged for the duration of the constructor
549 * call.
550 */
551 BigInteger(char[] val, int sign, int len) {
552 int cursor = 0, numDigits;
553
554 // Skip leading zeros and compute number of digits in magnitude
555 while (cursor < len && Character.digit(val[cursor], 10) == 0) {
556 cursor++;
557 }
558 if (cursor == len) {
559 signum = 0;
560 mag = ZERO.mag;
561 return;
562 }
563
564 numDigits = len - cursor;
565 signum = sign;
566 // Pre-allocate array of expected size
567 int numWords;
568 if (len < 10) {
569 numWords = 1;
570 } else {
571 long numBits = ((numDigits * bitsPerDigit[10]) >>> 10) + 1;
572 if (numBits + 31 >= (1L << 32)) {
573 reportOverflow();
574 }
575 numWords = (int) (numBits + 31) >>> 5;
576 }
577 int[] magnitude = new int[numWords];
578
579 // Process first (potentially short) digit group
580 int firstGroupLen = numDigits % digitsPerInt[10];
581 if (firstGroupLen == 0)
582 firstGroupLen = digitsPerInt[10];
583 magnitude[numWords - 1] = parseInt(val, cursor, cursor += firstGroupLen);
584
585 // Process remaining digit groups
586 while (cursor < len) {
587 int groupVal = parseInt(val, cursor, cursor += digitsPerInt[10]);
588 destructiveMulAdd(magnitude, intRadix[10], groupVal);
589 }
590 mag = trustedStripLeadingZeroInts(magnitude);
591 if (mag.length >= MAX_MAG_LENGTH) {
592 checkRange();
593 }
594 }
595
596 // Create an integer with the digits between the two indexes
597 // Assumes start < end. The result may be negative, but it
598 // is to be treated as an unsigned value.
599 private int parseInt(char[] source, int start, int end) {
600 int result = Character.digit(source[start++], 10);
601 if (result == -1)
602 throw new NumberFormatException(new String(source));
603
604 for (int index = start; index < end; index++) {
605 int nextVal = Character.digit(source[index], 10);
606 if (nextVal == -1)
607 throw new NumberFormatException(new String(source));
608 result = 10*result + nextVal;
609 }
610
611 return result;
612 }
613
614 // bitsPerDigit in the given radix times 1024
615 // Rounded up to avoid underallocation.
616 private static long bitsPerDigit[] = { 0, 0,
617 1024, 1624, 2048, 2378, 2648, 2875, 3072, 3247, 3402, 3543, 3672,
618 3790, 3899, 4001, 4096, 4186, 4271, 4350, 4426, 4498, 4567, 4633,
619 4696, 4756, 4814, 4870, 4923, 4975, 5025, 5074, 5120, 5166, 5210,
620 5253, 5295};
621
622 // Multiply x array times word y in place, and add word z
623 private static void destructiveMulAdd(int[] x, int y, int z) {
624 // Perform the multiplication word by word
625 long ylong = y & LONG_MASK;
626 long zlong = z & LONG_MASK;
627 int len = x.length;
628
629 long product = 0;
630 long carry = 0;
631 for (int i = len-1; i >= 0; i--) {
632 product = ylong * (x[i] & LONG_MASK) + carry;
633 x[i] = (int)product;
634 carry = product >>> 32;
635 }
636
637 // Perform the addition
638 long sum = (x[len-1] & LONG_MASK) + zlong;
639 x[len-1] = (int)sum;
640 carry = sum >>> 32;
641 for (int i = len-2; i >= 0; i--) {
642 sum = (x[i] & LONG_MASK) + carry;
643 x[i] = (int)sum;
644 carry = sum >>> 32;
645 }
646 }
647
648 /**
649 * Translates the decimal String representation of a BigInteger into a
650 * BigInteger. The String representation consists of an optional minus
651 * sign followed by a sequence of one or more decimal digits. The
652 * character-to-digit mapping is provided by {@code Character.digit}.
653 * The String may not contain any extraneous characters (whitespace, for
654 * example).
655 *
656 * @param val decimal String representation of BigInteger.
657 * @throws NumberFormatException {@code val} is not a valid representation
658 * of a BigInteger.
659 * @see Character#digit
660 */
661 public BigInteger(String val) {
662 this(val, 10);
663 }
664
665 /**
666 * Constructs a randomly generated BigInteger, uniformly distributed over
667 * the range 0 to (2<sup>{@code numBits}</sup> - 1), inclusive.
668 * The uniformity of the distribution assumes that a fair source of random
669 * bits is provided in {@code rnd}. Note that this constructor always
670 * constructs a non-negative BigInteger.
671 *
672 * @param numBits maximum bitLength of the new BigInteger.
673 * @param rnd source of randomness to be used in computing the new
674 * BigInteger.
675 * @throws IllegalArgumentException {@code numBits} is negative.
676 * @see #bitLength()
677 */
678 public BigInteger(int numBits, Random rnd) {
679 this(1, randomBits(numBits, rnd));
680 }
681
682 private static byte[] randomBits(int numBits, Random rnd) {
683 if (numBits < 0)
684 throw new IllegalArgumentException("numBits must be non-negative");
685 int numBytes = (int)(((long)numBits+7)/8); // avoid overflow
686 byte[] randomBits = new byte[numBytes];
687
688 // Generate random bytes and mask out any excess bits
689 if (numBytes > 0) {
690 rnd.nextBytes(randomBits);
691 int excessBits = 8*numBytes - numBits;
692 randomBits[0] &= (1 << (8-excessBits)) - 1;
693 }
694 return randomBits;
695 }
696
697 /**
698 * Constructs a randomly generated positive BigInteger that is probably
699 * prime, with the specified bitLength.
700 *
701 * <p>It is recommended that the {@link #probablePrime probablePrime}
702 * method be used in preference to this constructor unless there
703 * is a compelling need to specify a certainty.
704 *
705 * @param bitLength bitLength of the returned BigInteger.
706 * @param certainty a measure of the uncertainty that the caller is
707 * willing to tolerate. The probability that the new BigInteger
708 * represents a prime number will exceed
709 * (1 - 1/2<sup>{@code certainty}</sup>). The execution time of
710 * this constructor is proportional to the value of this parameter.
711 * @param rnd source of random bits used to select candidates to be
712 * tested for primality.
713 * @throws ArithmeticException {@code bitLength < 2} or {@code bitLength} is too large.
714 * @see #bitLength()
715 */
716 public BigInteger(int bitLength, int certainty, Random rnd) {
717 BigInteger prime;
718
719 if (bitLength < 2)
720 throw new ArithmeticException("bitLength < 2");
721 prime = (bitLength < SMALL_PRIME_THRESHOLD
722 ? smallPrime(bitLength, certainty, rnd)
723 : largePrime(bitLength, certainty, rnd));
724 signum = 1;
725 mag = prime.mag;
726 }
727
728 // Minimum size in bits that the requested prime number has
729 // before we use the large prime number generating algorithms.
730 // The cutoff of 95 was chosen empirically for best performance.
731 private static final int SMALL_PRIME_THRESHOLD = 95;
732
733 // Certainty required to meet the spec of probablePrime
734 private static final int DEFAULT_PRIME_CERTAINTY = 100;
735
736 /**
737 * Returns a positive BigInteger that is probably prime, with the
738 * specified bitLength. The probability that a BigInteger returned
739 * by this method is composite does not exceed 2<sup>-100</sup>.
740 *
741 * @param bitLength bitLength of the returned BigInteger.
742 * @param rnd source of random bits used to select candidates to be
743 * tested for primality.
744 * @return a BigInteger of {@code bitLength} bits that is probably prime
745 * @throws ArithmeticException {@code bitLength < 2} or {@code bitLength} is too large.
746 * @see #bitLength()
747 * @since 1.4
748 */
749 public static BigInteger probablePrime(int bitLength, Random rnd) {
750 if (bitLength < 2)
751 throw new ArithmeticException("bitLength < 2");
752
753 return (bitLength < SMALL_PRIME_THRESHOLD ?
754 smallPrime(bitLength, DEFAULT_PRIME_CERTAINTY, rnd) :
755 largePrime(bitLength, DEFAULT_PRIME_CERTAINTY, rnd));
756 }
757
758 /**
759 * Find a random number of the specified bitLength that is probably prime.
760 * This method is used for smaller primes, its performance degrades on
761 * larger bitlengths.
762 *
763 * This method assumes bitLength > 1.
764 */
765 private static BigInteger smallPrime(int bitLength, int certainty, Random rnd) {
766 int magLen = (bitLength + 31) >>> 5;
767 int temp[] = new int[magLen];
768 int highBit = 1 << ((bitLength+31) & 0x1f); // High bit of high int
769 int highMask = (highBit << 1) - 1; // Bits to keep in high int
770
771 while (true) {
772 // Construct a candidate
773 for (int i=0; i < magLen; i++)
774 temp[i] = rnd.nextInt();
775 temp[0] = (temp[0] & highMask) | highBit; // Ensure exact length
776 if (bitLength > 2)
777 temp[magLen-1] |= 1; // Make odd if bitlen > 2
778
779 BigInteger p = new BigInteger(temp, 1);
780
781 // Do cheap "pre-test" if applicable
782 if (bitLength > 6) {
783 long r = p.remainder(SMALL_PRIME_PRODUCT).longValue();
784 if ((r%3==0) || (r%5==0) || (r%7==0) || (r%11==0) ||
785 (r%13==0) || (r%17==0) || (r%19==0) || (r%23==0) ||
786 (r%29==0) || (r%31==0) || (r%37==0) || (r%41==0))
787 continue; // Candidate is composite; try another
788 }
789
790 // All candidates of bitLength 2 and 3 are prime by this point
791 if (bitLength < 4)
792 return p;
793
794 // Do expensive test if we survive pre-test (or it's inapplicable)
795 if (p.primeToCertainty(certainty, rnd))
796 return p;
797 }
798 }
799
800 private static final BigInteger SMALL_PRIME_PRODUCT
801 = valueOf(3L*5*7*11*13*17*19*23*29*31*37*41);
802
803 /**
804 * Find a random number of the specified bitLength that is probably prime.
805 * This method is more appropriate for larger bitlengths since it uses
806 * a sieve to eliminate most composites before using a more expensive
807 * test.
808 */
809 private static BigInteger largePrime(int bitLength, int certainty, Random rnd) {
810 BigInteger p;
811 p = new BigInteger(bitLength, rnd).setBit(bitLength-1);
812 p.mag[p.mag.length-1] &= 0xfffffffe;
813
814 // Use a sieve length likely to contain the next prime number
815 int searchLen = getPrimeSearchLen(bitLength);
816 BitSieve searchSieve = new BitSieve(p, searchLen);
817 BigInteger candidate = searchSieve.retrieve(p, certainty, rnd);
818
819 while ((candidate == null) || (candidate.bitLength() != bitLength)) {
820 p = p.add(BigInteger.valueOf(2*searchLen));
821 if (p.bitLength() != bitLength)
822 p = new BigInteger(bitLength, rnd).setBit(bitLength-1);
823 p.mag[p.mag.length-1] &= 0xfffffffe;
824 searchSieve = new BitSieve(p, searchLen);
825 candidate = searchSieve.retrieve(p, certainty, rnd);
826 }
827 return candidate;
828 }
829
830 /**
831 * Returns the first integer greater than this {@code BigInteger} that
832 * is probably prime. The probability that the number returned by this
833 * method is composite does not exceed 2<sup>-100</sup>. This method will
834 * never skip over a prime when searching: if it returns {@code p}, there
835 * is no prime {@code q} such that {@code this < q < p}.
836 *
837 * @return the first integer greater than this {@code BigInteger} that
838 * is probably prime.
839 * @throws ArithmeticException {@code this < 0} or {@code this} is too large.
840 * @since 1.5
841 */
842 public BigInteger nextProbablePrime() {
843 if (this.signum < 0)
844 throw new ArithmeticException("start < 0: " + this);
845
846 // Handle trivial cases
847 if ((this.signum == 0) || this.equals(ONE))
848 return TWO;
849
850 BigInteger result = this.add(ONE);
851
852 // Fastpath for small numbers
853 if (result.bitLength() < SMALL_PRIME_THRESHOLD) {
854
855 // Ensure an odd number
856 if (!result.testBit(0))
857 result = result.add(ONE);
858
859 while (true) {
860 // Do cheap "pre-test" if applicable
861 if (result.bitLength() > 6) {
862 long r = result.remainder(SMALL_PRIME_PRODUCT).longValue();
863 if ((r%3==0) || (r%5==0) || (r%7==0) || (r%11==0) ||
864 (r%13==0) || (r%17==0) || (r%19==0) || (r%23==0) ||
865 (r%29==0) || (r%31==0) || (r%37==0) || (r%41==0)) {
866 result = result.add(TWO);
867 continue; // Candidate is composite; try another
868 }
869 }
870
871 // All candidates of bitLength 2 and 3 are prime by this point
872 if (result.bitLength() < 4)
873 return result;
874
875 // The expensive test
876 if (result.primeToCertainty(DEFAULT_PRIME_CERTAINTY, null))
877 return result;
878
879 result = result.add(TWO);
880 }
881 }
882
883 // Start at previous even number
884 if (result.testBit(0))
885 result = result.subtract(ONE);
886
887 // Looking for the next large prime
888 int searchLen = getPrimeSearchLen(result.bitLength());
889
890 while (true) {
891 BitSieve searchSieve = new BitSieve(result, searchLen);
892 BigInteger candidate = searchSieve.retrieve(result,
893 DEFAULT_PRIME_CERTAINTY, null);
894 if (candidate != null)
895 return candidate;
896 result = result.add(BigInteger.valueOf(2 * searchLen));
897 }
898 }
899
900 private static int getPrimeSearchLen(int bitLength) {
901 if (bitLength > PRIME_SEARCH_BIT_LENGTH_LIMIT + 1) {
902 throw new ArithmeticException("Prime search implementation restriction on bitLength");
903 }
904 return bitLength / 20 * 64;
905 }
906
907 /**
908 * Returns {@code true} if this BigInteger is probably prime,
909 * {@code false} if it's definitely composite.
910 *
911 * This method assumes bitLength > 2.
912 *
913 * @param certainty a measure of the uncertainty that the caller is
914 * willing to tolerate: if the call returns {@code true}
915 * the probability that this BigInteger is prime exceeds
916 * {@code (1 - 1/2<sup>certainty</sup>)}. The execution time of
917 * this method is proportional to the value of this parameter.
918 * @return {@code true} if this BigInteger is probably prime,
919 * {@code false} if it's definitely composite.
920 */
921 boolean primeToCertainty(int certainty, Random random) {
922 int rounds = 0;
923 int n = (Math.min(certainty, Integer.MAX_VALUE-1)+1)/2;
924
925 // The relationship between the certainty and the number of rounds
926 // we perform is given in the draft standard ANSI X9.80, "PRIME
927 // NUMBER GENERATION, PRIMALITY TESTING, AND PRIMALITY CERTIFICATES".
928 int sizeInBits = this.bitLength();
929 if (sizeInBits < 100) {
930 rounds = 50;
931 rounds = n < rounds ? n : rounds;
932 return passesMillerRabin(rounds, random);
933 }
934
935 if (sizeInBits < 256) {
936 rounds = 27;
937 } else if (sizeInBits < 512) {
938 rounds = 15;
939 } else if (sizeInBits < 768) {
940 rounds = 8;
941 } else if (sizeInBits < 1024) {
942 rounds = 4;
943 } else {
944 rounds = 2;
945 }
946 rounds = n < rounds ? n : rounds;
947
948 return passesMillerRabin(rounds, random) && passesLucasLehmer();
949 }
950
951 /**
952 * Returns true iff this BigInteger is a Lucas-Lehmer probable prime.
953 *
954 * The following assumptions are made:
955 * This BigInteger is a positive, odd number.
956 */
957 private boolean passesLucasLehmer() {
958 BigInteger thisPlusOne = this.add(ONE);
959
960 // Step 1
961 int d = 5;
962 while (jacobiSymbol(d, this) != -1) {
963 // 5, -7, 9, -11, ...
964 d = (d < 0) ? Math.abs(d)+2 : -(d+2);
965 }
966
967 // Step 2
968 BigInteger u = lucasLehmerSequence(d, thisPlusOne, this);
969
970 // Step 3
971 return u.mod(this).equals(ZERO);
972 }
973
974 /**
975 * Computes Jacobi(p,n).
976 * Assumes n positive, odd, n>=3.
977 */
978 private static int jacobiSymbol(int p, BigInteger n) {
979 if (p == 0)
980 return 0;
981
982 // Algorithm and comments adapted from Colin Plumb's C library.
983 int j = 1;
984 int u = n.mag[n.mag.length-1];
985
986 // Make p positive
987 if (p < 0) {
988 p = -p;
989 int n8 = u & 7;
990 if ((n8 == 3) || (n8 == 7))
991 j = -j; // 3 (011) or 7 (111) mod 8
992 }
993
994 // Get rid of factors of 2 in p
995 while ((p & 3) == 0)
996 p >>= 2;
997 if ((p & 1) == 0) {
998 p >>= 1;
999 if (((u ^ (u>>1)) & 2) != 0)
1000 j = -j; // 3 (011) or 5 (101) mod 8
1001 }
1002 if (p == 1)
1003 return j;
1004 // Then, apply quadratic reciprocity
1005 if ((p & u & 2) != 0) // p = u = 3 (mod 4)?
1006 j = -j;
1007 // And reduce u mod p
1008 u = n.mod(BigInteger.valueOf(p)).intValue();
1009
1010 // Now compute Jacobi(u,p), u < p
1011 while (u != 0) {
1012 while ((u & 3) == 0)
1013 u >>= 2;
1014 if ((u & 1) == 0) {
1015 u >>= 1;
1016 if (((p ^ (p>>1)) & 2) != 0)
1017 j = -j; // 3 (011) or 5 (101) mod 8
1018 }
1019 if (u == 1)
1020 return j;
1021 // Now both u and p are odd, so use quadratic reciprocity
1022 assert (u < p);
1023 int t = u; u = p; p = t;
1024 if ((u & p & 2) != 0) // u = p = 3 (mod 4)?
1025 j = -j;
1026 // Now u >= p, so it can be reduced
1027 u %= p;
1028 }
1029 return 0;
1030 }
1031
1032 private static BigInteger lucasLehmerSequence(int z, BigInteger k, BigInteger n) {
1033 BigInteger d = BigInteger.valueOf(z);
1034 BigInteger u = ONE; BigInteger u2;
1035 BigInteger v = ONE; BigInteger v2;
1036
1037 for (int i=k.bitLength()-2; i >= 0; i--) {
1038 u2 = u.multiply(v).mod(n);
1039
1040 v2 = v.square().add(d.multiply(u.square())).mod(n);
1041 if (v2.testBit(0))
1042 v2 = v2.subtract(n);
1043
1044 v2 = v2.shiftRight(1);
1045
1046 u = u2; v = v2;
1047 if (k.testBit(i)) {
1048 u2 = u.add(v).mod(n);
1049 if (u2.testBit(0))
1050 u2 = u2.subtract(n);
1051
1052 u2 = u2.shiftRight(1);
1053 v2 = v.add(d.multiply(u)).mod(n);
1054 if (v2.testBit(0))
1055 v2 = v2.subtract(n);
1056 v2 = v2.shiftRight(1);
1057
1058 u = u2; v = v2;
1059 }
1060 }
1061 return u;
1062 }
1063
1064 /**
1065 * Returns true iff this BigInteger passes the specified number of
1066 * Miller-Rabin tests. This test is taken from the DSA spec (NIST FIPS
1067 * 186-2).
1068 *
1069 * The following assumptions are made:
1070 * This BigInteger is a positive, odd number greater than 2.
1071 * iterations<=50.
1072 */
1073 private boolean passesMillerRabin(int iterations, Random rnd) {
1074 // Find a and m such that m is odd and this == 1 + 2**a * m
1075 BigInteger thisMinusOne = this.subtract(ONE);
1076 BigInteger m = thisMinusOne;
1077 int a = m.getLowestSetBit();
1078 m = m.shiftRight(a);
1079
1080 // Do the tests
1081 if (rnd == null) {
1082 rnd = ThreadLocalRandom.current();
1083 }
1084 for (int i=0; i < iterations; i++) {
1085 // Generate a uniform random on (1, this)
1086 BigInteger b;
1087 do {
1088 b = new BigInteger(this.bitLength(), rnd);
1089 } while (b.compareTo(ONE) <= 0 || b.compareTo(this) >= 0);
1090
1091 int j = 0;
1092 BigInteger z = b.modPow(m, this);
1093 while (!((j == 0 && z.equals(ONE)) || z.equals(thisMinusOne))) {
1094 if (j > 0 && z.equals(ONE) || ++j == a)
1095 return false;
1096 z = z.modPow(TWO, this);
1097 }
1098 }
1099 return true;
1100 }
1101
1102 /**
1103 * This internal constructor differs from its public cousin
1104 * with the arguments reversed in two ways: it assumes that its
1105 * arguments are correct, and it doesn't copy the magnitude array.
1106 */
1107 BigInteger(int[] magnitude, int signum) {
1108 this.signum = (magnitude.length == 0 ? 0 : signum);
1109 this.mag = magnitude;
1110 if (mag.length >= MAX_MAG_LENGTH) {
1111 checkRange();
1112 }
1113 }
1114
1115 /**
1116 * This private constructor is for internal use and assumes that its
1117 * arguments are correct. The {@code magnitude} array is assumed to be
1118 * unchanged for the duration of the constructor call.
1119 */
1120 private BigInteger(byte[] magnitude, int signum) {
1121 this.signum = (magnitude.length == 0 ? 0 : signum);
1122 this.mag = stripLeadingZeroBytes(magnitude, 0, magnitude.length);
1123 if (mag.length >= MAX_MAG_LENGTH) {
1124 checkRange();
1125 }
1126 }
1127
1128 /**
1129 * Throws an {@code ArithmeticException} if the {@code BigInteger} would be
1130 * out of the supported range.
1131 *
1132 * @throws ArithmeticException if {@code this} exceeds the supported range.
1133 */
1134 private void checkRange() {
1135 if (mag.length > MAX_MAG_LENGTH || mag.length == MAX_MAG_LENGTH && mag[0] < 0) {
1136 reportOverflow();
1137 }
1138 }
1139
1140 private static void reportOverflow() {
1141 throw new ArithmeticException("BigInteger would overflow supported range");
1142 }
1143
1144 //Static Factory Methods
1145
1146 /**
1147 * Returns a BigInteger whose value is equal to that of the
1148 * specified {@code long}. This "static factory method" is
1149 * provided in preference to a ({@code long}) constructor
1150 * because it allows for reuse of frequently used BigIntegers.
1151 *
1152 * @param val value of the BigInteger to return.
1153 * @return a BigInteger with the specified value.
1154 */
1155 public static BigInteger valueOf(long val) {
1156 // If -MAX_CONSTANT < val < MAX_CONSTANT, return stashed constant
1157 if (val == 0)
1158 return ZERO;
1159 if (val > 0 && val <= MAX_CONSTANT)
1160 return posConst[(int) val];
1161 else if (val < 0 && val >= -MAX_CONSTANT)
1162 return negConst[(int) -val];
1163
1164 return new BigInteger(val);
1165 }
1166
1167 /**
1168 * Constructs a BigInteger with the specified value, which may not be zero.
1169 */
1170 private BigInteger(long val) {
1171 if (val < 0) {
1172 val = -val;
1173 signum = -1;
1174 } else {
1175 signum = 1;
1176 }
1177
1178 int highWord = (int)(val >>> 32);
1179 if (highWord == 0) {
1180 mag = new int[1];
1181 mag[0] = (int)val;
1182 } else {
1183 mag = new int[2];
1184 mag[0] = highWord;
1185 mag[1] = (int)val;
1186 }
1187 }
1188
1189 /**
1190 * Returns a BigInteger with the given two's complement representation.
1191 * Assumes that the input array will not be modified (the returned
1192 * BigInteger will reference the input array if feasible).
1193 */
1194 private static BigInteger valueOf(int val[]) {
1195 return (val[0] > 0 ? new BigInteger(val, 1) : new BigInteger(val));
1196 }
1197
1198 // Constants
1199
1200 /**
1201 * Initialize static constant array when class is loaded.
1202 */
1203 private final static int MAX_CONSTANT = 16;
1204 private static BigInteger posConst[] = new BigInteger[MAX_CONSTANT+1];
1205 private static BigInteger negConst[] = new BigInteger[MAX_CONSTANT+1];
1206
1207 /**
1208 * The cache of powers of each radix. This allows us to not have to
1209 * recalculate powers of radix^(2^n) more than once. This speeds
1210 * Schoenhage recursive base conversion significantly.
1211 */
1212 private static volatile BigInteger[][] powerCache;
1213
1214 /** The cache of logarithms of radices for base conversion. */
1215 private static final double[] logCache;
1216
1217 /** The natural log of 2. This is used in computing cache indices. */
1218 private static final double LOG_TWO = Math.log(2.0);
1219
1220 static {
1221 for (int i = 1; i <= MAX_CONSTANT; i++) {
1222 int[] magnitude = new int[1];
1223 magnitude[0] = i;
1224 posConst[i] = new BigInteger(magnitude, 1);
1225 negConst[i] = new BigInteger(magnitude, -1);
1226 }
1227
1228 /*
1229 * Initialize the cache of radix^(2^x) values used for base conversion
1230 * with just the very first value. Additional values will be created
1231 * on demand.
1232 */
1233 powerCache = new BigInteger[Character.MAX_RADIX+1][];
1234 logCache = new double[Character.MAX_RADIX+1];
1235
1236 for (int i=Character.MIN_RADIX; i <= Character.MAX_RADIX; i++) {
1237 powerCache[i] = new BigInteger[] { BigInteger.valueOf(i) };
1238 logCache[i] = Math.log(i);
1239 }
1240 }
1241
1242 /**
1243 * The BigInteger constant zero.
1244 *
1245 * @since 1.2
1246 */
1247 public static final BigInteger ZERO = new BigInteger(new int[0], 0);
1248
1249 /**
1250 * The BigInteger constant one.
1251 *
1252 * @since 1.2
1253 */
1254 public static final BigInteger ONE = valueOf(1);
1255
1256 /**
1257 * The BigInteger constant two. (Not exported.)
1258 */
1259 private static final BigInteger TWO = valueOf(2);
1260
1261 /**
1262 * The BigInteger constant -1. (Not exported.)
1263 */
1264 private static final BigInteger NEGATIVE_ONE = valueOf(-1);
1265
1266 /**
1267 * The BigInteger constant ten.
1268 *
1269 * @since 1.5
1270 */
1271 public static final BigInteger TEN = valueOf(10);
1272
1273 // Arithmetic Operations
1274
1275 /**
1276 * Returns a BigInteger whose value is {@code (this + val)}.
1277 *
1278 * @param val value to be added to this BigInteger.
1279 * @return {@code this + val}
1280 */
1281 public BigInteger add(BigInteger val) {
1282 if (val.signum == 0)
1283 return this;
1284 if (signum == 0)
1285 return val;
1286 if (val.signum == signum)
1287 return new BigInteger(add(mag, val.mag), signum);
1288
1289 int cmp = compareMagnitude(val);
1290 if (cmp == 0)
1291 return ZERO;
1292 int[] resultMag = (cmp > 0 ? subtract(mag, val.mag)
1293 : subtract(val.mag, mag));
1294 resultMag = trustedStripLeadingZeroInts(resultMag);
1295
1296 return new BigInteger(resultMag, cmp == signum ? 1 : -1);
1297 }
1298
1299 /**
1300 * Package private methods used by BigDecimal code to add a BigInteger
1301 * with a long. Assumes val is not equal to INFLATED.
1302 */
1303 BigInteger add(long val) {
1304 if (val == 0)
1305 return this;
1306 if (signum == 0)
1307 return valueOf(val);
1308 if (Long.signum(val) == signum)
1309 return new BigInteger(add(mag, Math.abs(val)), signum);
1310 int cmp = compareMagnitude(val);
1311 if (cmp == 0)
1312 return ZERO;
1313 int[] resultMag = (cmp > 0 ? subtract(mag, Math.abs(val)) : subtract(Math.abs(val), mag));
1314 resultMag = trustedStripLeadingZeroInts(resultMag);
1315 return new BigInteger(resultMag, cmp == signum ? 1 : -1);
1316 }
1317
1318 /**
1319 * Adds the contents of the int array x and long value val. This
1320 * method allocates a new int array to hold the answer and returns
1321 * a reference to that array. Assumes x.length > 0 and val is
1322 * non-negative
1323 */
1324 private static int[] add(int[] x, long val) {
1325 int[] y;
1326 long sum = 0;
1327 int xIndex = x.length;
1328 int[] result;
1329 int highWord = (int)(val >>> 32);
1330 if (highWord == 0) {
1331 result = new int[xIndex];
1332 sum = (x[--xIndex] & LONG_MASK) + val;
1333 result[xIndex] = (int)sum;
1334 } else {
1335 if (xIndex == 1) {
1336 result = new int[2];
1337 sum = val + (x[0] & LONG_MASK);
1338 result[1] = (int)sum;
1339 result[0] = (int)(sum >>> 32);
1340 return result;
1341 } else {
1342 result = new int[xIndex];
1343 sum = (x[--xIndex] & LONG_MASK) + (val & LONG_MASK);
1344 result[xIndex] = (int)sum;
1345 sum = (x[--xIndex] & LONG_MASK) + (highWord & LONG_MASK) + (sum >>> 32);
1346 result[xIndex] = (int)sum;
1347 }
1348 }
1349 // Copy remainder of longer number while carry propagation is required
1350 boolean carry = (sum >>> 32 != 0);
1351 while (xIndex > 0 && carry)
1352 carry = ((result[--xIndex] = x[xIndex] + 1) == 0);
1353 // Copy remainder of longer number
1354 while (xIndex > 0)
1355 result[--xIndex] = x[xIndex];
1356 // Grow result if necessary
1357 if (carry) {
1358 int bigger[] = new int[result.length + 1];
1359 System.arraycopy(result, 0, bigger, 1, result.length);
1360 bigger[0] = 0x01;
1361 return bigger;
1362 }
1363 return result;
1364 }
1365
1366 /**
1367 * Adds the contents of the int arrays x and y. This method allocates
1368 * a new int array to hold the answer and returns a reference to that
1369 * array.
1370 */
1371 private static int[] add(int[] x, int[] y) {
1372 // If x is shorter, swap the two arrays
1373 if (x.length < y.length) {
1374 int[] tmp = x;
1375 x = y;
1376 y = tmp;
1377 }
1378
1379 int xIndex = x.length;
1380 int yIndex = y.length;
1381 int result[] = new int[xIndex];
1382 long sum = 0;
1383 if (yIndex == 1) {
1384 sum = (x[--xIndex] & LONG_MASK) + (y[0] & LONG_MASK) ;
1385 result[xIndex] = (int)sum;
1386 } else {
1387 // Add common parts of both numbers
1388 while (yIndex > 0) {
1389 sum = (x[--xIndex] & LONG_MASK) +
1390 (y[--yIndex] & LONG_MASK) + (sum >>> 32);
1391 result[xIndex] = (int)sum;
1392 }
1393 }
1394 // Copy remainder of longer number while carry propagation is required
1395 boolean carry = (sum >>> 32 != 0);
1396 while (xIndex > 0 && carry)
1397 carry = ((result[--xIndex] = x[xIndex] + 1) == 0);
1398
1399 // Copy remainder of longer number
1400 while (xIndex > 0)
1401 result[--xIndex] = x[xIndex];
1402
1403 // Grow result if necessary
1404 if (carry) {
1405 int bigger[] = new int[result.length + 1];
1406 System.arraycopy(result, 0, bigger, 1, result.length);
1407 bigger[0] = 0x01;
1408 return bigger;
1409 }
1410 return result;
1411 }
1412
1413 private static int[] subtract(long val, int[] little) {
1414 int highWord = (int)(val >>> 32);
1415 if (highWord == 0) {
1416 int result[] = new int[1];
1417 result[0] = (int)(val - (little[0] & LONG_MASK));
1418 return result;
1419 } else {
1420 int result[] = new int[2];
1421 if (little.length == 1) {
1422 long difference = ((int)val & LONG_MASK) - (little[0] & LONG_MASK);
1423 result[1] = (int)difference;
1424 // Subtract remainder of longer number while borrow propagates
1425 boolean borrow = (difference >> 32 != 0);
1426 if (borrow) {
1427 result[0] = highWord - 1;
1428 } else { // Copy remainder of longer number
1429 result[0] = highWord;
1430 }
1431 return result;
1432 } else { // little.length == 2
1433 long difference = ((int)val & LONG_MASK) - (little[1] & LONG_MASK);
1434 result[1] = (int)difference;
1435 difference = (highWord & LONG_MASK) - (little[0] & LONG_MASK) + (difference >> 32);
1436 result[0] = (int)difference;
1437 return result;
1438 }
1439 }
1440 }
1441
1442 /**
1443 * Subtracts the contents of the second argument (val) from the
1444 * first (big). The first int array (big) must represent a larger number
1445 * than the second. This method allocates the space necessary to hold the
1446 * answer.
1447 * assumes val >= 0
1448 */
1449 private static int[] subtract(int[] big, long val) {
1450 int highWord = (int)(val >>> 32);
1451 int bigIndex = big.length;
1452 int result[] = new int[bigIndex];
1453 long difference = 0;
1454
1455 if (highWord == 0) {
1456 difference = (big[--bigIndex] & LONG_MASK) - val;
1457 result[bigIndex] = (int)difference;
1458 } else {
1459 difference = (big[--bigIndex] & LONG_MASK) - (val & LONG_MASK);
1460 result[bigIndex] = (int)difference;
1461 difference = (big[--bigIndex] & LONG_MASK) - (highWord & LONG_MASK) + (difference >> 32);
1462 result[bigIndex] = (int)difference;
1463 }
1464
1465 // Subtract remainder of longer number while borrow propagates
1466 boolean borrow = (difference >> 32 != 0);
1467 while (bigIndex > 0 && borrow)
1468 borrow = ((result[--bigIndex] = big[bigIndex] - 1) == -1);
1469
1470 // Copy remainder of longer number
1471 while (bigIndex > 0)
1472 result[--bigIndex] = big[bigIndex];
1473
1474 return result;
1475 }
1476
1477 /**
1478 * Returns a BigInteger whose value is {@code (this - val)}.
1479 *
1480 * @param val value to be subtracted from this BigInteger.
1481 * @return {@code this - val}
1482 */
1483 public BigInteger subtract(BigInteger val) {
1484 if (val.signum == 0)
1485 return this;
1486 if (signum == 0)
1487 return val.negate();
1488 if (val.signum != signum)
1489 return new BigInteger(add(mag, val.mag), signum);
1490
1491 int cmp = compareMagnitude(val);
1492 if (cmp == 0)
1493 return ZERO;
1494 int[] resultMag = (cmp > 0 ? subtract(mag, val.mag)
1495 : subtract(val.mag, mag));
1496 resultMag = trustedStripLeadingZeroInts(resultMag);
1497 return new BigInteger(resultMag, cmp == signum ? 1 : -1);
1498 }
1499
1500 /**
1501 * Subtracts the contents of the second int arrays (little) from the
1502 * first (big). The first int array (big) must represent a larger number
1503 * than the second. This method allocates the space necessary to hold the
1504 * answer.
1505 */
1506 private static int[] subtract(int[] big, int[] little) {
1507 int bigIndex = big.length;
1508 int result[] = new int[bigIndex];
1509 int littleIndex = little.length;
1510 long difference = 0;
1511
1512 // Subtract common parts of both numbers
1513 while (littleIndex > 0) {
1514 difference = (big[--bigIndex] & LONG_MASK) -
1515 (little[--littleIndex] & LONG_MASK) +
1516 (difference >> 32);
1517 result[bigIndex] = (int)difference;
1518 }
1519
1520 // Subtract remainder of longer number while borrow propagates
1521 boolean borrow = (difference >> 32 != 0);
1522 while (bigIndex > 0 && borrow)
1523 borrow = ((result[--bigIndex] = big[bigIndex] - 1) == -1);
1524
1525 // Copy remainder of longer number
1526 while (bigIndex > 0)
1527 result[--bigIndex] = big[bigIndex];
1528
1529 return result;
1530 }
1531
1532 /**
1533 * Returns a BigInteger whose value is {@code (this * val)}.
1534 *
1535 * @implNote An implementation may offer better algorithmic
1536 * performance when {@code val == this}.
1537 *
1538 * @param val value to be multiplied by this BigInteger.
1539 * @return {@code this * val}
1540 */
1541 public BigInteger multiply(BigInteger val) {
1542 if (val.signum == 0 || signum == 0)
1543 return ZERO;
1544
1545 int xlen = mag.length;
1546
1547 if (val == this && xlen > MULTIPLY_SQUARE_THRESHOLD) {
1548 return square();
1549 }
1550
1551 int ylen = val.mag.length;
1552
1553 if ((xlen < KARATSUBA_THRESHOLD) || (ylen < KARATSUBA_THRESHOLD)) {
1554 int resultSign = signum == val.signum ? 1 : -1;
1555 if (val.mag.length == 1) {
1556 return multiplyByInt(mag,val.mag[0], resultSign);
1557 }
1558 if (mag.length == 1) {
1559 return multiplyByInt(val.mag,mag[0], resultSign);
1560 }
1561 int[] result = multiplyToLen(mag, xlen,
1562 val.mag, ylen, null);
1563 result = trustedStripLeadingZeroInts(result);
1564 return new BigInteger(result, resultSign);
1565 } else {
1566 if ((xlen < TOOM_COOK_THRESHOLD) && (ylen < TOOM_COOK_THRESHOLD)) {
1567 return multiplyKaratsuba(this, val);
1568 } else {
1569 return multiplyToomCook3(this, val);
1570 }
1571 }
1572 }
1573
1574 private static BigInteger multiplyByInt(int[] x, int y, int sign) {
1575 if (Integer.bitCount(y) == 1) {
1576 return new BigInteger(shiftLeft(x,Integer.numberOfTrailingZeros(y)), sign);
1577 }
1578 int xlen = x.length;
1579 int[] rmag = new int[xlen + 1];
1580 long carry = 0;
1581 long yl = y & LONG_MASK;
1582 int rstart = rmag.length - 1;
1583 for (int i = xlen - 1; i >= 0; i--) {
1584 long product = (x[i] & LONG_MASK) * yl + carry;
1585 rmag[rstart--] = (int)product;
1586 carry = product >>> 32;
1587 }
1588 if (carry == 0L) {
1589 rmag = java.util.Arrays.copyOfRange(rmag, 1, rmag.length);
1590 } else {
1591 rmag[rstart] = (int)carry;
1592 }
1593 return new BigInteger(rmag, sign);
1594 }
1595
1596 /**
1597 * Package private methods used by BigDecimal code to multiply a BigInteger
1598 * with a long. Assumes v is not equal to INFLATED.
1599 */
1600 BigInteger multiply(long v) {
1601 if (v == 0 || signum == 0)
1602 return ZERO;
1603 if (v == BigDecimal.INFLATED)
1604 return multiply(BigInteger.valueOf(v));
1605 int rsign = (v > 0 ? signum : -signum);
1606 if (v < 0)
1607 v = -v;
1608 long dh = v >>> 32; // higher order bits
1609 long dl = v & LONG_MASK; // lower order bits
1610
1611 int xlen = mag.length;
1612 int[] value = mag;
1613 int[] rmag = (dh == 0L) ? (new int[xlen + 1]) : (new int[xlen + 2]);
1614 long carry = 0;
1615 int rstart = rmag.length - 1;
1616 for (int i = xlen - 1; i >= 0; i--) {
1617 long product = (value[i] & LONG_MASK) * dl + carry;
1618 rmag[rstart--] = (int)product;
1619 carry = product >>> 32;
1620 }
1621 rmag[rstart] = (int)carry;
1622 if (dh != 0L) {
1623 carry = 0;
1624 rstart = rmag.length - 2;
1625 for (int i = xlen - 1; i >= 0; i--) {
1626 long product = (value[i] & LONG_MASK) * dh +
1627 (rmag[rstart] & LONG_MASK) + carry;
1628 rmag[rstart--] = (int)product;
1629 carry = product >>> 32;
1630 }
1631 rmag[0] = (int)carry;
1632 }
1633 if (carry == 0L)
1634 rmag = java.util.Arrays.copyOfRange(rmag, 1, rmag.length);
1635 return new BigInteger(rmag, rsign);
1636 }
1637
1638 /**
1639 * Multiplies int arrays x and y to the specified lengths and places
1640 * the result into z. There will be no leading zeros in the resultant array.
1641 */
1642 private int[] multiplyToLen(int[] x, int xlen, int[] y, int ylen, int[] z) {
1643 int xstart = xlen - 1;
1644 int ystart = ylen - 1;
1645
1646 if (z == null || z.length < (xlen+ ylen))
1647 z = new int[xlen+ylen];
1648
1649 long carry = 0;
1650 for (int j=ystart, k=ystart+1+xstart; j >= 0; j--, k--) {
1651 long product = (y[j] & LONG_MASK) *
1652 (x[xstart] & LONG_MASK) + carry;
1653 z[k] = (int)product;
1654 carry = product >>> 32;
1655 }
1656 z[xstart] = (int)carry;
1657
1658 for (int i = xstart-1; i >= 0; i--) {
1659 carry = 0;
1660 for (int j=ystart, k=ystart+1+i; j >= 0; j--, k--) {
1661 long product = (y[j] & LONG_MASK) *
1662 (x[i] & LONG_MASK) +
1663 (z[k] & LONG_MASK) + carry;
1664 z[k] = (int)product;
1665 carry = product >>> 32;
1666 }
1667 z[i] = (int)carry;
1668 }
1669 return z;
1670 }
1671
1672 /**
1673 * Multiplies two BigIntegers using the Karatsuba multiplication
1674 * algorithm. This is a recursive divide-and-conquer algorithm which is
1675 * more efficient for large numbers than what is commonly called the
1676 * "grade-school" algorithm used in multiplyToLen. If the numbers to be
1677 * multiplied have length n, the "grade-school" algorithm has an
1678 * asymptotic complexity of O(n^2). In contrast, the Karatsuba algorithm
1679 * has complexity of O(n^(log2(3))), or O(n^1.585). It achieves this
1680 * increased performance by doing 3 multiplies instead of 4 when
1681 * evaluating the product. As it has some overhead, should be used when
1682 * both numbers are larger than a certain threshold (found
1683 * experimentally).
1684 *
1685 * See: http://en.wikipedia.org/wiki/Karatsuba_algorithm
1686 */
1687 private static BigInteger multiplyKaratsuba(BigInteger x, BigInteger y) {
1688 int xlen = x.mag.length;
1689 int ylen = y.mag.length;
1690
1691 // The number of ints in each half of the number.
1692 int half = (Math.max(xlen, ylen)+1) / 2;
1693
1694 // xl and yl are the lower halves of x and y respectively,
1695 // xh and yh are the upper halves.
1696 BigInteger xl = x.getLower(half);
1697 BigInteger xh = x.getUpper(half);
1698 BigInteger yl = y.getLower(half);
1699 BigInteger yh = y.getUpper(half);
1700
1701 BigInteger p1 = xh.multiply(yh); // p1 = xh*yh
1702 BigInteger p2 = xl.multiply(yl); // p2 = xl*yl
1703
1704 // p3=(xh+xl)*(yh+yl)
1705 BigInteger p3 = xh.add(xl).multiply(yh.add(yl));
1706
1707 // result = p1 * 2^(32*2*half) + (p3 - p1 - p2) * 2^(32*half) + p2
1708 BigInteger result = p1.shiftLeft(32*half).add(p3.subtract(p1).subtract(p2)).shiftLeft(32*half).add(p2);
1709
1710 if (x.signum != y.signum) {
1711 return result.negate();
1712 } else {
1713 return result;
1714 }
1715 }
1716
1717 /**
1718 * Multiplies two BigIntegers using a 3-way Toom-Cook multiplication
1719 * algorithm. This is a recursive divide-and-conquer algorithm which is
1720 * more efficient for large numbers than what is commonly called the
1721 * "grade-school" algorithm used in multiplyToLen. If the numbers to be
1722 * multiplied have length n, the "grade-school" algorithm has an
1723 * asymptotic complexity of O(n^2). In contrast, 3-way Toom-Cook has a
1724 * complexity of about O(n^1.465). It achieves this increased asymptotic
1725 * performance by breaking each number into three parts and by doing 5
1726 * multiplies instead of 9 when evaluating the product. Due to overhead
1727 * (additions, shifts, and one division) in the Toom-Cook algorithm, it
1728 * should only be used when both numbers are larger than a certain
1729 * threshold (found experimentally). This threshold is generally larger
1730 * than that for Karatsuba multiplication, so this algorithm is generally
1731 * only used when numbers become significantly larger.
1732 *
1733 * The algorithm used is the "optimal" 3-way Toom-Cook algorithm outlined
1734 * by Marco Bodrato.
1735 *
1736 * See: http://bodrato.it/toom-cook/
1737 * http://bodrato.it/papers/#WAIFI2007
1738 *
1739 * "Towards Optimal Toom-Cook Multiplication for Univariate and
1740 * Multivariate Polynomials in Characteristic 2 and 0." by Marco BODRATO;
1741 * In C.Carlet and B.Sunar, Eds., "WAIFI'07 proceedings", p. 116-133,
1742 * LNCS #4547. Springer, Madrid, Spain, June 21-22, 2007.
1743 *
1744 */
1745 private static BigInteger multiplyToomCook3(BigInteger a, BigInteger b) {
1746 int alen = a.mag.length;
1747 int blen = b.mag.length;
1748
1749 int largest = Math.max(alen, blen);
1750
1751 // k is the size (in ints) of the lower-order slices.
1752 int k = (largest+2)/3; // Equal to ceil(largest/3)
1753
1754 // r is the size (in ints) of the highest-order slice.
1755 int r = largest - 2*k;
1756
1757 // Obtain slices of the numbers. a2 and b2 are the most significant
1758 // bits of the numbers a and b, and a0 and b0 the least significant.
1759 BigInteger a0, a1, a2, b0, b1, b2;
1760 a2 = a.getToomSlice(k, r, 0, largest);
1761 a1 = a.getToomSlice(k, r, 1, largest);
1762 a0 = a.getToomSlice(k, r, 2, largest);
1763 b2 = b.getToomSlice(k, r, 0, largest);
1764 b1 = b.getToomSlice(k, r, 1, largest);
1765 b0 = b.getToomSlice(k, r, 2, largest);
1766
1767 BigInteger v0, v1, v2, vm1, vinf, t1, t2, tm1, da1, db1;
1768
1769 v0 = a0.multiply(b0);
1770 da1 = a2.add(a0);
1771 db1 = b2.add(b0);
1772 vm1 = da1.subtract(a1).multiply(db1.subtract(b1));
1773 da1 = da1.add(a1);
1774 db1 = db1.add(b1);
1775 v1 = da1.multiply(db1);
1776 v2 = da1.add(a2).shiftLeft(1).subtract(a0).multiply(
1777 db1.add(b2).shiftLeft(1).subtract(b0));
1778 vinf = a2.multiply(b2);
1779
1780 // The algorithm requires two divisions by 2 and one by 3.
1781 // All divisions are known to be exact, that is, they do not produce
1782 // remainders, and all results are positive. The divisions by 2 are
1783 // implemented as right shifts which are relatively efficient, leaving
1784 // only an exact division by 3, which is done by a specialized
1785 // linear-time algorithm.
1786 t2 = v2.subtract(vm1).exactDivideBy3();
1787 tm1 = v1.subtract(vm1).shiftRight(1);
1788 t1 = v1.subtract(v0);
1789 t2 = t2.subtract(t1).shiftRight(1);
1790 t1 = t1.subtract(tm1).subtract(vinf);
1791 t2 = t2.subtract(vinf.shiftLeft(1));
1792 tm1 = tm1.subtract(t2);
1793
1794 // Number of bits to shift left.
1795 int ss = k*32;
1796
1797 BigInteger result = vinf.shiftLeft(ss).add(t2).shiftLeft(ss).add(t1).shiftLeft(ss).add(tm1).shiftLeft(ss).add(v0);
1798
1799 if (a.signum != b.signum) {
1800 return result.negate();
1801 } else {
1802 return result;
1803 }
1804 }
1805
1806
1807 /**
1808 * Returns a slice of a BigInteger for use in Toom-Cook multiplication.
1809 *
1810 * @param lowerSize The size of the lower-order bit slices.
1811 * @param upperSize The size of the higher-order bit slices.
1812 * @param slice The index of which slice is requested, which must be a
1813 * number from 0 to size-1. Slice 0 is the highest-order bits, and slice
1814 * size-1 are the lowest-order bits. Slice 0 may be of different size than
1815 * the other slices.
1816 * @param fullsize The size of the larger integer array, used to align
1817 * slices to the appropriate position when multiplying different-sized
1818 * numbers.
1819 */
1820 private BigInteger getToomSlice(int lowerSize, int upperSize, int slice,
1821 int fullsize) {
1822 int start, end, sliceSize, len, offset;
1823
1824 len = mag.length;
1825 offset = fullsize - len;
1826
1827 if (slice == 0) {
1828 start = 0 - offset;
1829 end = upperSize - 1 - offset;
1830 } else {
1831 start = upperSize + (slice-1)*lowerSize - offset;
1832 end = start + lowerSize - 1;
1833 }
1834
1835 if (start < 0) {
1836 start = 0;
1837 }
1838 if (end < 0) {
1839 return ZERO;
1840 }
1841
1842 sliceSize = (end-start) + 1;
1843
1844 if (sliceSize <= 0) {
1845 return ZERO;
1846 }
1847
1848 // While performing Toom-Cook, all slices are positive and
1849 // the sign is adjusted when the final number is composed.
1850 if (start == 0 && sliceSize >= len) {
1851 return this.abs();
1852 }
1853
1854 int intSlice[] = new int[sliceSize];
1855 System.arraycopy(mag, start, intSlice, 0, sliceSize);
1856
1857 return new BigInteger(trustedStripLeadingZeroInts(intSlice), 1);
1858 }
1859
1860 /**
1861 * Does an exact division (that is, the remainder is known to be zero)
1862 * of the specified number by 3. This is used in Toom-Cook
1863 * multiplication. This is an efficient algorithm that runs in linear
1864 * time. If the argument is not exactly divisible by 3, results are
1865 * undefined. Note that this is expected to be called with positive
1866 * arguments only.
1867 */
1868 private BigInteger exactDivideBy3() {
1869 int len = mag.length;
1870 int[] result = new int[len];
1871 long x, w, q, borrow;
1872 borrow = 0L;
1873 for (int i=len-1; i >= 0; i--) {
1874 x = (mag[i] & LONG_MASK);
1875 w = x - borrow;
1876 if (borrow > x) { // Did we make the number go negative?
1877 borrow = 1L;
1878 } else {
1879 borrow = 0L;
1880 }
1881
1882 // 0xAAAAAAAB is the modular inverse of 3 (mod 2^32). Thus,
1883 // the effect of this is to divide by 3 (mod 2^32).
1884 // This is much faster than division on most architectures.
1885 q = (w * 0xAAAAAAABL) & LONG_MASK;
1886 result[i] = (int) q;
1887
1888 // Now check the borrow. The second check can of course be
1889 // eliminated if the first fails.
1890 if (q >= 0x55555556L) {
1891 borrow++;
1892 if (q >= 0xAAAAAAABL)
1893 borrow++;
1894 }
1895 }
1896 result = trustedStripLeadingZeroInts(result);
1897 return new BigInteger(result, signum);
1898 }
1899
1900 /**
1901 * Returns a new BigInteger representing n lower ints of the number.
1902 * This is used by Karatsuba multiplication and Karatsuba squaring.
1903 */
1904 private BigInteger getLower(int n) {
1905 int len = mag.length;
1906
1907 if (len <= n) {
1908 return abs();
1909 }
1910
1911 int lowerInts[] = new int[n];
1912 System.arraycopy(mag, len-n, lowerInts, 0, n);
1913
1914 return new BigInteger(trustedStripLeadingZeroInts(lowerInts), 1);
1915 }
1916
1917 /**
1918 * Returns a new BigInteger representing mag.length-n upper
1919 * ints of the number. This is used by Karatsuba multiplication and
1920 * Karatsuba squaring.
1921 */
1922 private BigInteger getUpper(int n) {
1923 int len = mag.length;
1924
1925 if (len <= n) {
1926 return ZERO;
1927 }
1928
1929 int upperLen = len - n;
1930 int upperInts[] = new int[upperLen];
1931 System.arraycopy(mag, 0, upperInts, 0, upperLen);
1932
1933 return new BigInteger(trustedStripLeadingZeroInts(upperInts), 1);
1934 }
1935
1936 // Squaring
1937
1938 /**
1939 * Returns a BigInteger whose value is {@code (this<sup>2</sup>)}.
1940 *
1941 * @return {@code this<sup>2</sup>}
1942 */
1943 private BigInteger square() {
1944 if (signum == 0) {
1945 return ZERO;
1946 }
1947 int len = mag.length;
1948
1949 if (len < KARATSUBA_SQUARE_THRESHOLD) {
1950 int[] z = squareToLen(mag, len, null);
1951 return new BigInteger(trustedStripLeadingZeroInts(z), 1);
1952 } else {
1953 if (len < TOOM_COOK_SQUARE_THRESHOLD) {
1954 return squareKaratsuba();
1955 } else {
1956 return squareToomCook3();
1957 }
1958 }
1959 }
1960
1961 /**
1962 * Squares the contents of the int array x. The result is placed into the
1963 * int array z. The contents of x are not changed.
1964 */
1965 private static final int[] squareToLen(int[] x, int len, int[] z) {
1966 /*
1967 * The algorithm used here is adapted from Colin Plumb's C library.
1968 * Technique: Consider the partial products in the multiplication
1969 * of "abcde" by itself:
1970 *
1971 * a b c d e
1972 * * a b c d e
1973 * ==================
1974 * ae be ce de ee
1975 * ad bd cd dd de
1976 * ac bc cc cd ce
1977 * ab bb bc bd be
1978 * aa ab ac ad ae
1979 *
1980 * Note that everything above the main diagonal:
1981 * ae be ce de = (abcd) * e
1982 * ad bd cd = (abc) * d
1983 * ac bc = (ab) * c
1984 * ab = (a) * b
1985 *
1986 * is a copy of everything below the main diagonal:
1987 * de
1988 * cd ce
1989 * bc bd be
1990 * ab ac ad ae
1991 *
1992 * Thus, the sum is 2 * (off the diagonal) + diagonal.
1993 *
1994 * This is accumulated beginning with the diagonal (which
1995 * consist of the squares of the digits of the input), which is then
1996 * divided by two, the off-diagonal added, and multiplied by two
1997 * again. The low bit is simply a copy of the low bit of the
1998 * input, so it doesn't need special care.
1999 */
2000 int zlen = len << 1;
2001 if (z == null || z.length < zlen)
2002 z = new int[zlen];
2003
2004 // Store the squares, right shifted one bit (i.e., divided by 2)
2005 int lastProductLowWord = 0;
2006 for (int j=0, i=0; j < len; j++) {
2007 long piece = (x[j] & LONG_MASK);
2008 long product = piece * piece;
2009 z[i++] = (lastProductLowWord << 31) | (int)(product >>> 33);
2010 z[i++] = (int)(product >>> 1);
2011 lastProductLowWord = (int)product;
2012 }
2013
2014 // Add in off-diagonal sums
2015 for (int i=len, offset=1; i > 0; i--, offset+=2) {
2016 int t = x[i-1];
2017 t = mulAdd(z, x, offset, i-1, t);
2018 addOne(z, offset-1, i, t);
2019 }
2020
2021 // Shift back up and set low bit
2022 primitiveLeftShift(z, zlen, 1);
2023 z[zlen-1] |= x[len-1] & 1;
2024
2025 return z;
2026 }
2027
2028 /**
2029 * Squares a BigInteger using the Karatsuba squaring algorithm. It should
2030 * be used when both numbers are larger than a certain threshold (found
2031 * experimentally). It is a recursive divide-and-conquer algorithm that
2032 * has better asymptotic performance than the algorithm used in
2033 * squareToLen.
2034 */
2035 private BigInteger squareKaratsuba() {
2036 int half = (mag.length+1) / 2;
2037
2038 BigInteger xl = getLower(half);
2039 BigInteger xh = getUpper(half);
2040
2041 BigInteger xhs = xh.square(); // xhs = xh^2
2042 BigInteger xls = xl.square(); // xls = xl^2
2043
2044 // xh^2 << 64 + (((xl+xh)^2 - (xh^2 + xl^2)) << 32) + xl^2
2045 return xhs.shiftLeft(half*32).add(xl.add(xh).square().subtract(xhs.add(xls))).shiftLeft(half*32).add(xls);
2046 }
2047
2048 /**
2049 * Squares a BigInteger using the 3-way Toom-Cook squaring algorithm. It
2050 * should be used when both numbers are larger than a certain threshold
2051 * (found experimentally). It is a recursive divide-and-conquer algorithm
2052 * that has better asymptotic performance than the algorithm used in
2053 * squareToLen or squareKaratsuba.
2054 */
2055 private BigInteger squareToomCook3() {
2056 int len = mag.length;
2057
2058 // k is the size (in ints) of the lower-order slices.
2059 int k = (len+2)/3; // Equal to ceil(largest/3)
2060
2061 // r is the size (in ints) of the highest-order slice.
2062 int r = len - 2*k;
2063
2064 // Obtain slices of the numbers. a2 is the most significant
2065 // bits of the number, and a0 the least significant.
2066 BigInteger a0, a1, a2;
2067 a2 = getToomSlice(k, r, 0, len);
2068 a1 = getToomSlice(k, r, 1, len);
2069 a0 = getToomSlice(k, r, 2, len);
2070 BigInteger v0, v1, v2, vm1, vinf, t1, t2, tm1, da1;
2071
2072 v0 = a0.square();
2073 da1 = a2.add(a0);
2074 vm1 = da1.subtract(a1).square();
2075 da1 = da1.add(a1);
2076 v1 = da1.square();
2077 vinf = a2.square();
2078 v2 = da1.add(a2).shiftLeft(1).subtract(a0).square();
2079
2080 // The algorithm requires two divisions by 2 and one by 3.
2081 // All divisions are known to be exact, that is, they do not produce
2082 // remainders, and all results are positive. The divisions by 2 are
2083 // implemented as right shifts which are relatively efficient, leaving
2084 // only a division by 3.
2085 // The division by 3 is done by an optimized algorithm for this case.
2086 t2 = v2.subtract(vm1).exactDivideBy3();
2087 tm1 = v1.subtract(vm1).shiftRight(1);
2088 t1 = v1.subtract(v0);
2089 t2 = t2.subtract(t1).shiftRight(1);
2090 t1 = t1.subtract(tm1).subtract(vinf);
2091 t2 = t2.subtract(vinf.shiftLeft(1));
2092 tm1 = tm1.subtract(t2);
2093
2094 // Number of bits to shift left.
2095 int ss = k*32;
2096
2097 return vinf.shiftLeft(ss).add(t2).shiftLeft(ss).add(t1).shiftLeft(ss).add(tm1).shiftLeft(ss).add(v0);
2098 }
2099
2100 // Division
2101
2102 /**
2103 * Returns a BigInteger whose value is {@code (this / val)}.
2104 *
2105 * @param val value by which this BigInteger is to be divided.
2106 * @return {@code this / val}
2107 * @throws ArithmeticException if {@code val} is zero.
2108 */
2109 public BigInteger divide(BigInteger val) {
2110 if (val.mag.length < BURNIKEL_ZIEGLER_THRESHOLD ||
2111 mag.length - val.mag.length < BURNIKEL_ZIEGLER_OFFSET) {
2112 return divideKnuth(val);
2113 } else {
2114 return divideBurnikelZiegler(val);
2115 }
2116 }
2117
2118 /**
2119 * Returns a BigInteger whose value is {@code (this / val)} using an O(n^2) algorithm from Knuth.
2120 *
2121 * @param val value by which this BigInteger is to be divided.
2122 * @return {@code this / val}
2123 * @throws ArithmeticException if {@code val} is zero.
2124 * @see MutableBigInteger#divideKnuth(MutableBigInteger, MutableBigInteger, boolean)
2125 */
2126 private BigInteger divideKnuth(BigInteger val) {
2127 MutableBigInteger q = new MutableBigInteger(),
2128 a = new MutableBigInteger(this.mag),
2129 b = new MutableBigInteger(val.mag);
2130
2131 a.divideKnuth(b, q, false);
2132 return q.toBigInteger(this.signum * val.signum);
2133 }
2134
2135 /**
2136 * Returns an array of two BigIntegers containing {@code (this / val)}
2137 * followed by {@code (this % val)}.
2138 *
2139 * @param val value by which this BigInteger is to be divided, and the
2140 * remainder computed.
2141 * @return an array of two BigIntegers: the quotient {@code (this / val)}
2142 * is the initial element, and the remainder {@code (this % val)}
2143 * is the final element.
2144 * @throws ArithmeticException if {@code val} is zero.
2145 */
2146 public BigInteger[] divideAndRemainder(BigInteger val) {
2147 if (val.mag.length < BURNIKEL_ZIEGLER_THRESHOLD ||
2148 mag.length - val.mag.length < BURNIKEL_ZIEGLER_OFFSET) {
2149 return divideAndRemainderKnuth(val);
2150 } else {
2151 return divideAndRemainderBurnikelZiegler(val);
2152 }
2153 }
2154
2155 /** Long division */
2156 private BigInteger[] divideAndRemainderKnuth(BigInteger val) {
2157 BigInteger[] result = new BigInteger[2];
2158 MutableBigInteger q = new MutableBigInteger(),
2159 a = new MutableBigInteger(this.mag),
2160 b = new MutableBigInteger(val.mag);
2161 MutableBigInteger r = a.divideKnuth(b, q);
2162 result[0] = q.toBigInteger(this.signum == val.signum ? 1 : -1);
2163 result[1] = r.toBigInteger(this.signum);
2164 return result;
2165 }
2166
2167 /**
2168 * Returns a BigInteger whose value is {@code (this % val)}.
2169 *
2170 * @param val value by which this BigInteger is to be divided, and the
2171 * remainder computed.
2172 * @return {@code this % val}
2173 * @throws ArithmeticException if {@code val} is zero.
2174 */
2175 public BigInteger remainder(BigInteger val) {
2176 if (val.mag.length < BURNIKEL_ZIEGLER_THRESHOLD ||
2177 mag.length - val.mag.length < BURNIKEL_ZIEGLER_OFFSET) {
2178 return remainderKnuth(val);
2179 } else {
2180 return remainderBurnikelZiegler(val);
2181 }
2182 }
2183
2184 /** Long division */
2185 private BigInteger remainderKnuth(BigInteger val) {
2186 MutableBigInteger q = new MutableBigInteger(),
2187 a = new MutableBigInteger(this.mag),
2188 b = new MutableBigInteger(val.mag);
2189
2190 return a.divideKnuth(b, q).toBigInteger(this.signum);
2191 }
2192
2193 /**
2194 * Calculates {@code this / val} using the Burnikel-Ziegler algorithm.
2195 * @param val the divisor
2196 * @return {@code this / val}
2197 */
2198 private BigInteger divideBurnikelZiegler(BigInteger val) {
2199 return divideAndRemainderBurnikelZiegler(val)[0];
2200 }
2201
2202 /**
2203 * Calculates {@code this % val} using the Burnikel-Ziegler algorithm.
2204 * @param val the divisor
2205 * @return {@code this % val}
2206 */
2207 private BigInteger remainderBurnikelZiegler(BigInteger val) {
2208 return divideAndRemainderBurnikelZiegler(val)[1];
2209 }
2210
2211 /**
2212 * Computes {@code this / val} and {@code this % val} using the
2213 * Burnikel-Ziegler algorithm.
2214 * @param val the divisor
2215 * @return an array containing the quotient and remainder
2216 */
2217 private BigInteger[] divideAndRemainderBurnikelZiegler(BigInteger val) {
2218 MutableBigInteger q = new MutableBigInteger();
2219 MutableBigInteger r = new MutableBigInteger(this).divideAndRemainderBurnikelZiegler(new MutableBigInteger(val), q);
2220 BigInteger qBigInt = q.isZero() ? ZERO : q.toBigInteger(signum*val.signum);
2221 BigInteger rBigInt = r.isZero() ? ZERO : r.toBigInteger(signum);
2222 return new BigInteger[] {qBigInt, rBigInt};
2223 }
2224
2225 /**
2226 * Returns a BigInteger whose value is <tt>(this<sup>exponent</sup>)</tt>.
2227 * Note that {@code exponent} is an integer rather than a BigInteger.
2228 *
2229 * @param exponent exponent to which this BigInteger is to be raised.
2230 * @return <tt>this<sup>exponent</sup></tt>
2231 * @throws ArithmeticException {@code exponent} is negative. (This would
2232 * cause the operation to yield a non-integer value.)
2233 */
2234 public BigInteger pow(int exponent) {
2235 if (exponent < 0) {
2236 throw new ArithmeticException("Negative exponent");
2237 }
2238 if (signum == 0) {
2239 return (exponent == 0 ? ONE : this);
2240 }
2241
2242 BigInteger partToSquare = this.abs();
2243
2244 // Factor out powers of two from the base, as the exponentiation of
2245 // these can be done by left shifts only.
2246 // The remaining part can then be exponentiated faster. The
2247 // powers of two will be multiplied back at the end.
2248 int powersOfTwo = partToSquare.getLowestSetBit();
2249 long bitsToShift = (long)powersOfTwo * exponent;
2250 if (bitsToShift > Integer.MAX_VALUE) {
2251 reportOverflow();
2252 }
2253
2254 int remainingBits;
2255
2256 // Factor the powers of two out quickly by shifting right, if needed.
2257 if (powersOfTwo > 0) {
2258 partToSquare = partToSquare.shiftRight(powersOfTwo);
2259 remainingBits = partToSquare.bitLength();
2260 if (remainingBits == 1) { // Nothing left but +/- 1?
2261 if (signum < 0 && (exponent&1) == 1) {
2262 return NEGATIVE_ONE.shiftLeft(powersOfTwo*exponent);
2263 } else {
2264 return ONE.shiftLeft(powersOfTwo*exponent);
2265 }
2266 }
2267 } else {
2268 remainingBits = partToSquare.bitLength();
2269 if (remainingBits == 1) { // Nothing left but +/- 1?
2270 if (signum < 0 && (exponent&1) == 1) {
2271 return NEGATIVE_ONE;
2272 } else {
2273 return ONE;
2274 }
2275 }
2276 }
2277
2278 // This is a quick way to approximate the size of the result,
2279 // similar to doing log2[n] * exponent. This will give an upper bound
2280 // of how big the result can be, and which algorithm to use.
2281 long scaleFactor = (long)remainingBits * exponent;
2282
2283 // Use slightly different algorithms for small and large operands.
2284 // See if the result will safely fit into a long. (Largest 2^63-1)
2285 if (partToSquare.mag.length == 1 && scaleFactor <= 62) {
2286 // Small number algorithm. Everything fits into a long.
2287 int newSign = (signum <0 && (exponent&1) == 1 ? -1 : 1);
2288 long result = 1;
2289 long baseToPow2 = partToSquare.mag[0] & LONG_MASK;
2290
2291 int workingExponent = exponent;
2292
2293 // Perform exponentiation using repeated squaring trick
2294 while (workingExponent != 0) {
2295 if ((workingExponent & 1) == 1) {
2296 result = result * baseToPow2;
2297 }
2298
2299 if ((workingExponent >>>= 1) != 0) {
2300 baseToPow2 = baseToPow2 * baseToPow2;
2301 }
2302 }
2303
2304 // Multiply back the powers of two (quickly, by shifting left)
2305 if (powersOfTwo > 0) {
2306 if (bitsToShift + scaleFactor <= 62) { // Fits in long?
2307 return valueOf((result << bitsToShift) * newSign);
2308 } else {
2309 return valueOf(result*newSign).shiftLeft((int) bitsToShift);
2310 }
2311 }
2312 else {
2313 return valueOf(result*newSign);
2314 }
2315 } else {
2316 // Large number algorithm. This is basically identical to
2317 // the algorithm above, but calls multiply() and square()
2318 // which may use more efficient algorithms for large numbers.
2319 BigInteger answer = ONE;
2320
2321 int workingExponent = exponent;
2322 // Perform exponentiation using repeated squaring trick
2323 while (workingExponent != 0) {
2324 if ((workingExponent & 1) == 1) {
2325 answer = answer.multiply(partToSquare);
2326 }
2327
2328 if ((workingExponent >>>= 1) != 0) {
2329 partToSquare = partToSquare.square();
2330 }
2331 }
2332 // Multiply back the (exponentiated) powers of two (quickly,
2333 // by shifting left)
2334 if (powersOfTwo > 0) {
2335 answer = answer.shiftLeft(powersOfTwo*exponent);
2336 }
2337
2338 if (signum < 0 && (exponent&1) == 1) {
2339 return answer.negate();
2340 } else {
2341 return answer;
2342 }
2343 }
2344 }
2345
2346 /**
2347 * Returns a BigInteger whose value is the greatest common divisor of
2348 * {@code abs(this)} and {@code abs(val)}. Returns 0 if
2349 * {@code this == 0 && val == 0}.
2350 *
2351 * @param val value with which the GCD is to be computed.
2352 * @return {@code GCD(abs(this), abs(val))}
2353 */
2354 public BigInteger gcd(BigInteger val) {
2355 if (val.signum == 0)
2356 return this.abs();
2357 else if (this.signum == 0)
2358 return val.abs();
2359
2360 MutableBigInteger a = new MutableBigInteger(this);
2361 MutableBigInteger b = new MutableBigInteger(val);
2362
2363 MutableBigInteger result = a.hybridGCD(b);
2364
2365 return result.toBigInteger(1);
2366 }
2367
2368 /**
2369 * Package private method to return bit length for an integer.
2370 */
2371 static int bitLengthForInt(int n) {
2372 return 32 - Integer.numberOfLeadingZeros(n);
2373 }
2374
2375 /**
2376 * Left shift int array a up to len by n bits. Returns the array that
2377 * results from the shift since space may have to be reallocated.
2378 */
2379 private static int[] leftShift(int[] a, int len, int n) {
2380 int nInts = n >>> 5;
2381 int nBits = n&0x1F;
2382 int bitsInHighWord = bitLengthForInt(a[0]);
2383
2384 // If shift can be done without recopy, do so
2385 if (n <= (32-bitsInHighWord)) {
2386 primitiveLeftShift(a, len, nBits);
2387 return a;
2388 } else { // Array must be resized
2389 if (nBits <= (32-bitsInHighWord)) {
2390 int result[] = new int[nInts+len];
2391 System.arraycopy(a, 0, result, 0, len);
2392 primitiveLeftShift(result, result.length, nBits);
2393 return result;
2394 } else {
2395 int result[] = new int[nInts+len+1];
2396 System.arraycopy(a, 0, result, 0, len);
2397 primitiveRightShift(result, result.length, 32 - nBits);
2398 return result;
2399 }
2400 }
2401 }
2402
2403 // shifts a up to len right n bits assumes no leading zeros, 0<n<32
2404 static void primitiveRightShift(int[] a, int len, int n) {
2405 int n2 = 32 - n;
2406 for (int i=len-1, c=a[i]; i > 0; i--) {
2407 int b = c;
2408 c = a[i-1];
2409 a[i] = (c << n2) | (b >>> n);
2410 }
2411 a[0] >>>= n;
2412 }
2413
2414 // shifts a up to len left n bits assumes no leading zeros, 0<=n<32
2415 static void primitiveLeftShift(int[] a, int len, int n) {
2416 if (len == 0 || n == 0)
2417 return;
2418
2419 int n2 = 32 - n;
2420 for (int i=0, c=a[i], m=i+len-1; i < m; i++) {
2421 int b = c;
2422 c = a[i+1];
2423 a[i] = (b << n) | (c >>> n2);
2424 }
2425 a[len-1] <<= n;
2426 }
2427
2428 /**
2429 * Calculate bitlength of contents of the first len elements an int array,
2430 * assuming there are no leading zero ints.
2431 */
2432 private static int bitLength(int[] val, int len) {
2433 if (len == 0)
2434 return 0;
2435 return ((len - 1) << 5) + bitLengthForInt(val[0]);
2436 }
2437
2438 /**
2439 * Returns a BigInteger whose value is the absolute value of this
2440 * BigInteger.
2441 *
2442 * @return {@code abs(this)}
2443 */
2444 public BigInteger abs() {
2445 return (signum >= 0 ? this : this.negate());
2446 }
2447
2448 /**
2449 * Returns a BigInteger whose value is {@code (-this)}.
2450 *
2451 * @return {@code -this}
2452 */
2453 public BigInteger negate() {
2454 return new BigInteger(this.mag, -this.signum);
2455 }
2456
2457 /**
2458 * Returns the signum function of this BigInteger.
2459 *
2460 * @return -1, 0 or 1 as the value of this BigInteger is negative, zero or
2461 * positive.
2462 */
2463 public int signum() {
2464 return this.signum;
2465 }
2466
2467 // Modular Arithmetic Operations
2468
2469 /**
2470 * Returns a BigInteger whose value is {@code (this mod m}). This method
2471 * differs from {@code remainder} in that it always returns a
2472 * <i>non-negative</i> BigInteger.
2473 *
2474 * @param m the modulus.
2475 * @return {@code this mod m}
2476 * @throws ArithmeticException {@code m} ≤ 0
2477 * @see #remainder
2478 */
2479 public BigInteger mod(BigInteger m) {
2480 if (m.signum <= 0)
2481 throw new ArithmeticException("BigInteger: modulus not positive");
2482
2483 BigInteger result = this.remainder(m);
2484 return (result.signum >= 0 ? result : result.add(m));
2485 }
2486
2487 /**
2488 * Returns a BigInteger whose value is
2489 * <tt>(this<sup>exponent</sup> mod m)</tt>. (Unlike {@code pow}, this
2490 * method permits negative exponents.)
2491 *
2492 * @param exponent the exponent.
2493 * @param m the modulus.
2494 * @return <tt>this<sup>exponent</sup> mod m</tt>
2495 * @throws ArithmeticException {@code m} ≤ 0 or the exponent is
2496 * negative and this BigInteger is not <i>relatively
2497 * prime</i> to {@code m}.
2498 * @see #modInverse
2499 */
2500 public BigInteger modPow(BigInteger exponent, BigInteger m) {
2501 if (m.signum <= 0)
2502 throw new ArithmeticException("BigInteger: modulus not positive");
2503
2504 // Trivial cases
2505 if (exponent.signum == 0)
2506 return (m.equals(ONE) ? ZERO : ONE);
2507
2508 if (this.equals(ONE))
2509 return (m.equals(ONE) ? ZERO : ONE);
2510
2511 if (this.equals(ZERO) && exponent.signum >= 0)
2512 return ZERO;
2513
2514 if (this.equals(negConst[1]) && (!exponent.testBit(0)))
2515 return (m.equals(ONE) ? ZERO : ONE);
2516
2517 boolean invertResult;
2518 if ((invertResult = (exponent.signum < 0)))
2519 exponent = exponent.negate();
2520
2521 BigInteger base = (this.signum < 0 || this.compareTo(m) >= 0
2522 ? this.mod(m) : this);
2523 BigInteger result;
2524 if (m.testBit(0)) { // odd modulus
2525 result = base.oddModPow(exponent, m);
2526 } else {
2527 /*
2528 * Even modulus. Tear it into an "odd part" (m1) and power of two
2529 * (m2), exponentiate mod m1, manually exponentiate mod m2, and
2530 * use Chinese Remainder Theorem to combine results.
2531 */
2532
2533 // Tear m apart into odd part (m1) and power of 2 (m2)
2534 int p = m.getLowestSetBit(); // Max pow of 2 that divides m
2535
2536 BigInteger m1 = m.shiftRight(p); // m/2**p
2537 BigInteger m2 = ONE.shiftLeft(p); // 2**p
2538
2539 // Calculate new base from m1
2540 BigInteger base2 = (this.signum < 0 || this.compareTo(m1) >= 0
2541 ? this.mod(m1) : this);
2542
2543 // Caculate (base ** exponent) mod m1.
2544 BigInteger a1 = (m1.equals(ONE) ? ZERO :
2545 base2.oddModPow(exponent, m1));
2546
2547 // Calculate (this ** exponent) mod m2
2548 BigInteger a2 = base.modPow2(exponent, p);
2549
2550 // Combine results using Chinese Remainder Theorem
2551 BigInteger y1 = m2.modInverse(m1);
2552 BigInteger y2 = m1.modInverse(m2);
2553
2554 if (m.mag.length < MAX_MAG_LENGTH / 2) {
2555 result = a1.multiply(m2).multiply(y1).add(a2.multiply(m1).multiply(y2)).mod(m);
2556 } else {
2557 MutableBigInteger t1 = new MutableBigInteger();
2558 new MutableBigInteger(a1.multiply(m2)).multiply(new MutableBigInteger(y1), t1);
2559 MutableBigInteger t2 = new MutableBigInteger();
2560 new MutableBigInteger(a2.multiply(m1)).multiply(new MutableBigInteger(y2), t2);
2561 t1.add(t2);
2562 MutableBigInteger q = new MutableBigInteger();
2563 result = t1.divide(new MutableBigInteger(m), q).toBigInteger();
2564 }
2565 }
2566
2567 return (invertResult ? result.modInverse(m) : result);
2568 }
2569
2570 static int[] bnExpModThreshTable = {7, 25, 81, 241, 673, 1793,
2571 Integer.MAX_VALUE}; // Sentinel
2572
2573 /**
2574 * Returns a BigInteger whose value is x to the power of y mod z.
2575 * Assumes: z is odd && x < z.
2576 */
2577 private BigInteger oddModPow(BigInteger y, BigInteger z) {
2578 /*
2579 * The algorithm is adapted from Colin Plumb's C library.
2580 *
2581 * The window algorithm:
2582 * The idea is to keep a running product of b1 = n^(high-order bits of exp)
2583 * and then keep appending exponent bits to it. The following patterns
2584 * apply to a 3-bit window (k = 3):
2585 * To append 0: square
2586 * To append 1: square, multiply by n^1
2587 * To append 10: square, multiply by n^1, square
2588 * To append 11: square, square, multiply by n^3
2589 * To append 100: square, multiply by n^1, square, square
2590 * To append 101: square, square, square, multiply by n^5
2591 * To append 110: square, square, multiply by n^3, square
2592 * To append 111: square, square, square, multiply by n^7
2593 *
2594 * Since each pattern involves only one multiply, the longer the pattern
2595 * the better, except that a 0 (no multiplies) can be appended directly.
2596 * We precompute a table of odd powers of n, up to 2^k, and can then
2597 * multiply k bits of exponent at a time. Actually, assuming random
2598 * exponents, there is on average one zero bit between needs to
2599 * multiply (1/2 of the time there's none, 1/4 of the time there's 1,
2600 * 1/8 of the time, there's 2, 1/32 of the time, there's 3, etc.), so
2601 * you have to do one multiply per k+1 bits of exponent.
2602 *
2603 * The loop walks down the exponent, squaring the result buffer as
2604 * it goes. There is a wbits+1 bit lookahead buffer, buf, that is
2605 * filled with the upcoming exponent bits. (What is read after the
2606 * end of the exponent is unimportant, but it is filled with zero here.)
2607 * When the most-significant bit of this buffer becomes set, i.e.
2608 * (buf & tblmask) != 0, we have to decide what pattern to multiply
2609 * by, and when to do it. We decide, remember to do it in future
2610 * after a suitable number of squarings have passed (e.g. a pattern
2611 * of "100" in the buffer requires that we multiply by n^1 immediately;
2612 * a pattern of "110" calls for multiplying by n^3 after one more
2613 * squaring), clear the buffer, and continue.
2614 *
2615 * When we start, there is one more optimization: the result buffer
2616 * is implcitly one, so squaring it or multiplying by it can be
2617 * optimized away. Further, if we start with a pattern like "100"
2618 * in the lookahead window, rather than placing n into the buffer
2619 * and then starting to square it, we have already computed n^2
2620 * to compute the odd-powers table, so we can place that into
2621 * the buffer and save a squaring.
2622 *
2623 * This means that if you have a k-bit window, to compute n^z,
2624 * where z is the high k bits of the exponent, 1/2 of the time
2625 * it requires no squarings. 1/4 of the time, it requires 1
2626 * squaring, ... 1/2^(k-1) of the time, it reqires k-2 squarings.
2627 * And the remaining 1/2^(k-1) of the time, the top k bits are a
2628 * 1 followed by k-1 0 bits, so it again only requires k-2
2629 * squarings, not k-1. The average of these is 1. Add that
2630 * to the one squaring we have to do to compute the table,
2631 * and you'll see that a k-bit window saves k-2 squarings
2632 * as well as reducing the multiplies. (It actually doesn't
2633 * hurt in the case k = 1, either.)
2634 */
2635 // Special case for exponent of one
2636 if (y.equals(ONE))
2637 return this;
2638
2639 // Special case for base of zero
2640 if (signum == 0)
2641 return ZERO;
2642
2643 int[] base = mag.clone();
2644 int[] exp = y.mag;
2645 int[] mod = z.mag;
2646 int modLen = mod.length;
2647
2648 // Select an appropriate window size
2649 int wbits = 0;
2650 int ebits = bitLength(exp, exp.length);
2651 // if exponent is 65537 (0x10001), use minimum window size
2652 if ((ebits != 17) || (exp[0] != 65537)) {
2653 while (ebits > bnExpModThreshTable[wbits]) {
2654 wbits++;
2655 }
2656 }
2657
2658 // Calculate appropriate table size
2659 int tblmask = 1 << wbits;
2660
2661 // Allocate table for precomputed odd powers of base in Montgomery form
2662 int[][] table = new int[tblmask][];
2663 for (int i=0; i < tblmask; i++)
2664 table[i] = new int[modLen];
2665
2666 // Compute the modular inverse
2667 int inv = -MutableBigInteger.inverseMod32(mod[modLen-1]);
2668
2669 // Convert base to Montgomery form
2670 int[] a = leftShift(base, base.length, modLen << 5);
2671
2672 MutableBigInteger q = new MutableBigInteger(),
2673 a2 = new MutableBigInteger(a),
2674 b2 = new MutableBigInteger(mod);
2675
2676 MutableBigInteger r= a2.divide(b2, q);
2677 table[0] = r.toIntArray();
2678
2679 // Pad table[0] with leading zeros so its length is at least modLen
2680 if (table[0].length < modLen) {
2681 int offset = modLen - table[0].length;
2682 int[] t2 = new int[modLen];
2683 for (int i=0; i < table[0].length; i++)
2684 t2[i+offset] = table[0][i];
2685 table[0] = t2;
2686 }
2687
2688 // Set b to the square of the base
2689 int[] b = squareToLen(table[0], modLen, null);
2690 b = montReduce(b, mod, modLen, inv);
2691
2692 // Set t to high half of b
2693 int[] t = Arrays.copyOf(b, modLen);
2694
2695 // Fill in the table with odd powers of the base
2696 for (int i=1; i < tblmask; i++) {
2697 int[] prod = multiplyToLen(t, modLen, table[i-1], modLen, null);
2698 table[i] = montReduce(prod, mod, modLen, inv);
2699 }
2700
2701 // Pre load the window that slides over the exponent
2702 int bitpos = 1 << ((ebits-1) & (32-1));
2703
2704 int buf = 0;
2705 int elen = exp.length;
2706 int eIndex = 0;
2707 for (int i = 0; i <= wbits; i++) {
2708 buf = (buf << 1) | (((exp[eIndex] & bitpos) != 0)?1:0);
2709 bitpos >>>= 1;
2710 if (bitpos == 0) {
2711 eIndex++;
2712 bitpos = 1 << (32-1);
2713 elen--;
2714 }
2715 }
2716
2717 int multpos = ebits;
2718
2719 // The first iteration, which is hoisted out of the main loop
2720 ebits--;
2721 boolean isone = true;
2722
2723 multpos = ebits - wbits;
2724 while ((buf & 1) == 0) {
2725 buf >>>= 1;
2726 multpos++;
2727 }
2728
2729 int[] mult = table[buf >>> 1];
2730
2731 buf = 0;
2732 if (multpos == ebits)
2733 isone = false;
2734
2735 // The main loop
2736 while (true) {
2737 ebits--;
2738 // Advance the window
2739 buf <<= 1;
2740
2741 if (elen != 0) {
2742 buf |= ((exp[eIndex] & bitpos) != 0) ? 1 : 0;
2743 bitpos >>>= 1;
2744 if (bitpos == 0) {
2745 eIndex++;
2746 bitpos = 1 << (32-1);
2747 elen--;
2748 }
2749 }
2750
2751 // Examine the window for pending multiplies
2752 if ((buf & tblmask) != 0) {
2753 multpos = ebits - wbits;
2754 while ((buf & 1) == 0) {
2755 buf >>>= 1;
2756 multpos++;
2757 }
2758 mult = table[buf >>> 1];
2759 buf = 0;
2760 }
2761
2762 // Perform multiply
2763 if (ebits == multpos) {
2764 if (isone) {
2765 b = mult.clone();
2766 isone = false;
2767 } else {
2768 t = b;
2769 a = multiplyToLen(t, modLen, mult, modLen, a);
2770 a = montReduce(a, mod, modLen, inv);
2771 t = a; a = b; b = t;
2772 }
2773 }
2774
2775 // Check if done
2776 if (ebits == 0)
2777 break;
2778
2779 // Square the input
2780 if (!isone) {
2781 t = b;
2782 a = squareToLen(t, modLen, a);
2783 a = montReduce(a, mod, modLen, inv);
2784 t = a; a = b; b = t;
2785 }
2786 }
2787
2788 // Convert result out of Montgomery form and return
2789 int[] t2 = new int[2*modLen];
2790 System.arraycopy(b, 0, t2, modLen, modLen);
2791
2792 b = montReduce(t2, mod, modLen, inv);
2793
2794 t2 = Arrays.copyOf(b, modLen);
2795
2796 return new BigInteger(1, t2);
2797 }
2798
2799 /**
2800 * Montgomery reduce n, modulo mod. This reduces modulo mod and divides
2801 * by 2^(32*mlen). Adapted from Colin Plumb's C library.
2802 */
2803 private static int[] montReduce(int[] n, int[] mod, int mlen, int inv) {
2804 int c=0;
2805 int len = mlen;
2806 int offset=0;
2807
2808 do {
2809 int nEnd = n[n.length-1-offset];
2810 int carry = mulAdd(n, mod, offset, mlen, inv * nEnd);
2811 c += addOne(n, offset, mlen, carry);
2812 offset++;
2813 } while (--len > 0);
2814
2815 while (c > 0)
2816 c += subN(n, mod, mlen);
2817
2818 while (intArrayCmpToLen(n, mod, mlen) >= 0)
2819 subN(n, mod, mlen);
2820
2821 return n;
2822 }
2823
2824
2825 /*
2826 * Returns -1, 0 or +1 as big-endian unsigned int array arg1 is less than,
2827 * equal to, or greater than arg2 up to length len.
2828 */
2829 private static int intArrayCmpToLen(int[] arg1, int[] arg2, int len) {
2830 for (int i=0; i < len; i++) {
2831 long b1 = arg1[i] & LONG_MASK;
2832 long b2 = arg2[i] & LONG_MASK;
2833 if (b1 < b2)
2834 return -1;
2835 if (b1 > b2)
2836 return 1;
2837 }
2838 return 0;
2839 }
2840
2841 /**
2842 * Subtracts two numbers of same length, returning borrow.
2843 */
2844 private static int subN(int[] a, int[] b, int len) {
2845 long sum = 0;
2846
2847 while (--len >= 0) {
2848 sum = (a[len] & LONG_MASK) -
2849 (b[len] & LONG_MASK) + (sum >> 32);
2850 a[len] = (int)sum;
2851 }
2852
2853 return (int)(sum >> 32);
2854 }
2855
2856 /**
2857 * Multiply an array by one word k and add to result, return the carry
2858 */
2859 static int mulAdd(int[] out, int[] in, int offset, int len, int k) {
2860 long kLong = k & LONG_MASK;
2861 long carry = 0;
2862
2863 offset = out.length-offset - 1;
2864 for (int j=len-1; j >= 0; j--) {
2865 long product = (in[j] & LONG_MASK) * kLong +
2866 (out[offset] & LONG_MASK) + carry;
2867 out[offset--] = (int)product;
2868 carry = product >>> 32;
2869 }
2870 return (int)carry;
2871 }
2872
2873 /**
2874 * Add one word to the number a mlen words into a. Return the resulting
2875 * carry.
2876 */
2877 static int addOne(int[] a, int offset, int mlen, int carry) {
2878 offset = a.length-1-mlen-offset;
2879 long t = (a[offset] & LONG_MASK) + (carry & LONG_MASK);
2880
2881 a[offset] = (int)t;
2882 if ((t >>> 32) == 0)
2883 return 0;
2884 while (--mlen >= 0) {
2885 if (--offset < 0) { // Carry out of number
2886 return 1;
2887 } else {
2888 a[offset]++;
2889 if (a[offset] != 0)
2890 return 0;
2891 }
2892 }
2893 return 1;
2894 }
2895
2896 /**
2897 * Returns a BigInteger whose value is (this ** exponent) mod (2**p)
2898 */
2899 private BigInteger modPow2(BigInteger exponent, int p) {
2900 /*
2901 * Perform exponentiation using repeated squaring trick, chopping off
2902 * high order bits as indicated by modulus.
2903 */
2904 BigInteger result = ONE;
2905 BigInteger baseToPow2 = this.mod2(p);
2906 int expOffset = 0;
2907
2908 int limit = exponent.bitLength();
2909
2910 if (this.testBit(0))
2911 limit = (p-1) < limit ? (p-1) : limit;
2912
2913 while (expOffset < limit) {
2914 if (exponent.testBit(expOffset))
2915 result = result.multiply(baseToPow2).mod2(p);
2916 expOffset++;
2917 if (expOffset < limit)
2918 baseToPow2 = baseToPow2.square().mod2(p);
2919 }
2920
2921 return result;
2922 }
2923
2924 /**
2925 * Returns a BigInteger whose value is this mod(2**p).
2926 * Assumes that this {@code BigInteger >= 0} and {@code p > 0}.
2927 */
2928 private BigInteger mod2(int p) {
2929 if (bitLength() <= p)
2930 return this;
2931
2932 // Copy remaining ints of mag
2933 int numInts = (p + 31) >>> 5;
2934 int[] mag = new int[numInts];
2935 System.arraycopy(this.mag, (this.mag.length - numInts), mag, 0, numInts);
2936
2937 // Mask out any excess bits
2938 int excessBits = (numInts << 5) - p;
2939 mag[0] &= (1L << (32-excessBits)) - 1;
2940
2941 return (mag[0] == 0 ? new BigInteger(1, mag) : new BigInteger(mag, 1));
2942 }
2943
2944 /**
2945 * Returns a BigInteger whose value is {@code (this}<sup>-1</sup> {@code mod m)}.
2946 *
2947 * @param m the modulus.
2948 * @return {@code this}<sup>-1</sup> {@code mod m}.
2949 * @throws ArithmeticException {@code m} ≤ 0, or this BigInteger
2950 * has no multiplicative inverse mod m (that is, this BigInteger
2951 * is not <i>relatively prime</i> to m).
2952 */
2953 public BigInteger modInverse(BigInteger m) {
2954 if (m.signum != 1)
2955 throw new ArithmeticException("BigInteger: modulus not positive");
2956
2957 if (m.equals(ONE))
2958 return ZERO;
2959
2960 // Calculate (this mod m)
2961 BigInteger modVal = this;
2962 if (signum < 0 || (this.compareMagnitude(m) >= 0))
2963 modVal = this.mod(m);
2964
2965 if (modVal.equals(ONE))
2966 return ONE;
2967
2968 MutableBigInteger a = new MutableBigInteger(modVal);
2969 MutableBigInteger b = new MutableBigInteger(m);
2970
2971 MutableBigInteger result = a.mutableModInverse(b);
2972 return result.toBigInteger(1);
2973 }
2974
2975 // Shift Operations
2976
2977 /**
2978 * Returns a BigInteger whose value is {@code (this << n)}.
2979 * The shift distance, {@code n}, may be negative, in which case
2980 * this method performs a right shift.
2981 * (Computes <tt>floor(this * 2<sup>n</sup>)</tt>.)
2982 *
2983 * @param n shift distance, in bits.
2984 * @return {@code this << n}
2985 * @see #shiftRight
2986 */
2987 public BigInteger shiftLeft(int n) {
2988 if (signum == 0)
2989 return ZERO;
2990 if (n > 0) {
2991 return new BigInteger(shiftLeft(mag, n), signum);
2992 } else if (n == 0) {
2993 return this;
2994 } else {
2995 // Possible int overflow in (-n) is not a trouble,
2996 // because shiftRightImpl considers its argument unsigned
2997 return shiftRightImpl(-n);
2998 }
2999 }
3000
3001 /**
3002 * Returns a magnitude array whose value is {@code (mag << n)}.
3003 * The shift distance, {@code n}, is considered unnsigned.
3004 * (Computes <tt>this * 2<sup>n</sup></tt>.)
3005 *
3006 * @param mag magnitude, the most-significant int ({@code mag[0]}) must be non-zero.
3007 * @param n unsigned shift distance, in bits.
3008 * @return {@code mag << n}
3009 */
3010 private static int[] shiftLeft(int[] mag, int n) {
3011 int nInts = n >>> 5;
3012 int nBits = n & 0x1f;
3013 int magLen = mag.length;
3014 int newMag[] = null;
3015
3016 if (nBits == 0) {
3017 newMag = new int[magLen + nInts];
3018 System.arraycopy(mag, 0, newMag, 0, magLen);
3019 } else {
3020 int i = 0;
3021 int nBits2 = 32 - nBits;
3022 int highBits = mag[0] >>> nBits2;
3023 if (highBits != 0) {
3024 newMag = new int[magLen + nInts + 1];
3025 newMag[i++] = highBits;
3026 } else {
3027 newMag = new int[magLen + nInts];
3028 }
3029 int j=0;
3030 while (j < magLen-1)
3031 newMag[i++] = mag[j++] << nBits | mag[j] >>> nBits2;
3032 newMag[i] = mag[j] << nBits;
3033 }
3034 return newMag;
3035 }
3036
3037 /**
3038 * Returns a BigInteger whose value is {@code (this >> n)}. Sign
3039 * extension is performed. The shift distance, {@code n}, may be
3040 * negative, in which case this method performs a left shift.
3041 * (Computes <tt>floor(this / 2<sup>n</sup>)</tt>.)
3042 *
3043 * @param n shift distance, in bits.
3044 * @return {@code this >> n}
3045 * @see #shiftLeft
3046 */
3047 public BigInteger shiftRight(int n) {
3048 if (signum == 0)
3049 return ZERO;
3050 if (n > 0) {
3051 return shiftRightImpl(n);
3052 } else if (n == 0) {
3053 return this;
3054 } else {
3055 // Possible int overflow in {@code -n} is not a trouble,
3056 // because shiftLeft considers its argument unsigned
3057 return new BigInteger(shiftLeft(mag, -n), signum);
3058 }
3059 }
3060
3061 /**
3062 * Returns a BigInteger whose value is {@code (this >> n)}. The shift
3063 * distance, {@code n}, is considered unsigned.
3064 * (Computes <tt>floor(this * 2<sup>-n</sup>)</tt>.)
3065 *
3066 * @param n unsigned shift distance, in bits.
3067 * @return {@code this >> n}
3068 */
3069 private BigInteger shiftRightImpl(int n) {
3070 int nInts = n >>> 5;
3071 int nBits = n & 0x1f;
3072 int magLen = mag.length;
3073 int newMag[] = null;
3074
3075 // Special case: entire contents shifted off the end
3076 if (nInts >= magLen)
3077 return (signum >= 0 ? ZERO : negConst[1]);
3078
3079 if (nBits == 0) {
3080 int newMagLen = magLen - nInts;
3081 newMag = Arrays.copyOf(mag, newMagLen);
3082 } else {
3083 int i = 0;
3084 int highBits = mag[0] >>> nBits;
3085 if (highBits != 0) {
3086 newMag = new int[magLen - nInts];
3087 newMag[i++] = highBits;
3088 } else {
3089 newMag = new int[magLen - nInts -1];
3090 }
3091
3092 int nBits2 = 32 - nBits;
3093 int j=0;
3094 while (j < magLen - nInts - 1)
3095 newMag[i++] = (mag[j++] << nBits2) | (mag[j] >>> nBits);
3096 }
3097
3098 if (signum < 0) {
3099 // Find out whether any one-bits were shifted off the end.
3100 boolean onesLost = false;
3101 for (int i=magLen-1, j=magLen-nInts; i >= j && !onesLost; i--)
3102 onesLost = (mag[i] != 0);
3103 if (!onesLost && nBits != 0)
3104 onesLost = (mag[magLen - nInts - 1] << (32 - nBits) != 0);
3105
3106 if (onesLost)
3107 newMag = javaIncrement(newMag);
3108 }
3109
3110 return new BigInteger(newMag, signum);
3111 }
3112
3113 int[] javaIncrement(int[] val) {
3114 int lastSum = 0;
3115 for (int i=val.length-1; i >= 0 && lastSum == 0; i--)
3116 lastSum = (val[i] += 1);
3117 if (lastSum == 0) {
3118 val = new int[val.length+1];
3119 val[0] = 1;
3120 }
3121 return val;
3122 }
3123
3124 // Bitwise Operations
3125
3126 /**
3127 * Returns a BigInteger whose value is {@code (this & val)}. (This
3128 * method returns a negative BigInteger if and only if this and val are
3129 * both negative.)
3130 *
3131 * @param val value to be AND'ed with this BigInteger.
3132 * @return {@code this & val}
3133 */
3134 public BigInteger and(BigInteger val) {
3135 int[] result = new int[Math.max(intLength(), val.intLength())];
3136 for (int i=0; i < result.length; i++)
3137 result[i] = (getInt(result.length-i-1)
3138 & val.getInt(result.length-i-1));
3139
3140 return valueOf(result);
3141 }
3142
3143 /**
3144 * Returns a BigInteger whose value is {@code (this | val)}. (This method
3145 * returns a negative BigInteger if and only if either this or val is
3146 * negative.)
3147 *
3148 * @param val value to be OR'ed with this BigInteger.
3149 * @return {@code this | val}
3150 */
3151 public BigInteger or(BigInteger val) {
3152 int[] result = new int[Math.max(intLength(), val.intLength())];
3153 for (int i=0; i < result.length; i++)
3154 result[i] = (getInt(result.length-i-1)
3155 | val.getInt(result.length-i-1));
3156
3157 return valueOf(result);
3158 }
3159
3160 /**
3161 * Returns a BigInteger whose value is {@code (this ^ val)}. (This method
3162 * returns a negative BigInteger if and only if exactly one of this and
3163 * val are negative.)
3164 *
3165 * @param val value to be XOR'ed with this BigInteger.
3166 * @return {@code this ^ val}
3167 */
3168 public BigInteger xor(BigInteger val) {
3169 int[] result = new int[Math.max(intLength(), val.intLength())];
3170 for (int i=0; i < result.length; i++)
3171 result[i] = (getInt(result.length-i-1)
3172 ^ val.getInt(result.length-i-1));
3173
3174 return valueOf(result);
3175 }
3176
3177 /**
3178 * Returns a BigInteger whose value is {@code (~this)}. (This method
3179 * returns a negative value if and only if this BigInteger is
3180 * non-negative.)
3181 *
3182 * @return {@code ~this}
3183 */
3184 public BigInteger not() {
3185 int[] result = new int[intLength()];
3186 for (int i=0; i < result.length; i++)
3187 result[i] = ~getInt(result.length-i-1);
3188
3189 return valueOf(result);
3190 }
3191
3192 /**
3193 * Returns a BigInteger whose value is {@code (this & ~val)}. This
3194 * method, which is equivalent to {@code and(val.not())}, is provided as
3195 * a convenience for masking operations. (This method returns a negative
3196 * BigInteger if and only if {@code this} is negative and {@code val} is
3197 * positive.)
3198 *
3199 * @param val value to be complemented and AND'ed with this BigInteger.
3200 * @return {@code this & ~val}
3201 */
3202 public BigInteger andNot(BigInteger val) {
3203 int[] result = new int[Math.max(intLength(), val.intLength())];
3204 for (int i=0; i < result.length; i++)
3205 result[i] = (getInt(result.length-i-1)
3206 & ~val.getInt(result.length-i-1));
3207
3208 return valueOf(result);
3209 }
3210
3211
3212 // Single Bit Operations
3213
3214 /**
3215 * Returns {@code true} if and only if the designated bit is set.
3216 * (Computes {@code ((this & (1<<n)) != 0)}.)
3217 *
3218 * @param n index of bit to test.
3219 * @return {@code true} if and only if the designated bit is set.
3220 * @throws ArithmeticException {@code n} is negative.
3221 */
3222 public boolean testBit(int n) {
3223 if (n < 0)
3224 throw new ArithmeticException("Negative bit address");
3225
3226 return (getInt(n >>> 5) & (1 << (n & 31))) != 0;
3227 }
3228
3229 /**
3230 * Returns a BigInteger whose value is equivalent to this BigInteger
3231 * with the designated bit set. (Computes {@code (this | (1<<n))}.)
3232 *
3233 * @param n index of bit to set.
3234 * @return {@code this | (1<<n)}
3235 * @throws ArithmeticException {@code n} is negative.
3236 */
3237 public BigInteger setBit(int n) {
3238 if (n < 0)
3239 throw new ArithmeticException("Negative bit address");
3240
3241 int intNum = n >>> 5;
3242 int[] result = new int[Math.max(intLength(), intNum+2)];
3243
3244 for (int i=0; i < result.length; i++)
3245 result[result.length-i-1] = getInt(i);
3246
3247 result[result.length-intNum-1] |= (1 << (n & 31));
3248
3249 return valueOf(result);
3250 }
3251
3252 /**
3253 * Returns a BigInteger whose value is equivalent to this BigInteger
3254 * with the designated bit cleared.
3255 * (Computes {@code (this & ~(1<<n))}.)
3256 *
3257 * @param n index of bit to clear.
3258 * @return {@code this & ~(1<<n)}
3259 * @throws ArithmeticException {@code n} is negative.
3260 */
3261 public BigInteger clearBit(int n) {
3262 if (n < 0)
3263 throw new ArithmeticException("Negative bit address");
3264
3265 int intNum = n >>> 5;
3266 int[] result = new int[Math.max(intLength(), ((n + 1) >>> 5) + 1)];
3267
3268 for (int i=0; i < result.length; i++)
3269 result[result.length-i-1] = getInt(i);
3270
3271 result[result.length-intNum-1] &= ~(1 << (n & 31));
3272
3273 return valueOf(result);
3274 }
3275
3276 /**
3277 * Returns a BigInteger whose value is equivalent to this BigInteger
3278 * with the designated bit flipped.
3279 * (Computes {@code (this ^ (1<<n))}.)
3280 *
3281 * @param n index of bit to flip.
3282 * @return {@code this ^ (1<<n)}
3283 * @throws ArithmeticException {@code n} is negative.
3284 */
3285 public BigInteger flipBit(int n) {
3286 if (n < 0)
3287 throw new ArithmeticException("Negative bit address");
3288
3289 int intNum = n >>> 5;
3290 int[] result = new int[Math.max(intLength(), intNum+2)];
3291
3292 for (int i=0; i < result.length; i++)
3293 result[result.length-i-1] = getInt(i);
3294
3295 result[result.length-intNum-1] ^= (1 << (n & 31));
3296
3297 return valueOf(result);
3298 }
3299
3300 /**
3301 * Returns the index of the rightmost (lowest-order) one bit in this
3302 * BigInteger (the number of zero bits to the right of the rightmost
3303 * one bit). Returns -1 if this BigInteger contains no one bits.
3304 * (Computes {@code (this == 0? -1 : log2(this & -this))}.)
3305 *
3306 * @return index of the rightmost one bit in this BigInteger.
3307 */
3308 public int getLowestSetBit() {
3309 int lsb = lowestSetBitPlusTwo - 2;
3310 if (lsb == -2) { // lowestSetBit not initialized yet
3311 lsb = 0;
3312 if (signum == 0) {
3313 lsb -= 1;
3314 } else {
3315 // Search for lowest order nonzero int
3316 int i,b;
3317 for (i=0; (b = getInt(i)) == 0; i++)
3318 ;
3319 lsb += (i << 5) + Integer.numberOfTrailingZeros(b);
3320 }
3321 lowestSetBitPlusTwo = lsb + 2;
3322 }
3323 return lsb;
3324 }
3325
3326
3327 // Miscellaneous Bit Operations
3328
3329 /**
3330 * Returns the number of bits in the minimal two's-complement
3331 * representation of this BigInteger, <i>excluding</i> a sign bit.
3332 * For positive BigIntegers, this is equivalent to the number of bits in
3333 * the ordinary binary representation. (Computes
3334 * {@code (ceil(log2(this < 0 ? -this : this+1)))}.)
3335 *
3336 * @return number of bits in the minimal two's-complement
3337 * representation of this BigInteger, <i>excluding</i> a sign bit.
3338 */
3339 public int bitLength() {
3340 int n = bitLengthPlusOne - 1;
3341 if (n == -1) { // bitLength not initialized yet
3342 int[] m = mag;
3343 int len = m.length;
3344 if (len == 0) {
3345 n = 0; // offset by one to initialize
3346 } else {
3347 // Calculate the bit length of the magnitude
3348 int magBitLength = ((len - 1) << 5) + bitLengthForInt(mag[0]);
3349 if (signum < 0) {
3350 // Check if magnitude is a power of two
3351 boolean pow2 = (Integer.bitCount(mag[0]) == 1);
3352 for (int i=1; i< len && pow2; i++)
3353 pow2 = (mag[i] == 0);
3354
3355 n = (pow2 ? magBitLength -1 : magBitLength);
3356 } else {
3357 n = magBitLength;
3358 }
3359 }
3360 bitLengthPlusOne = n + 1;
3361 }
3362 return n;
3363 }
3364
3365 /**
3366 * Returns the number of bits in the two's complement representation
3367 * of this BigInteger that differ from its sign bit. This method is
3368 * useful when implementing bit-vector style sets atop BigIntegers.
3369 *
3370 * @return number of bits in the two's complement representation
3371 * of this BigInteger that differ from its sign bit.
3372 */
3373 public int bitCount() {
3374 int bc = bitCountPlusOne - 1;
3375 if (bc == -1) { // bitCount not initialized yet
3376 bc = 0; // offset by one to initialize
3377 // Count the bits in the magnitude
3378 for (int i=0; i < mag.length; i++)
3379 bc += Integer.bitCount(mag[i]);
3380 if (signum < 0) {
3381 // Count the trailing zeros in the magnitude
3382 int magTrailingZeroCount = 0, j;
3383 for (j=mag.length-1; mag[j] == 0; j--)
3384 magTrailingZeroCount += 32;
3385 magTrailingZeroCount += Integer.numberOfTrailingZeros(mag[j]);
3386 bc += magTrailingZeroCount - 1;
3387 }
3388 bitCountPlusOne = bc + 1;
3389 }
3390 return bc;
3391 }
3392
3393 // Primality Testing
3394
3395 /**
3396 * Returns {@code true} if this BigInteger is probably prime,
3397 * {@code false} if it's definitely composite. If
3398 * {@code certainty} is ≤ 0, {@code true} is
3399 * returned.
3400 *
3401 * @param certainty a measure of the uncertainty that the caller is
3402 * willing to tolerate: if the call returns {@code true}
3403 * the probability that this BigInteger is prime exceeds
3404 * (1 - 1/2<sup>{@code certainty}</sup>). The execution time of
3405 * this method is proportional to the value of this parameter.
3406 * @return {@code true} if this BigInteger is probably prime,
3407 * {@code false} if it's definitely composite.
3408 */
3409 public boolean isProbablePrime(int certainty) {
3410 if (certainty <= 0)
3411 return true;
3412 BigInteger w = this.abs();
3413 if (w.equals(TWO))
3414 return true;
3415 if (!w.testBit(0) || w.equals(ONE))
3416 return false;
3417
3418 return w.primeToCertainty(certainty, null);
3419 }
3420
3421 // Comparison Operations
3422
3423 /**
3424 * Compares this BigInteger with the specified BigInteger. This
3425 * method is provided in preference to individual methods for each
3426 * of the six boolean comparison operators ({@literal <}, ==,
3427 * {@literal >}, {@literal >=}, !=, {@literal <=}). The suggested
3428 * idiom for performing these comparisons is: {@code
3429 * (x.compareTo(y)} <<i>op</i>> {@code 0)}, where
3430 * <<i>op</i>> is one of the six comparison operators.
3431 *
3432 * @param val BigInteger to which this BigInteger is to be compared.
3433 * @return -1, 0 or 1 as this BigInteger is numerically less than, equal
3434 * to, or greater than {@code val}.
3435 */
3436 public int compareTo(BigInteger val) {
3437 if (signum == val.signum) {
3438 switch (signum) {
3439 case 1:
3440 return compareMagnitude(val);
3441 case -1:
3442 return val.compareMagnitude(this);
3443 default:
3444 return 0;
3445 }
3446 }
3447 return signum > val.signum ? 1 : -1;
3448 }
3449
3450 /**
3451 * Compares the magnitude array of this BigInteger with the specified
3452 * BigInteger's. This is the version of compareTo ignoring sign.
3453 *
3454 * @param val BigInteger whose magnitude array to be compared.
3455 * @return -1, 0 or 1 as this magnitude array is less than, equal to or
3456 * greater than the magnitude aray for the specified BigInteger's.
3457 */
3458 final int compareMagnitude(BigInteger val) {
3459 int[] m1 = mag;
3460 int len1 = m1.length;
3461 int[] m2 = val.mag;
3462 int len2 = m2.length;
3463 if (len1 < len2)
3464 return -1;
3465 if (len1 > len2)
3466 return 1;
3467 for (int i = 0; i < len1; i++) {
3468 int a = m1[i];
3469 int b = m2[i];
3470 if (a != b)
3471 return ((a & LONG_MASK) < (b & LONG_MASK)) ? -1 : 1;
3472 }
3473 return 0;
3474 }
3475
3476 /**
3477 * Version of compareMagnitude that compares magnitude with long value.
3478 * val can't be Long.MIN_VALUE.
3479 */
3480 final int compareMagnitude(long val) {
3481 assert val != Long.MIN_VALUE;
3482 int[] m1 = mag;
3483 int len = m1.length;
3484 if (len > 2) {
3485 return 1;
3486 }
3487 if (val < 0) {
3488 val = -val;
3489 }
3490 int highWord = (int)(val >>> 32);
3491 if (highWord == 0) {
3492 if (len < 1)
3493 return -1;
3494 if (len > 1)
3495 return 1;
3496 int a = m1[0];
3497 int b = (int)val;
3498 if (a != b) {
3499 return ((a & LONG_MASK) < (b & LONG_MASK))? -1 : 1;
3500 }
3501 return 0;
3502 } else {
3503 if (len < 2)
3504 return -1;
3505 int a = m1[0];
3506 int b = highWord;
3507 if (a != b) {
3508 return ((a & LONG_MASK) < (b & LONG_MASK))? -1 : 1;
3509 }
3510 a = m1[1];
3511 b = (int)val;
3512 if (a != b) {
3513 return ((a & LONG_MASK) < (b & LONG_MASK))? -1 : 1;
3514 }
3515 return 0;
3516 }
3517 }
3518
3519 /**
3520 * Compares this BigInteger with the specified Object for equality.
3521 *
3522 * @param x Object to which this BigInteger is to be compared.
3523 * @return {@code true} if and only if the specified Object is a
3524 * BigInteger whose value is numerically equal to this BigInteger.
3525 */
3526 public boolean equals(Object x) {
3527 // This test is just an optimization, which may or may not help
3528 if (x == this)
3529 return true;
3530
3531 if (!(x instanceof BigInteger))
3532 return false;
3533
3534 BigInteger xInt = (BigInteger) x;
3535 if (xInt.signum != signum)
3536 return false;
3537
3538 int[] m = mag;
3539 int len = m.length;
3540 int[] xm = xInt.mag;
3541 if (len != xm.length)
3542 return false;
3543
3544 for (int i = 0; i < len; i++)
3545 if (xm[i] != m[i])
3546 return false;
3547
3548 return true;
3549 }
3550
3551 /**
3552 * Returns the minimum of this BigInteger and {@code val}.
3553 *
3554 * @param val value with which the minimum is to be computed.
3555 * @return the BigInteger whose value is the lesser of this BigInteger and
3556 * {@code val}. If they are equal, either may be returned.
3557 */
3558 public BigInteger min(BigInteger val) {
3559 return (compareTo(val) < 0 ? this : val);
3560 }
3561
3562 /**
3563 * Returns the maximum of this BigInteger and {@code val}.
3564 *
3565 * @param val value with which the maximum is to be computed.
3566 * @return the BigInteger whose value is the greater of this and
3567 * {@code val}. If they are equal, either may be returned.
3568 */
3569 public BigInteger max(BigInteger val) {
3570 return (compareTo(val) > 0 ? this : val);
3571 }
3572
3573
3574 // Hash Function
3575
3576 /**
3577 * Returns the hash code for this BigInteger.
3578 *
3579 * @return hash code for this BigInteger.
3580 */
3581 public int hashCode() {
3582 int hashCode = 0;
3583
3584 for (int i=0; i < mag.length; i++)
3585 hashCode = (int)(31*hashCode + (mag[i] & LONG_MASK));
3586
3587 return hashCode * signum;
3588 }
3589
3590 /**
3591 * Returns the String representation of this BigInteger in the
3592 * given radix. If the radix is outside the range from {@link
3593 * Character#MIN_RADIX} to {@link Character#MAX_RADIX} inclusive,
3594 * it will default to 10 (as is the case for
3595 * {@code Integer.toString}). The digit-to-character mapping
3596 * provided by {@code Character.forDigit} is used, and a minus
3597 * sign is prepended if appropriate. (This representation is
3598 * compatible with the {@link #BigInteger(String, int) (String,
3599 * int)} constructor.)
3600 *
3601 * @param radix radix of the String representation.
3602 * @return String representation of this BigInteger in the given radix.
3603 * @see Integer#toString
3604 * @see Character#forDigit
3605 * @see #BigInteger(java.lang.String, int)
3606 */
3607 public String toString(int radix) {
3608 if (signum == 0)
3609 return "0";
3610 if (radix < Character.MIN_RADIX || radix > Character.MAX_RADIX)
3611 radix = 10;
3612
3613 // If it's small enough, use smallToString.
3614 if (mag.length <= SCHOENHAGE_BASE_CONVERSION_THRESHOLD)
3615 return smallToString(radix);
3616
3617 // Otherwise use recursive toString, which requires positive arguments.
3618 // The results will be concatenated into this StringBuilder
3619 StringBuilder sb = new StringBuilder();
3620 if (signum < 0) {
3621 toString(this.negate(), sb, radix, 0);
3622 sb.insert(0, '-');
3623 }
3624 else
3625 toString(this, sb, radix, 0);
3626
3627 return sb.toString();
3628 }
3629
3630 /** This method is used to perform toString when arguments are small. */
3631 private String smallToString(int radix) {
3632 if (signum == 0) {
3633 return "0";
3634 }
3635
3636 // Compute upper bound on number of digit groups and allocate space
3637 int maxNumDigitGroups = (4*mag.length + 6)/7;
3638 String digitGroup[] = new String[maxNumDigitGroups];
3639
3640 // Translate number to string, a digit group at a time
3641 BigInteger tmp = this.abs();
3642 int numGroups = 0;
3643 while (tmp.signum != 0) {
3644 BigInteger d = longRadix[radix];
3645
3646 MutableBigInteger q = new MutableBigInteger(),
3647 a = new MutableBigInteger(tmp.mag),
3648 b = new MutableBigInteger(d.mag);
3649 MutableBigInteger r = a.divide(b, q);
3650 BigInteger q2 = q.toBigInteger(tmp.signum * d.signum);
3651 BigInteger r2 = r.toBigInteger(tmp.signum * d.signum);
3652
3653 digitGroup[numGroups++] = Long.toString(r2.longValue(), radix);
3654 tmp = q2;
3655 }
3656
3657 // Put sign (if any) and first digit group into result buffer
3658 StringBuilder buf = new StringBuilder(numGroups*digitsPerLong[radix]+1);
3659 if (signum < 0) {
3660 buf.append('-');
3661 }
3662 buf.append(digitGroup[numGroups-1]);
3663
3664 // Append remaining digit groups padded with leading zeros
3665 for (int i=numGroups-2; i >= 0; i--) {
3666 // Prepend (any) leading zeros for this digit group
3667 int numLeadingZeros = digitsPerLong[radix]-digitGroup[i].length();
3668 if (numLeadingZeros != 0) {
3669 buf.append(zeros[numLeadingZeros]);
3670 }
3671 buf.append(digitGroup[i]);
3672 }
3673 return buf.toString();
3674 }
3675
3676 /**
3677 * Converts the specified BigInteger to a string and appends to
3678 * {@code sb}. This implements the recursive Schoenhage algorithm
3679 * for base conversions.
3680 * <p>
3681 * See Knuth, Donald, _The Art of Computer Programming_, Vol. 2,
3682 * Answers to Exercises (4.4) Question 14.
3683 *
3684 * @param u The number to convert to a string.
3685 * @param sb The StringBuilder that will be appended to in place.
3686 * @param radix The base to convert to.
3687 * @param digits The minimum number of digits to pad to.
3688 */
3689 private static void toString(BigInteger u, StringBuilder sb, int radix,
3690 int digits) {
3691 // If we're smaller than a certain threshold, use the smallToString
3692 // method, padding with leading zeroes when necessary.
3693 if (u.mag.length <= SCHOENHAGE_BASE_CONVERSION_THRESHOLD) {
3694 String s = u.smallToString(radix);
3695
3696 // Pad with internal zeros if necessary.
3697 // Don't pad if we're at the beginning of the string.
3698 if ((s.length() < digits) && (sb.length() > 0)) {
3699 for (int i=s.length(); i < digits; i++) {
3700 sb.append('0');
3701 }
3702 }
3703
3704 sb.append(s);
3705 return;
3706 }
3707
3708 int b, n;
3709 b = u.bitLength();
3710
3711 // Calculate a value for n in the equation radix^(2^n) = u
3712 // and subtract 1 from that value. This is used to find the
3713 // cache index that contains the best value to divide u.
3714 n = (int) Math.round(Math.log(b * LOG_TWO / logCache[radix]) / LOG_TWO - 1.0);
3715 BigInteger v = getRadixConversionCache(radix, n);
3716 BigInteger[] results;
3717 results = u.divideAndRemainder(v);
3718
3719 int expectedDigits = 1 << n;
3720
3721 // Now recursively build the two halves of each number.
3722 toString(results[0], sb, radix, digits-expectedDigits);
3723 toString(results[1], sb, radix, expectedDigits);
3724 }
3725
3726 /**
3727 * Returns the value radix^(2^exponent) from the cache.
3728 * If this value doesn't already exist in the cache, it is added.
3729 * <p>
3730 * This could be changed to a more complicated caching method using
3731 * {@code Future}.
3732 */
3733 private static BigInteger getRadixConversionCache(int radix, int exponent) {
3734 BigInteger[] cacheLine = powerCache[radix]; // volatile read
3735 if (exponent < cacheLine.length) {
3736 return cacheLine[exponent];
3737 }
3738
3739 int oldLength = cacheLine.length;
3740 cacheLine = Arrays.copyOf(cacheLine, exponent + 1);
3741 for (int i = oldLength; i <= exponent; i++) {
3742 cacheLine[i] = cacheLine[i - 1].pow(2);
3743 }
3744
3745 BigInteger[][] pc = powerCache; // volatile read again
3746 if (exponent >= pc[radix].length) {
3747 pc = pc.clone();
3748 pc[radix] = cacheLine;
3749 powerCache = pc; // volatile write, publish
3750 }
3751 return cacheLine[exponent];
3752 }
3753
3754 /* zero[i] is a string of i consecutive zeros. */
3755 private static String zeros[] = new String[64];
3756 static {
3757 zeros[63] =
3758 "000000000000000000000000000000000000000000000000000000000000000";
3759 for (int i=0; i < 63; i++)
3760 zeros[i] = zeros[63].substring(0, i);
3761 }
3762
3763 /**
3764 * Returns the decimal String representation of this BigInteger.
3765 * The digit-to-character mapping provided by
3766 * {@code Character.forDigit} is used, and a minus sign is
3767 * prepended if appropriate. (This representation is compatible
3768 * with the {@link #BigInteger(String) (String)} constructor, and
3769 * allows for String concatenation with Java's + operator.)
3770 *
3771 * @return decimal String representation of this BigInteger.
3772 * @see Character#forDigit
3773 * @see #BigInteger(java.lang.String)
3774 */
3775 public String toString() {
3776 return toString(10);
3777 }
3778
3779 /**
3780 * Returns a byte array containing the two's-complement
3781 * representation of this BigInteger. The byte array will be in
3782 * <i>big-endian</i> byte-order: the most significant byte is in
3783 * the zeroth element. The array will contain the minimum number
3784 * of bytes required to represent this BigInteger, including at
3785 * least one sign bit, which is {@code (ceil((this.bitLength() +
3786 * 1)/8))}. (This representation is compatible with the
3787 * {@link #BigInteger(byte[]) (byte[])} constructor.)
3788 *
3789 * @return a byte array containing the two's-complement representation of
3790 * this BigInteger.
3791 * @see #BigInteger(byte[])
3792 */
3793 public byte[] toByteArray() {
3794 int byteLen = bitLength()/8 + 1;
3795 byte[] byteArray = new byte[byteLen];
3796
3797 for (int i=byteLen-1, bytesCopied=4, nextInt=0, intIndex=0; i >= 0; i--) {
3798 if (bytesCopied == 4) {
3799 nextInt = getInt(intIndex++);
3800 bytesCopied = 1;
3801 } else {
3802 nextInt >>>= 8;
3803 bytesCopied++;
3804 }
3805 byteArray[i] = (byte)nextInt;
3806 }
3807 return byteArray;
3808 }
3809
3810 /**
3811 * Converts this BigInteger to an {@code int}. This
3812 * conversion is analogous to a
3813 * <i>narrowing primitive conversion</i> from {@code long} to
3814 * {@code int} as defined in section 5.1.3 of
3815 * <cite>The Java™ Language Specification</cite>:
3816 * if this BigInteger is too big to fit in an
3817 * {@code int}, only the low-order 32 bits are returned.
3818 * Note that this conversion can lose information about the
3819 * overall magnitude of the BigInteger value as well as return a
3820 * result with the opposite sign.
3821 *
3822 * @return this BigInteger converted to an {@code int}.
3823 * @see #intValueExact()
3824 */
3825 public int intValue() {
3826 int result = 0;
3827 result = getInt(0);
3828 return result;
3829 }
3830
3831 /**
3832 * Converts this BigInteger to a {@code long}. This
3833 * conversion is analogous to a
3834 * <i>narrowing primitive conversion</i> from {@code long} to
3835 * {@code int} as defined in section 5.1.3 of
3836 * <cite>The Java™ Language Specification</cite>:
3837 * if this BigInteger is too big to fit in a
3838 * {@code long}, only the low-order 64 bits are returned.
3839 * Note that this conversion can lose information about the
3840 * overall magnitude of the BigInteger value as well as return a
3841 * result with the opposite sign.
3842 *
3843 * @return this BigInteger converted to a {@code long}.
3844 * @see #longValueExact()
3845 */
3846 public long longValue() {
3847 long result = 0;
3848
3849 for (int i=1; i >= 0; i--)
3850 result = (result << 32) + (getInt(i) & LONG_MASK);
3851 return result;
3852 }
3853
3854 /**
3855 * Converts this BigInteger to a {@code float}. This
3856 * conversion is similar to the
3857 * <i>narrowing primitive conversion</i> from {@code double} to
3858 * {@code float} as defined in section 5.1.3 of
3859 * <cite>The Java™ Language Specification</cite>:
3860 * if this BigInteger has too great a magnitude
3861 * to represent as a {@code float}, it will be converted to
3862 * {@link Float#NEGATIVE_INFINITY} or {@link
3863 * Float#POSITIVE_INFINITY} as appropriate. Note that even when
3864 * the return value is finite, this conversion can lose
3865 * information about the precision of the BigInteger value.
3866 *
3867 * @return this BigInteger converted to a {@code float}.
3868 */
3869 public float floatValue() {
3870 if (signum == 0) {
3871 return 0.0f;
3872 }
3873
3874 int exponent = ((mag.length - 1) << 5) + bitLengthForInt(mag[0]) - 1;
3875
3876 // exponent == floor(log2(abs(this)))
3877 if (exponent < Long.SIZE - 1) {
3878 return longValue();
3879 } else if (exponent > Float.MAX_EXPONENT) {
3880 return signum > 0 ? Float.POSITIVE_INFINITY : Float.NEGATIVE_INFINITY;
3881 }
3882
3883 /*
3884 * We need the top SIGNIFICAND_WIDTH bits, including the "implicit"
3885 * one bit. To make rounding easier, we pick out the top
3886 * SIGNIFICAND_WIDTH + 1 bits, so we have one to help us round up or
3887 * down. twiceSignifFloor will contain the top SIGNIFICAND_WIDTH + 1
3888 * bits, and signifFloor the top SIGNIFICAND_WIDTH.
3889 *
3890 * It helps to consider the real number signif = abs(this) *
3891 * 2^(SIGNIFICAND_WIDTH - 1 - exponent).
3892 */
3893 int shift = exponent - FloatConsts.SIGNIFICAND_WIDTH;
3894
3895 int twiceSignifFloor;
3896 // twiceSignifFloor will be == abs().shiftRight(shift).intValue()
3897 // We do the shift into an int directly to improve performance.
3898
3899 int nBits = shift & 0x1f;
3900 int nBits2 = 32 - nBits;
3901
3902 if (nBits == 0) {
3903 twiceSignifFloor = mag[0];
3904 } else {
3905 twiceSignifFloor = mag[0] >>> nBits;
3906 if (twiceSignifFloor == 0) {
3907 twiceSignifFloor = (mag[0] << nBits2) | (mag[1] >>> nBits);
3908 }
3909 }
3910
3911 int signifFloor = twiceSignifFloor >> 1;
3912 signifFloor &= FloatConsts.SIGNIF_BIT_MASK; // remove the implied bit
3913
3914 /*
3915 * We round up if either the fractional part of signif is strictly
3916 * greater than 0.5 (which is true if the 0.5 bit is set and any lower
3917 * bit is set), or if the fractional part of signif is >= 0.5 and
3918 * signifFloor is odd (which is true if both the 0.5 bit and the 1 bit
3919 * are set). This is equivalent to the desired HALF_EVEN rounding.
3920 */
3921 boolean increment = (twiceSignifFloor & 1) != 0
3922 && ((signifFloor & 1) != 0 || abs().getLowestSetBit() < shift);
3923 int signifRounded = increment ? signifFloor + 1 : signifFloor;
3924 int bits = ((exponent + FloatConsts.EXP_BIAS))
3925 << (FloatConsts.SIGNIFICAND_WIDTH - 1);
3926 bits += signifRounded;
3927 /*
3928 * If signifRounded == 2^24, we'd need to set all of the significand
3929 * bits to zero and add 1 to the exponent. This is exactly the behavior
3930 * we get from just adding signifRounded to bits directly. If the
3931 * exponent is Float.MAX_EXPONENT, we round up (correctly) to
3932 * Float.POSITIVE_INFINITY.
3933 */
3934 bits |= signum & FloatConsts.SIGN_BIT_MASK;
3935 return Float.intBitsToFloat(bits);
3936 }
3937
3938 /**
3939 * Converts this BigInteger to a {@code double}. This
3940 * conversion is similar to the
3941 * <i>narrowing primitive conversion</i> from {@code double} to
3942 * {@code float} as defined in section 5.1.3 of
3943 * <cite>The Java™ Language Specification</cite>:
3944 * if this BigInteger has too great a magnitude
3945 * to represent as a {@code double}, it will be converted to
3946 * {@link Double#NEGATIVE_INFINITY} or {@link
3947 * Double#POSITIVE_INFINITY} as appropriate. Note that even when
3948 * the return value is finite, this conversion can lose
3949 * information about the precision of the BigInteger value.
3950 *
3951 * @return this BigInteger converted to a {@code double}.
3952 */
3953 public double doubleValue() {
3954 if (signum == 0) {
3955 return 0.0;
3956 }
3957
3958 int exponent = ((mag.length - 1) << 5) + bitLengthForInt(mag[0]) - 1;
3959
3960 // exponent == floor(log2(abs(this))Double)
3961 if (exponent < Long.SIZE - 1) {
3962 return longValue();
3963 } else if (exponent > Double.MAX_EXPONENT) {
3964 return signum > 0 ? Double.POSITIVE_INFINITY : Double.NEGATIVE_INFINITY;
3965 }
3966
3967 /*
3968 * We need the top SIGNIFICAND_WIDTH bits, including the "implicit"
3969 * one bit. To make rounding easier, we pick out the top
3970 * SIGNIFICAND_WIDTH + 1 bits, so we have one to help us round up or
3971 * down. twiceSignifFloor will contain the top SIGNIFICAND_WIDTH + 1
3972 * bits, and signifFloor the top SIGNIFICAND_WIDTH.
3973 *
3974 * It helps to consider the real number signif = abs(this) *
3975 * 2^(SIGNIFICAND_WIDTH - 1 - exponent).
3976 */
3977 int shift = exponent - DoubleConsts.SIGNIFICAND_WIDTH;
3978
3979 long twiceSignifFloor;
3980 // twiceSignifFloor will be == abs().shiftRight(shift).longValue()
3981 // We do the shift into a long directly to improve performance.
3982
3983 int nBits = shift & 0x1f;
3984 int nBits2 = 32 - nBits;
3985
3986 int highBits;
3987 int lowBits;
3988 if (nBits == 0) {
3989 highBits = mag[0];
3990 lowBits = mag[1];
3991 } else {
3992 highBits = mag[0] >>> nBits;
3993 lowBits = (mag[0] << nBits2) | (mag[1] >>> nBits);
3994 if (highBits == 0) {
3995 highBits = lowBits;
3996 lowBits = (mag[1] << nBits2) | (mag[2] >>> nBits);
3997 }
3998 }
3999
4000 twiceSignifFloor = ((highBits & LONG_MASK) << 32)
4001 | (lowBits & LONG_MASK);
4002
4003 long signifFloor = twiceSignifFloor >> 1;
4004 signifFloor &= DoubleConsts.SIGNIF_BIT_MASK; // remove the implied bit
4005
4006 /*
4007 * We round up if either the fractional part of signif is strictly
4008 * greater than 0.5 (which is true if the 0.5 bit is set and any lower
4009 * bit is set), or if the fractional part of signif is >= 0.5 and
4010 * signifFloor is odd (which is true if both the 0.5 bit and the 1 bit
4011 * are set). This is equivalent to the desired HALF_EVEN rounding.
4012 */
4013 boolean increment = (twiceSignifFloor & 1) != 0
4014 && ((signifFloor & 1) != 0 || abs().getLowestSetBit() < shift);
4015 long signifRounded = increment ? signifFloor + 1 : signifFloor;
4016 long bits = (long) ((exponent + DoubleConsts.EXP_BIAS))
4017 << (DoubleConsts.SIGNIFICAND_WIDTH - 1);
4018 bits += signifRounded;
4019 /*
4020 * If signifRounded == 2^53, we'd need to set all of the significand
4021 * bits to zero and add 1 to the exponent. This is exactly the behavior
4022 * we get from just adding signifRounded to bits directly. If the
4023 * exponent is Double.MAX_EXPONENT, we round up (correctly) to
4024 * Double.POSITIVE_INFINITY.
4025 */
4026 bits |= signum & DoubleConsts.SIGN_BIT_MASK;
4027 return Double.longBitsToDouble(bits);
4028 }
4029
4030 /**
4031 * Returns a copy of the input array stripped of any leading zero bytes.
4032 */
4033 private static int[] stripLeadingZeroInts(int val[]) {
4034 int vlen = val.length;
4035 int keep;
4036
4037 // Find first nonzero byte
4038 for (keep = 0; keep < vlen && val[keep] == 0; keep++)
4039 ;
4040 return java.util.Arrays.copyOfRange(val, keep, vlen);
4041 }
4042
4043 /**
4044 * Returns the input array stripped of any leading zero bytes.
4045 * Since the source is trusted the copying may be skipped.
4046 */
4047 private static int[] trustedStripLeadingZeroInts(int val[]) {
4048 int vlen = val.length;
4049 int keep;
4050
4051 // Find first nonzero byte
4052 for (keep = 0; keep < vlen && val[keep] == 0; keep++)
4053 ;
4054 return keep == 0 ? val : java.util.Arrays.copyOfRange(val, keep, vlen);
4055 }
4056
4057 /**
4058 * Returns a copy of the input array stripped of any leading zero bytes.
4059 */
4060 private static int[] stripLeadingZeroBytes(byte a[], int off, int len) {
4061 int indexBound = off + len;
4062 int keep;
4063
4064 // Find first nonzero byte
4065 for (keep = off; keep < indexBound && a[keep] == 0; keep++)
4066 ;
4067
4068 // Allocate new array and copy relevant part of input array
4069 int intLength = ((indexBound - keep) + 3) >>> 2;
4070 int[] result = new int[intLength];
4071 int b = indexBound - 1;
4072 for (int i = intLength-1; i >= 0; i--) {
4073 result[i] = a[b--] & 0xff;
4074 int bytesRemaining = b - keep + 1;
4075 int bytesToTransfer = Math.min(3, bytesRemaining);
4076 for (int j=8; j <= (bytesToTransfer << 3); j += 8)
4077 result[i] |= ((a[b--] & 0xff) << j);
4078 }
4079 return result;
4080 }
4081
4082 /**
4083 * Takes an array a representing a negative 2's-complement number and
4084 * returns the minimal (no leading zero bytes) unsigned whose value is -a.
4085 */
4086 private static int[] makePositive(byte a[], int off, int len) {
4087 int keep, k;
4088 int indexBound = off + len;
4089
4090 // Find first non-sign (0xff) byte of input
4091 for (keep=off; keep < indexBound && a[keep] == -1; keep++)
4092 ;
4093
4094
4095 /* Allocate output array. If all non-sign bytes are 0x00, we must
4096 * allocate space for one extra output byte. */
4097 for (k=keep; k < indexBound && a[k] == 0; k++)
4098 ;
4099
4100 int extraByte = (k == indexBound) ? 1 : 0;
4101 int intLength = ((indexBound - keep + extraByte) + 3) >>> 2;
4102 int result[] = new int[intLength];
4103
4104 /* Copy one's complement of input into output, leaving extra
4105 * byte (if it exists) == 0x00 */
4106 int b = indexBound - 1;
4107 for (int i = intLength-1; i >= 0; i--) {
4108 result[i] = a[b--] & 0xff;
4109 int numBytesToTransfer = Math.min(3, b-keep+1);
4110 if (numBytesToTransfer < 0)
4111 numBytesToTransfer = 0;
4112 for (int j=8; j <= 8*numBytesToTransfer; j += 8)
4113 result[i] |= ((a[b--] & 0xff) << j);
4114
4115 // Mask indicates which bits must be complemented
4116 int mask = -1 >>> (8*(3-numBytesToTransfer));
4117 result[i] = ~result[i] & mask;
4118 }
4119
4120 // Add one to one's complement to generate two's complement
4121 for (int i=result.length-1; i >= 0; i--) {
4122 result[i] = (int)((result[i] & LONG_MASK) + 1);
4123 if (result[i] != 0)
4124 break;
4125 }
4126
4127 return result;
4128 }
4129
4130 /**
4131 * Takes an array a representing a negative 2's-complement number and
4132 * returns the minimal (no leading zero ints) unsigned whose value is -a.
4133 */
4134 private static int[] makePositive(int a[]) {
4135 int keep, j;
4136
4137 // Find first non-sign (0xffffffff) int of input
4138 for (keep=0; keep < a.length && a[keep] == -1; keep++)
4139 ;
4140
4141 /* Allocate output array. If all non-sign ints are 0x00, we must
4142 * allocate space for one extra output int. */
4143 for (j=keep; j < a.length && a[j] == 0; j++)
4144 ;
4145 int extraInt = (j == a.length ? 1 : 0);
4146 int result[] = new int[a.length - keep + extraInt];
4147
4148 /* Copy one's complement of input into output, leaving extra
4149 * int (if it exists) == 0x00 */
4150 for (int i = keep; i < a.length; i++)
4151 result[i - keep + extraInt] = ~a[i];
4152
4153 // Add one to one's complement to generate two's complement
4154 for (int i=result.length-1; ++result[i] == 0; i--)
4155 ;
4156
4157 return result;
4158 }
4159
4160 /*
4161 * The following two arrays are used for fast String conversions. Both
4162 * are indexed by radix. The first is the number of digits of the given
4163 * radix that can fit in a Java long without "going negative", i.e., the
4164 * highest integer n such that radix**n < 2**63. The second is the
4165 * "long radix" that tears each number into "long digits", each of which
4166 * consists of the number of digits in the corresponding element in
4167 * digitsPerLong (longRadix[i] = i**digitPerLong[i]). Both arrays have
4168 * nonsense values in their 0 and 1 elements, as radixes 0 and 1 are not
4169 * used.
4170 */
4171 private static int digitsPerLong[] = {0, 0,
4172 62, 39, 31, 27, 24, 22, 20, 19, 18, 18, 17, 17, 16, 16, 15, 15, 15, 14,
4173 14, 14, 14, 13, 13, 13, 13, 13, 13, 12, 12, 12, 12, 12, 12, 12, 12};
4174
4175 private static BigInteger longRadix[] = {null, null,
4176 valueOf(0x4000000000000000L), valueOf(0x383d9170b85ff80bL),
4177 valueOf(0x4000000000000000L), valueOf(0x6765c793fa10079dL),
4178 valueOf(0x41c21cb8e1000000L), valueOf(0x3642798750226111L),
4179 valueOf(0x1000000000000000L), valueOf(0x12bf307ae81ffd59L),
4180 valueOf( 0xde0b6b3a7640000L), valueOf(0x4d28cb56c33fa539L),
4181 valueOf(0x1eca170c00000000L), valueOf(0x780c7372621bd74dL),
4182 valueOf(0x1e39a5057d810000L), valueOf(0x5b27ac993df97701L),
4183 valueOf(0x1000000000000000L), valueOf(0x27b95e997e21d9f1L),
4184 valueOf(0x5da0e1e53c5c8000L), valueOf( 0xb16a458ef403f19L),
4185 valueOf(0x16bcc41e90000000L), valueOf(0x2d04b7fdd9c0ef49L),
4186 valueOf(0x5658597bcaa24000L), valueOf( 0x6feb266931a75b7L),
4187 valueOf( 0xc29e98000000000L), valueOf(0x14adf4b7320334b9L),
4188 valueOf(0x226ed36478bfa000L), valueOf(0x383d9170b85ff80bL),
4189 valueOf(0x5a3c23e39c000000L), valueOf( 0x4e900abb53e6b71L),
4190 valueOf( 0x7600ec618141000L), valueOf( 0xaee5720ee830681L),
4191 valueOf(0x1000000000000000L), valueOf(0x172588ad4f5f0981L),
4192 valueOf(0x211e44f7d02c1000L), valueOf(0x2ee56725f06e5c71L),
4193 valueOf(0x41c21cb8e1000000L)};
4194
4195 /*
4196 * These two arrays are the integer analogue of above.
4197 */
4198 private static int digitsPerInt[] = {0, 0, 30, 19, 15, 13, 11,
4199 11, 10, 9, 9, 8, 8, 8, 8, 7, 7, 7, 7, 7, 7, 7, 6, 6, 6, 6,
4200 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 5};
4201
4202 private static int intRadix[] = {0, 0,
4203 0x40000000, 0x4546b3db, 0x40000000, 0x48c27395, 0x159fd800,
4204 0x75db9c97, 0x40000000, 0x17179149, 0x3b9aca00, 0xcc6db61,
4205 0x19a10000, 0x309f1021, 0x57f6c100, 0xa2f1b6f, 0x10000000,
4206 0x18754571, 0x247dbc80, 0x3547667b, 0x4c4b4000, 0x6b5a6e1d,
4207 0x6c20a40, 0x8d2d931, 0xb640000, 0xe8d4a51, 0x1269ae40,
4208 0x17179149, 0x1cb91000, 0x23744899, 0x2b73a840, 0x34e63b41,
4209 0x40000000, 0x4cfa3cc1, 0x5c13d840, 0x6d91b519, 0x39aa400
4210 };
4211
4212 /**
4213 * These routines provide access to the two's complement representation
4214 * of BigIntegers.
4215 */
4216
4217 /**
4218 * Returns the length of the two's complement representation in ints,
4219 * including space for at least one sign bit.
4220 */
4221 private int intLength() {
4222 return (bitLength() >>> 5) + 1;
4223 }
4224
4225 /* Returns sign bit */
4226 private int signBit() {
4227 return signum < 0 ? 1 : 0;
4228 }
4229
4230 /* Returns an int of sign bits */
4231 private int signInt() {
4232 return signum < 0 ? -1 : 0;
4233 }
4234
4235 /**
4236 * Returns the specified int of the little-endian two's complement
4237 * representation (int 0 is the least significant). The int number can
4238 * be arbitrarily high (values are logically preceded by infinitely many
4239 * sign ints).
4240 */
4241 private int getInt(int n) {
4242 if (n < 0)
4243 return 0;
4244 if (n >= mag.length)
4245 return signInt();
4246
4247 int magInt = mag[mag.length-n-1];
4248
4249 return (signum >= 0 ? magInt :
4250 (n <= firstNonzeroIntNum() ? -magInt : ~magInt));
4251 }
4252
4253 /**
4254 * Returns the index of the int that contains the first nonzero int in the
4255 * little-endian binary representation of the magnitude (int 0 is the
4256 * least significant). If the magnitude is zero, return value is undefined.
4257 *
4258 * <p>Note: never used for a BigInteger with a magnitude of zero.
4259 * @see #getInt.
4260 */
4261 private int firstNonzeroIntNum() {
4262 int fn = firstNonzeroIntNumPlusTwo - 2;
4263 if (fn == -2) { // firstNonzeroIntNum not initialized yet
4264 // Search for the first nonzero int
4265 int i;
4266 int mlen = mag.length;
4267 for (i = mlen - 1; i >= 0 && mag[i] == 0; i--)
4268 ;
4269 fn = mlen - i - 1;
4270 firstNonzeroIntNumPlusTwo = fn + 2; // offset by two to initialize
4271 }
4272 return fn;
4273 }
4274
4275 /** use serialVersionUID from JDK 1.1. for interoperability */
4276 private static final long serialVersionUID = -8287574255936472291L;
4277
4278 /**
4279 * Serializable fields for BigInteger.
4280 *
4281 * @serialField signum int
4282 * signum of this BigInteger
4283 * @serialField magnitude byte[]
4284 * magnitude array of this BigInteger
4285 * @serialField bitCount int
4286 * appears in the serialized form for backward compatibility
4287 * @serialField bitLength int
4288 * appears in the serialized form for backward compatibility
4289 * @serialField firstNonzeroByteNum int
4290 * appears in the serialized form for backward compatibility
4291 * @serialField lowestSetBit int
4292 * appears in the serialized form for backward compatibility
4293 */
4294 private static final ObjectStreamField[] serialPersistentFields = {
4295 new ObjectStreamField("signum", Integer.TYPE),
4296 new ObjectStreamField("magnitude", byte[].class),
4297 new ObjectStreamField("bitCount", Integer.TYPE),
4298 new ObjectStreamField("bitLength", Integer.TYPE),
4299 new ObjectStreamField("firstNonzeroByteNum", Integer.TYPE),
4300 new ObjectStreamField("lowestSetBit", Integer.TYPE)
4301 };
4302
4303 /**
4304 * Reconstitute the {@code BigInteger} instance from a stream (that is,
4305 * deserialize it). The magnitude is read in as an array of bytes
4306 * for historical reasons, but it is converted to an array of ints
4307 * and the byte array is discarded.
4308 * Note:
4309 * The current convention is to initialize the cache fields, bitCountPlusOne,
4310 * bitLengthPlusOne and lowestSetBitPlusTwo, to 0 rather than some other
4311 * marker value. Therefore, no explicit action to set these fields needs to
4312 * be taken in readObject because those fields already have a 0 value by
4313 * default since defaultReadObject is not being used.
4314 */
4315 private void readObject(java.io.ObjectInputStream s)
4316 throws java.io.IOException, ClassNotFoundException {
4317 // prepare to read the alternate persistent fields
4318 ObjectInputStream.GetField fields = s.readFields();
4319
4320 // Read the alternate persistent fields that we care about
4321 int sign = fields.get("signum", -2);
4322 byte[] magnitude = (byte[])fields.get("magnitude", null);
4323
4324 // Validate signum
4325 if (sign < -1 || sign > 1) {
4326 String message = "BigInteger: Invalid signum value";
4327 if (fields.defaulted("signum"))
4328 message = "BigInteger: Signum not present in stream";
4329 throw new java.io.StreamCorruptedException(message);
4330 }
4331 int[] mag = stripLeadingZeroBytes(magnitude, 0, magnitude.length);
4332 if ((mag.length == 0) != (sign == 0)) {
4333 String message = "BigInteger: signum-magnitude mismatch";
4334 if (fields.defaulted("magnitude"))
4335 message = "BigInteger: Magnitude not present in stream";
4336 throw new java.io.StreamCorruptedException(message);
4337 }
4338
4339 // Commit final fields via Unsafe
4340 UnsafeHolder.putSign(this, sign);
4341
4342 // Calculate mag field from magnitude and discard magnitude
4343 UnsafeHolder.putMag(this, mag);
4344 if (mag.length >= MAX_MAG_LENGTH) {
4345 try {
4346 checkRange();
4347 } catch (ArithmeticException e) {
4348 throw new java.io.StreamCorruptedException("BigInteger: Out of the supported range");
4349 }
4350 }
4351 }
4352
4353 // Support for resetting final fields while deserializing
4354 private static class UnsafeHolder {
4355 private static final sun.misc.Unsafe unsafe;
4356 private static final long signumOffset;
4357 private static final long magOffset;
4358 static {
4359 try {
4360 unsafe = sun.misc.Unsafe.getUnsafe();
4361 signumOffset = unsafe.objectFieldOffset
4362 (BigInteger.class.getDeclaredField("signum"));
4363 magOffset = unsafe.objectFieldOffset
4364 (BigInteger.class.getDeclaredField("mag"));
4365 } catch (Exception ex) {
4366 throw new ExceptionInInitializerError(ex);
4367 }
4368 }
4369
4370 static void putSign(BigInteger bi, int sign) {
4371 unsafe.putInt(bi, signumOffset, sign);
4372 }
4373
4374 static void putMag(BigInteger bi, int[] magnitude) {
4375 unsafe.putObject(bi, magOffset, magnitude);
4376 }
4377 }
4378
4379 /**
4380 * Save the {@code BigInteger} instance to a stream. The magnitude of a
4381 * {@code BigInteger} is serialized as a byte array for historical reasons.
4382 * To maintain compatibility with older implementations, the integers
4383 * -1, -1, -2, and -2 are written as the values of the obsolete fields
4384 * {@code bitCount}, {@code bitLength}, {@code lowestSetBit}, and
4385 * {@code firstNonzeroByteNum}, respectively. These values are compatible
4386 * with older implementations, but will be ignored by current
4387 * implementations.
4388 */
4389 private void writeObject(ObjectOutputStream s) throws IOException {
4390 // set the values of the Serializable fields
4391 ObjectOutputStream.PutField fields = s.putFields();
4392 fields.put("signum", signum);
4393 fields.put("magnitude", magSerializedForm());
4394 // The values written for cached fields are compatible with older
4395 // versions, but are ignored in readObject so don't otherwise matter.
4396 fields.put("bitCount", -1);
4397 fields.put("bitLength", -1);
4398 fields.put("lowestSetBit", -2);
4399 fields.put("firstNonzeroByteNum", -2);
4400
4401 // save them
4402 s.writeFields();
4403 }
4404
4405 /**
4406 * Returns the mag array as an array of bytes.
4407 */
4408 private byte[] magSerializedForm() {
4409 int len = mag.length;
4410
4411 int bitLen = (len == 0 ? 0 : ((len - 1) << 5) + bitLengthForInt(mag[0]));
4412 int byteLen = (bitLen + 7) >>> 3;
4413 byte[] result = new byte[byteLen];
4414
4415 for (int i = byteLen - 1, bytesCopied = 4, intIndex = len - 1, nextInt = 0;
4416 i >= 0; i--) {
4417 if (bytesCopied == 4) {
4418 nextInt = mag[intIndex--];
4419 bytesCopied = 1;
4420 } else {
4421 nextInt >>>= 8;
4422 bytesCopied++;
4423 }
4424 result[i] = (byte)nextInt;
4425 }
4426 return result;
4427 }
4428
4429 /**
4430 * Converts this {@code BigInteger} to a {@code long}, checking
4431 * for lost information. If the value of this {@code BigInteger}
4432 * is out of the range of the {@code long} type, then an
4433 * {@code ArithmeticException} is thrown.
4434 *
4435 * @return this {@code BigInteger} converted to a {@code long}.
4436 * @throws ArithmeticException if the value of {@code this} will
4437 * not exactly fit in a {@code long}.
4438 * @see BigInteger#longValue
4439 * @since 1.8
4440 */
4441 public long longValueExact() {
4442 if (mag.length <= 2 && bitLength() <= 63)
4443 return longValue();
4444 else
4445 throw new ArithmeticException("BigInteger out of long range");
4446 }
4447
4448 /**
4449 * Converts this {@code BigInteger} to an {@code int}, checking
4450 * for lost information. If the value of this {@code BigInteger}
4451 * is out of the range of the {@code int} type, then an
4452 * {@code ArithmeticException} is thrown.
4453 *
4454 * @return this {@code BigInteger} converted to an {@code int}.
4455 * @throws ArithmeticException if the value of {@code this} will
4456 * not exactly fit in a {@code int}.
4457 * @see BigInteger#intValue
4458 * @since 1.8
4459 */
4460 public int intValueExact() {
4461 if (mag.length <= 1 && bitLength() <= 31)
4462 return intValue();
4463 else
4464 throw new ArithmeticException("BigInteger out of int range");
4465 }
4466
4467 /**
4468 * Converts this {@code BigInteger} to a {@code short}, checking
4469 * for lost information. If the value of this {@code BigInteger}
4470 * is out of the range of the {@code short} type, then an
4471 * {@code ArithmeticException} is thrown.
4472 *
4473 * @return this {@code BigInteger} converted to a {@code short}.
4474 * @throws ArithmeticException if the value of {@code this} will
4475 * not exactly fit in a {@code short}.
4476 * @see BigInteger#shortValue
4477 * @since 1.8
4478 */
4479 public short shortValueExact() {
4480 if (mag.length <= 1 && bitLength() <= 31) {
4481 int value = intValue();
4482 if (value >= Short.MIN_VALUE && value <= Short.MAX_VALUE)
4483 return shortValue();
4484 }
4485 throw new ArithmeticException("BigInteger out of short range");
4486 }
4487
4488 /**
4489 * Converts this {@code BigInteger} to a {@code byte}, checking
4490 * for lost information. If the value of this {@code BigInteger}
4491 * is out of the range of the {@code byte} type, then an
4492 * {@code ArithmeticException} is thrown.
4493 *
4494 * @return this {@code BigInteger} converted to a {@code byte}.
4495 * @throws ArithmeticException if the value of {@code this} will
4496 * not exactly fit in a {@code byte}.
4497 * @see BigInteger#byteValue
4498 * @since 1.8
4499 */
4500 public byte byteValueExact() {
4501 if (mag.length <= 1 && bitLength() <= 31) {
4502 int value = intValue();
4503 if (value >= Byte.MIN_VALUE && value <= Byte.MAX_VALUE)
4504 return byteValue();
4505 }
4506 throw new ArithmeticException("BigInteger out of byte range");
4507 }
4508 }