1 /*
2 * Copyright (c) 1996, 2013, 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 JDK1.1
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 * @serial
131 */
132 final int signum;
133
134 /**
135 * The magnitude of this BigInteger, in <i>big-endian</i> order: the
136 * zeroth element of this array is the most-significant int of the
137 * magnitude. The magnitude must be "minimal" in that the most-significant
138 * int ({@code mag[0]}) must be non-zero. This is necessary to
139 * ensure that there is exactly one representation for each BigInteger
140 * value. Note that this implies that the BigInteger zero has a
141 * zero-length mag array.
142 */
143 final int[] mag;
144
145 // These "redundant fields" are initialized with recognizable nonsense
146 // values, and cached the first time they are needed (or never, if they
147 // aren't needed).
148
149 /**
150 * One plus the bitCount of this BigInteger. Zeros means unitialized.
151 *
152 * @serial
153 * @see #bitCount
154 * @deprecated Deprecated since logical value is offset from stored
155 * value and correction factor is applied in accessor method.
156 */
157 @Deprecated
158 private int bitCount;
159
160 /**
161 * One plus the bitLength of this BigInteger. Zeros means unitialized.
162 * (either value is acceptable).
163 *
164 * @serial
165 * @see #bitLength()
166 * @deprecated Deprecated since logical value is offset from stored
167 * value and correction factor is applied in accessor method.
168 */
169 @Deprecated
170 private int bitLength;
171
172 /**
173 * Two plus the lowest set bit of this BigInteger, as returned by
174 * getLowestSetBit().
175 *
176 * @serial
177 * @see #getLowestSetBit
178 * @deprecated Deprecated since logical value is offset from stored
179 * value and correction factor is applied in accessor method.
180 */
181 @Deprecated
182 private int lowestSetBit;
183
184 /**
185 * Two plus the index of the lowest-order int in the magnitude of this
186 * BigInteger that contains a nonzero int, or -2 (either value is acceptable).
187 * The least significant int has int-number 0, the next int in order of
188 * increasing significance has int-number 1, and so forth.
189 * @deprecated Deprecated since logical value is offset from stored
190 * value and correction factor is applied in accessor method.
191 */
192 @Deprecated
193 private int firstNonzeroIntNum;
194
195 /**
196 * This mask is used to obtain the value of an int as if it were unsigned.
197 */
198 final static long LONG_MASK = 0xffffffffL;
199
200 /**
201 * This constant limits {@code mag.length} of BigIntegers to the supported
202 * range.
203 */
204 private static final int MAX_MAG_LENGTH = Integer.MAX_VALUE / Integer.SIZE + 1; // (1 << 26)
205
206 /**
207 * Bit lengths larger than this constant can cause overflow in searchLen
208 * calculation and in BitSieve.singleSearch method.
209 */
210 private static final int PRIME_SEARCH_BIT_LENGTH_LIMIT = 500000000;
211
212 /**
213 * The threshold value for using Karatsuba multiplication. If the number
214 * of ints in both mag arrays are greater than this number, then
215 * Karatsuba multiplication will be used. This value is found
216 * experimentally to work well.
217 */
218 private static final int KARATSUBA_THRESHOLD = 80;
219
220 /**
221 * The threshold value for using 3-way Toom-Cook multiplication.
222 * If the number of ints in each mag array is greater than the
223 * Karatsuba threshold, and the number of ints in at least one of
224 * the mag arrays is greater than this threshold, then Toom-Cook
225 * multiplication will be used.
226 */
227 private static final int TOOM_COOK_THRESHOLD = 240;
228
229 /**
230 * The threshold value for using Karatsuba squaring. If the number
231 * of ints in the number are larger than this value,
232 * Karatsuba squaring will be used. This value is found
233 * experimentally to work well.
234 */
235 private static final int KARATSUBA_SQUARE_THRESHOLD = 128;
236
237 /**
238 * The threshold value for using Toom-Cook squaring. If the number
239 * of ints in the number are larger than this value,
240 * Toom-Cook squaring will be used. This value is found
241 * experimentally to work well.
242 */
243 private static final int TOOM_COOK_SQUARE_THRESHOLD = 216;
244
245 /**
246 * The threshold value for using Burnikel-Ziegler division. If the number
247 * of ints in the divisor are larger than this value, Burnikel-Ziegler
248 * division may be used. This value is found experimentally to work well.
249 */
250 static final int BURNIKEL_ZIEGLER_THRESHOLD = 80;
251
252 /**
253 * The offset value for using Burnikel-Ziegler division. If the number
254 * of ints in the divisor exceeds the Burnikel-Ziegler threshold, and the
255 * number of ints in the dividend is greater than the number of ints in the
256 * divisor plus this value, Burnikel-Ziegler division will be used. This
257 * value is found experimentally to work well.
258 */
259 static final int BURNIKEL_ZIEGLER_OFFSET = 40;
260
261 /**
262 * The threshold value for using Schoenhage recursive base conversion. If
263 * the number of ints in the number are larger than this value,
264 * the Schoenhage algorithm will be used. In practice, it appears that the
265 * Schoenhage routine is faster for any threshold down to 2, and is
266 * relatively flat for thresholds between 2-25, so this choice may be
267 * varied within this range for very small effect.
268 */
269 private static final int SCHOENHAGE_BASE_CONVERSION_THRESHOLD = 20;
270
271 /**
272 * The threshold value for using squaring code to perform multiplication
273 * of a {@code BigInteger} instance by itself. If the number of ints in
274 * the number are larger than this value, {@code multiply(this)} will
275 * return {@code square()}.
276 */
277 private static final int MULTIPLY_SQUARE_THRESHOLD = 20;
278
279 // Constructors
280
281 /**
282 * Translates a byte array containing the two's-complement binary
283 * representation of a BigInteger into a BigInteger. The input array is
284 * assumed to be in <i>big-endian</i> byte-order: the most significant
285 * byte is in the zeroth element.
286 *
287 * @param val big-endian two's-complement binary representation of
288 * BigInteger.
289 * @throws NumberFormatException {@code val} is zero bytes long.
290 */
291 public BigInteger(byte[] val) {
292 if (val.length == 0)
293 throw new NumberFormatException("Zero length BigInteger");
294
295 if (val[0] < 0) {
296 mag = makePositive(val);
297 signum = -1;
298 } else {
299 mag = stripLeadingZeroBytes(val);
300 signum = (mag.length == 0 ? 0 : 1);
301 }
302 if (mag.length >= MAX_MAG_LENGTH) {
303 checkRange();
304 }
305 }
306
307 /**
308 * This private constructor translates an int array containing the
309 * two's-complement binary representation of a BigInteger into a
310 * BigInteger. The input array is assumed to be in <i>big-endian</i>
311 * int-order: the most significant int is in the zeroth element.
312 */
313 private BigInteger(int[] val) {
314 if (val.length == 0)
315 throw new NumberFormatException("Zero length BigInteger");
316
317 if (val[0] < 0) {
318 mag = makePositive(val);
319 signum = -1;
320 } else {
321 mag = trustedStripLeadingZeroInts(val);
322 signum = (mag.length == 0 ? 0 : 1);
323 }
324 if (mag.length >= MAX_MAG_LENGTH) {
325 checkRange();
326 }
327 }
328
329 /**
330 * Translates the sign-magnitude representation of a BigInteger into a
331 * BigInteger. The sign is represented as an integer signum value: -1 for
332 * negative, 0 for zero, or 1 for positive. The magnitude is a byte array
333 * in <i>big-endian</i> byte-order: the most significant byte is in the
334 * zeroth element. A zero-length magnitude array is permissible, and will
335 * result in a BigInteger value of 0, whether signum is -1, 0 or 1.
336 *
337 * @param signum signum of the number (-1 for negative, 0 for zero, 1
338 * for positive).
339 * @param magnitude big-endian binary representation of the magnitude of
340 * the number.
341 * @throws NumberFormatException {@code signum} is not one of the three
342 * legal values (-1, 0, and 1), or {@code signum} is 0 and
343 * {@code magnitude} contains one or more non-zero bytes.
344 */
345 public BigInteger(int signum, byte[] magnitude) {
346 this.mag = stripLeadingZeroBytes(magnitude);
347
348 if (signum < -1 || signum > 1)
349 throw(new NumberFormatException("Invalid signum value"));
350
351 if (this.mag.length == 0) {
352 this.signum = 0;
353 } else {
354 if (signum == 0)
355 throw(new NumberFormatException("signum-magnitude mismatch"));
356 this.signum = signum;
357 }
358 if (mag.length >= MAX_MAG_LENGTH) {
359 checkRange();
360 }
361 }
362
363 /**
364 * A constructor for internal use that translates the sign-magnitude
365 * representation of a BigInteger into a BigInteger. It checks the
366 * arguments and copies the magnitude so this constructor would be
367 * safe for external use.
368 */
369 private BigInteger(int signum, int[] magnitude) {
370 this.mag = stripLeadingZeroInts(magnitude);
371
372 if (signum < -1 || signum > 1)
373 throw(new NumberFormatException("Invalid signum value"));
374
375 if (this.mag.length == 0) {
376 this.signum = 0;
377 } else {
378 if (signum == 0)
379 throw(new NumberFormatException("signum-magnitude mismatch"));
380 this.signum = signum;
381 }
382 if (mag.length >= MAX_MAG_LENGTH) {
383 checkRange();
384 }
385 }
386
387 /**
388 * Translates the String representation of a BigInteger in the
389 * specified radix into a BigInteger. The String representation
390 * consists of an optional minus or plus sign followed by a
391 * sequence of one or more digits in the specified radix. The
392 * character-to-digit mapping is provided by {@code
393 * Character.digit}. The String may not contain any extraneous
394 * characters (whitespace, for example).
395 *
396 * @param val String representation of BigInteger.
397 * @param radix radix to be used in interpreting {@code val}.
398 * @throws NumberFormatException {@code val} is not a valid representation
399 * of a BigInteger in the specified radix, or {@code radix} is
400 * outside the range from {@link Character#MIN_RADIX} to
401 * {@link Character#MAX_RADIX}, inclusive.
402 * @see Character#digit
403 */
404 public BigInteger(String val, int radix) {
405 int cursor = 0, numDigits;
406 final int len = val.length();
407
408 if (radix < Character.MIN_RADIX || radix > Character.MAX_RADIX)
409 throw new NumberFormatException("Radix out of range");
410 if (len == 0)
411 throw new NumberFormatException("Zero length BigInteger");
412
413 // Check for at most one leading sign
414 int sign = 1;
415 int index1 = val.lastIndexOf('-');
416 int index2 = val.lastIndexOf('+');
417 if (index1 >= 0) {
418 if (index1 != 0 || index2 >= 0) {
419 throw new NumberFormatException("Illegal embedded sign character");
420 }
421 sign = -1;
422 cursor = 1;
423 } else if (index2 >= 0) {
424 if (index2 != 0) {
425 throw new NumberFormatException("Illegal embedded sign character");
426 }
427 cursor = 1;
428 }
429 if (cursor == len)
430 throw new NumberFormatException("Zero length BigInteger");
431
432 // Skip leading zeros and compute number of digits in magnitude
433 while (cursor < len &&
434 Character.digit(val.charAt(cursor), radix) == 0) {
435 cursor++;
436 }
437
438 if (cursor == len) {
439 signum = 0;
440 mag = ZERO.mag;
441 return;
442 }
443
444 numDigits = len - cursor;
445 signum = sign;
446
447 // Pre-allocate array of expected size. May be too large but can
448 // never be too small. Typically exact.
449 long numBits = ((numDigits * bitsPerDigit[radix]) >>> 10) + 1;
450 if (numBits + 31 >= (1L << 32)) {
451 reportOverflow();
452 }
453 int numWords = (int) (numBits + 31) >>> 5;
454 int[] magnitude = new int[numWords];
455
456 // Process first (potentially short) digit group
457 int firstGroupLen = numDigits % digitsPerInt[radix];
458 if (firstGroupLen == 0)
459 firstGroupLen = digitsPerInt[radix];
460 String group = val.substring(cursor, cursor += firstGroupLen);
461 magnitude[numWords - 1] = Integer.parseInt(group, radix);
462 if (magnitude[numWords - 1] < 0)
463 throw new NumberFormatException("Illegal digit");
464
465 // Process remaining digit groups
466 int superRadix = intRadix[radix];
467 int groupVal = 0;
468 while (cursor < len) {
469 group = val.substring(cursor, cursor += digitsPerInt[radix]);
470 groupVal = Integer.parseInt(group, radix);
471 if (groupVal < 0)
472 throw new NumberFormatException("Illegal digit");
473 destructiveMulAdd(magnitude, superRadix, groupVal);
474 }
475 // Required for cases where the array was overallocated.
476 mag = trustedStripLeadingZeroInts(magnitude);
477 if (mag.length >= MAX_MAG_LENGTH) {
478 checkRange();
479 }
480 }
481
482 /*
483 * Constructs a new BigInteger using a char array with radix=10.
484 * Sign is precalculated outside and not allowed in the val.
485 */
486 BigInteger(char[] val, int sign, int len) {
487 int cursor = 0, numDigits;
488
489 // Skip leading zeros and compute number of digits in magnitude
490 while (cursor < len && Character.digit(val[cursor], 10) == 0) {
491 cursor++;
492 }
493 if (cursor == len) {
494 signum = 0;
495 mag = ZERO.mag;
496 return;
497 }
498
499 numDigits = len - cursor;
500 signum = sign;
501 // Pre-allocate array of expected size
502 int numWords;
503 if (len < 10) {
504 numWords = 1;
505 } else {
506 long numBits = ((numDigits * bitsPerDigit[10]) >>> 10) + 1;
507 if (numBits + 31 >= (1L << 32)) {
508 reportOverflow();
509 }
510 numWords = (int) (numBits + 31) >>> 5;
511 }
512 int[] magnitude = new int[numWords];
513
514 // Process first (potentially short) digit group
515 int firstGroupLen = numDigits % digitsPerInt[10];
516 if (firstGroupLen == 0)
517 firstGroupLen = digitsPerInt[10];
518 magnitude[numWords - 1] = parseInt(val, cursor, cursor += firstGroupLen);
519
520 // Process remaining digit groups
521 while (cursor < len) {
522 int groupVal = parseInt(val, cursor, cursor += digitsPerInt[10]);
523 destructiveMulAdd(magnitude, intRadix[10], groupVal);
524 }
525 mag = trustedStripLeadingZeroInts(magnitude);
526 if (mag.length >= MAX_MAG_LENGTH) {
527 checkRange();
528 }
529 }
530
531 // Create an integer with the digits between the two indexes
532 // Assumes start < end. The result may be negative, but it
533 // is to be treated as an unsigned value.
534 private int parseInt(char[] source, int start, int end) {
535 int result = Character.digit(source[start++], 10);
536 if (result == -1)
537 throw new NumberFormatException(new String(source));
538
539 for (int index = start; index < end; index++) {
540 int nextVal = Character.digit(source[index], 10);
541 if (nextVal == -1)
542 throw new NumberFormatException(new String(source));
543 result = 10*result + nextVal;
544 }
545
546 return result;
547 }
548
549 // bitsPerDigit in the given radix times 1024
550 // Rounded up to avoid underallocation.
551 private static long bitsPerDigit[] = { 0, 0,
552 1024, 1624, 2048, 2378, 2648, 2875, 3072, 3247, 3402, 3543, 3672,
553 3790, 3899, 4001, 4096, 4186, 4271, 4350, 4426, 4498, 4567, 4633,
554 4696, 4756, 4814, 4870, 4923, 4975, 5025, 5074, 5120, 5166, 5210,
555 5253, 5295};
556
557 // Multiply x array times word y in place, and add word z
558 private static void destructiveMulAdd(int[] x, int y, int z) {
559 // Perform the multiplication word by word
560 long ylong = y & LONG_MASK;
561 long zlong = z & LONG_MASK;
562 int len = x.length;
563
564 long product = 0;
565 long carry = 0;
566 for (int i = len-1; i >= 0; i--) {
567 product = ylong * (x[i] & LONG_MASK) + carry;
568 x[i] = (int)product;
569 carry = product >>> 32;
570 }
571
572 // Perform the addition
573 long sum = (x[len-1] & LONG_MASK) + zlong;
574 x[len-1] = (int)sum;
575 carry = sum >>> 32;
576 for (int i = len-2; i >= 0; i--) {
577 sum = (x[i] & LONG_MASK) + carry;
578 x[i] = (int)sum;
579 carry = sum >>> 32;
580 }
581 }
582
583 /**
584 * Translates the decimal String representation of a BigInteger into a
585 * BigInteger. The String representation consists of an optional minus
586 * sign followed by a sequence of one or more decimal digits. The
587 * character-to-digit mapping is provided by {@code Character.digit}.
588 * The String may not contain any extraneous characters (whitespace, for
589 * example).
590 *
591 * @param val decimal String representation of BigInteger.
592 * @throws NumberFormatException {@code val} is not a valid representation
593 * of a BigInteger.
594 * @see Character#digit
595 */
596 public BigInteger(String val) {
597 this(val, 10);
598 }
599
600 /**
601 * Constructs a randomly generated BigInteger, uniformly distributed over
602 * the range 0 to (2<sup>{@code numBits}</sup> - 1), inclusive.
603 * The uniformity of the distribution assumes that a fair source of random
604 * bits is provided in {@code rnd}. Note that this constructor always
605 * constructs a non-negative BigInteger.
606 *
607 * @param numBits maximum bitLength of the new BigInteger.
608 * @param rnd source of randomness to be used in computing the new
609 * BigInteger.
610 * @throws IllegalArgumentException {@code numBits} is negative.
611 * @see #bitLength()
612 */
613 public BigInteger(int numBits, Random rnd) {
614 this(1, randomBits(numBits, rnd));
615 }
616
617 private static byte[] randomBits(int numBits, Random rnd) {
618 if (numBits < 0)
619 throw new IllegalArgumentException("numBits must be non-negative");
620 int numBytes = (int)(((long)numBits+7)/8); // avoid overflow
621 byte[] randomBits = new byte[numBytes];
622
623 // Generate random bytes and mask out any excess bits
624 if (numBytes > 0) {
625 rnd.nextBytes(randomBits);
626 int excessBits = 8*numBytes - numBits;
627 randomBits[0] &= (1 << (8-excessBits)) - 1;
628 }
629 return randomBits;
630 }
631
632 /**
633 * Constructs a randomly generated positive BigInteger that is probably
634 * prime, with the specified bitLength.
635 *
636 * <p>It is recommended that the {@link #probablePrime probablePrime}
637 * method be used in preference to this constructor unless there
638 * is a compelling need to specify a certainty.
639 *
640 * @param bitLength bitLength of the returned BigInteger.
641 * @param certainty a measure of the uncertainty that the caller is
642 * willing to tolerate. The probability that the new BigInteger
643 * represents a prime number will exceed
644 * (1 - 1/2<sup>{@code certainty}</sup>). The execution time of
645 * this constructor is proportional to the value of this parameter.
646 * @param rnd source of random bits used to select candidates to be
647 * tested for primality.
648 * @throws ArithmeticException {@code bitLength < 2} or {@code bitLength} is too large.
649 * @see #bitLength()
650 */
651 public BigInteger(int bitLength, int certainty, Random rnd) {
652 BigInteger prime;
653
654 if (bitLength < 2)
655 throw new ArithmeticException("bitLength < 2");
656 prime = (bitLength < SMALL_PRIME_THRESHOLD
657 ? smallPrime(bitLength, certainty, rnd)
658 : largePrime(bitLength, certainty, rnd));
659 signum = 1;
660 mag = prime.mag;
661 }
662
663 // Minimum size in bits that the requested prime number has
664 // before we use the large prime number generating algorithms.
665 // The cutoff of 95 was chosen empirically for best performance.
666 private static final int SMALL_PRIME_THRESHOLD = 95;
667
668 // Certainty required to meet the spec of probablePrime
669 private static final int DEFAULT_PRIME_CERTAINTY = 100;
670
671 /**
672 * Returns a positive BigInteger that is probably prime, with the
673 * specified bitLength. The probability that a BigInteger returned
674 * by this method is composite does not exceed 2<sup>-100</sup>.
675 *
676 * @param bitLength bitLength of the returned BigInteger.
677 * @param rnd source of random bits used to select candidates to be
678 * tested for primality.
679 * @return a BigInteger of {@code bitLength} bits that is probably prime
680 * @throws ArithmeticException {@code bitLength < 2} or {@code bitLength} is too large.
681 * @see #bitLength()
682 * @since 1.4
683 */
684 public static BigInteger probablePrime(int bitLength, Random rnd) {
685 if (bitLength < 2)
686 throw new ArithmeticException("bitLength < 2");
687
688 return (bitLength < SMALL_PRIME_THRESHOLD ?
689 smallPrime(bitLength, DEFAULT_PRIME_CERTAINTY, rnd) :
690 largePrime(bitLength, DEFAULT_PRIME_CERTAINTY, rnd));
691 }
692
693 /**
694 * Find a random number of the specified bitLength that is probably prime.
695 * This method is used for smaller primes, its performance degrades on
696 * larger bitlengths.
697 *
698 * This method assumes bitLength > 1.
699 */
700 private static BigInteger smallPrime(int bitLength, int certainty, Random rnd) {
701 int magLen = (bitLength + 31) >>> 5;
702 int temp[] = new int[magLen];
703 int highBit = 1 << ((bitLength+31) & 0x1f); // High bit of high int
704 int highMask = (highBit << 1) - 1; // Bits to keep in high int
705
706 while (true) {
707 // Construct a candidate
708 for (int i=0; i < magLen; i++)
709 temp[i] = rnd.nextInt();
710 temp[0] = (temp[0] & highMask) | highBit; // Ensure exact length
711 if (bitLength > 2)
712 temp[magLen-1] |= 1; // Make odd if bitlen > 2
713
714 BigInteger p = new BigInteger(temp, 1);
715
716 // Do cheap "pre-test" if applicable
717 if (bitLength > 6) {
718 long r = p.remainder(SMALL_PRIME_PRODUCT).longValue();
719 if ((r%3==0) || (r%5==0) || (r%7==0) || (r%11==0) ||
720 (r%13==0) || (r%17==0) || (r%19==0) || (r%23==0) ||
721 (r%29==0) || (r%31==0) || (r%37==0) || (r%41==0))
722 continue; // Candidate is composite; try another
723 }
724
725 // All candidates of bitLength 2 and 3 are prime by this point
726 if (bitLength < 4)
727 return p;
728
729 // Do expensive test if we survive pre-test (or it's inapplicable)
730 if (p.primeToCertainty(certainty, rnd))
731 return p;
732 }
733 }
734
735 private static final BigInteger SMALL_PRIME_PRODUCT
736 = valueOf(3L*5*7*11*13*17*19*23*29*31*37*41);
737
738 /**
739 * Find a random number of the specified bitLength that is probably prime.
740 * This method is more appropriate for larger bitlengths since it uses
741 * a sieve to eliminate most composites before using a more expensive
742 * test.
743 */
744 private static BigInteger largePrime(int bitLength, int certainty, Random rnd) {
745 BigInteger p;
746 p = new BigInteger(bitLength, rnd).setBit(bitLength-1);
747 p.mag[p.mag.length-1] &= 0xfffffffe;
748
749 // Use a sieve length likely to contain the next prime number
750 int searchLen = getPrimeSearchLen(bitLength);
751 BitSieve searchSieve = new BitSieve(p, searchLen);
752 BigInteger candidate = searchSieve.retrieve(p, certainty, rnd);
753
754 while ((candidate == null) || (candidate.bitLength() != bitLength)) {
755 p = p.add(BigInteger.valueOf(2*searchLen));
756 if (p.bitLength() != bitLength)
757 p = new BigInteger(bitLength, rnd).setBit(bitLength-1);
758 p.mag[p.mag.length-1] &= 0xfffffffe;
759 searchSieve = new BitSieve(p, searchLen);
760 candidate = searchSieve.retrieve(p, certainty, rnd);
761 }
762 return candidate;
763 }
764
765 /**
766 * Returns the first integer greater than this {@code BigInteger} that
767 * is probably prime. The probability that the number returned by this
768 * method is composite does not exceed 2<sup>-100</sup>. This method will
769 * never skip over a prime when searching: if it returns {@code p}, there
770 * is no prime {@code q} such that {@code this < q < p}.
771 *
772 * @return the first integer greater than this {@code BigInteger} that
773 * is probably prime.
774 * @throws ArithmeticException {@code this < 0} or {@code this} is too large.
775 * @since 1.5
776 */
777 public BigInteger nextProbablePrime() {
778 if (this.signum < 0)
779 throw new ArithmeticException("start < 0: " + this);
780
781 // Handle trivial cases
782 if ((this.signum == 0) || this.equals(ONE))
783 return TWO;
784
785 BigInteger result = this.add(ONE);
786
787 // Fastpath for small numbers
788 if (result.bitLength() < SMALL_PRIME_THRESHOLD) {
789
790 // Ensure an odd number
791 if (!result.testBit(0))
792 result = result.add(ONE);
793
794 while (true) {
795 // Do cheap "pre-test" if applicable
796 if (result.bitLength() > 6) {
797 long r = result.remainder(SMALL_PRIME_PRODUCT).longValue();
798 if ((r%3==0) || (r%5==0) || (r%7==0) || (r%11==0) ||
799 (r%13==0) || (r%17==0) || (r%19==0) || (r%23==0) ||
800 (r%29==0) || (r%31==0) || (r%37==0) || (r%41==0)) {
801 result = result.add(TWO);
802 continue; // Candidate is composite; try another
803 }
804 }
805
806 // All candidates of bitLength 2 and 3 are prime by this point
807 if (result.bitLength() < 4)
808 return result;
809
810 // The expensive test
811 if (result.primeToCertainty(DEFAULT_PRIME_CERTAINTY, null))
812 return result;
813
814 result = result.add(TWO);
815 }
816 }
817
818 // Start at previous even number
819 if (result.testBit(0))
820 result = result.subtract(ONE);
821
822 // Looking for the next large prime
823 int searchLen = getPrimeSearchLen(result.bitLength());
824
825 while (true) {
826 BitSieve searchSieve = new BitSieve(result, searchLen);
827 BigInteger candidate = searchSieve.retrieve(result,
828 DEFAULT_PRIME_CERTAINTY, null);
829 if (candidate != null)
830 return candidate;
831 result = result.add(BigInteger.valueOf(2 * searchLen));
832 }
833 }
834
835 private static int getPrimeSearchLen(int bitLength) {
836 if (bitLength > PRIME_SEARCH_BIT_LENGTH_LIMIT + 1) {
837 throw new ArithmeticException("Prime search implementation restriction on bitLength");
838 }
839 return bitLength / 20 * 64;
840 }
841
842 /**
843 * Returns {@code true} if this BigInteger is probably prime,
844 * {@code false} if it's definitely composite.
845 *
846 * This method assumes bitLength > 2.
847 *
848 * @param certainty a measure of the uncertainty that the caller is
849 * willing to tolerate: if the call returns {@code true}
850 * the probability that this BigInteger is prime exceeds
851 * {@code (1 - 1/2<sup>certainty</sup>)}. The execution time of
852 * this method is proportional to the value of this parameter.
853 * @return {@code true} if this BigInteger is probably prime,
854 * {@code false} if it's definitely composite.
855 */
856 boolean primeToCertainty(int certainty, Random random) {
857 int rounds = 0;
858 int n = (Math.min(certainty, Integer.MAX_VALUE-1)+1)/2;
859
860 // The relationship between the certainty and the number of rounds
861 // we perform is given in the draft standard ANSI X9.80, "PRIME
862 // NUMBER GENERATION, PRIMALITY TESTING, AND PRIMALITY CERTIFICATES".
863 int sizeInBits = this.bitLength();
864 if (sizeInBits < 100) {
865 rounds = 50;
866 rounds = n < rounds ? n : rounds;
867 return passesMillerRabin(rounds, random);
868 }
869
870 if (sizeInBits < 256) {
871 rounds = 27;
872 } else if (sizeInBits < 512) {
873 rounds = 15;
874 } else if (sizeInBits < 768) {
875 rounds = 8;
876 } else if (sizeInBits < 1024) {
877 rounds = 4;
878 } else {
879 rounds = 2;
880 }
881 rounds = n < rounds ? n : rounds;
882
883 return passesMillerRabin(rounds, random) && passesLucasLehmer();
884 }
885
886 /**
887 * Returns true iff this BigInteger is a Lucas-Lehmer probable prime.
888 *
889 * The following assumptions are made:
890 * This BigInteger is a positive, odd number.
891 */
892 private boolean passesLucasLehmer() {
893 BigInteger thisPlusOne = this.add(ONE);
894
895 // Step 1
896 int d = 5;
897 while (jacobiSymbol(d, this) != -1) {
898 // 5, -7, 9, -11, ...
899 d = (d < 0) ? Math.abs(d)+2 : -(d+2);
900 }
901
902 // Step 2
903 BigInteger u = lucasLehmerSequence(d, thisPlusOne, this);
904
905 // Step 3
906 return u.mod(this).equals(ZERO);
907 }
908
909 /**
910 * Computes Jacobi(p,n).
911 * Assumes n positive, odd, n>=3.
912 */
913 private static int jacobiSymbol(int p, BigInteger n) {
914 if (p == 0)
915 return 0;
916
917 // Algorithm and comments adapted from Colin Plumb's C library.
918 int j = 1;
919 int u = n.mag[n.mag.length-1];
920
921 // Make p positive
922 if (p < 0) {
923 p = -p;
924 int n8 = u & 7;
925 if ((n8 == 3) || (n8 == 7))
926 j = -j; // 3 (011) or 7 (111) mod 8
927 }
928
929 // Get rid of factors of 2 in p
930 while ((p & 3) == 0)
931 p >>= 2;
932 if ((p & 1) == 0) {
933 p >>= 1;
934 if (((u ^ (u>>1)) & 2) != 0)
935 j = -j; // 3 (011) or 5 (101) mod 8
936 }
937 if (p == 1)
938 return j;
939 // Then, apply quadratic reciprocity
940 if ((p & u & 2) != 0) // p = u = 3 (mod 4)?
941 j = -j;
942 // And reduce u mod p
943 u = n.mod(BigInteger.valueOf(p)).intValue();
944
945 // Now compute Jacobi(u,p), u < p
946 while (u != 0) {
947 while ((u & 3) == 0)
948 u >>= 2;
949 if ((u & 1) == 0) {
950 u >>= 1;
951 if (((p ^ (p>>1)) & 2) != 0)
952 j = -j; // 3 (011) or 5 (101) mod 8
953 }
954 if (u == 1)
955 return j;
956 // Now both u and p are odd, so use quadratic reciprocity
957 assert (u < p);
958 int t = u; u = p; p = t;
959 if ((u & p & 2) != 0) // u = p = 3 (mod 4)?
960 j = -j;
961 // Now u >= p, so it can be reduced
962 u %= p;
963 }
964 return 0;
965 }
966
967 private static BigInteger lucasLehmerSequence(int z, BigInteger k, BigInteger n) {
968 BigInteger d = BigInteger.valueOf(z);
969 BigInteger u = ONE; BigInteger u2;
970 BigInteger v = ONE; BigInteger v2;
971
972 for (int i=k.bitLength()-2; i >= 0; i--) {
973 u2 = u.multiply(v).mod(n);
974
975 v2 = v.square().add(d.multiply(u.square())).mod(n);
976 if (v2.testBit(0))
977 v2 = v2.subtract(n);
978
979 v2 = v2.shiftRight(1);
980
981 u = u2; v = v2;
982 if (k.testBit(i)) {
983 u2 = u.add(v).mod(n);
984 if (u2.testBit(0))
985 u2 = u2.subtract(n);
986
987 u2 = u2.shiftRight(1);
988 v2 = v.add(d.multiply(u)).mod(n);
989 if (v2.testBit(0))
990 v2 = v2.subtract(n);
991 v2 = v2.shiftRight(1);
992
993 u = u2; v = v2;
994 }
995 }
996 return u;
997 }
998
999 /**
1000 * Returns true iff this BigInteger passes the specified number of
1001 * Miller-Rabin tests. This test is taken from the DSA spec (NIST FIPS
1002 * 186-2).
1003 *
1004 * The following assumptions are made:
1005 * This BigInteger is a positive, odd number greater than 2.
1006 * iterations<=50.
1007 */
1008 private boolean passesMillerRabin(int iterations, Random rnd) {
1009 // Find a and m such that m is odd and this == 1 + 2**a * m
1010 BigInteger thisMinusOne = this.subtract(ONE);
1011 BigInteger m = thisMinusOne;
1012 int a = m.getLowestSetBit();
1013 m = m.shiftRight(a);
1014
1015 // Do the tests
1016 if (rnd == null) {
1017 rnd = ThreadLocalRandom.current();
1018 }
1019 for (int i=0; i < iterations; i++) {
1020 // Generate a uniform random on (1, this)
1021 BigInteger b;
1022 do {
1023 b = new BigInteger(this.bitLength(), rnd);
1024 } while (b.compareTo(ONE) <= 0 || b.compareTo(this) >= 0);
1025
1026 int j = 0;
1027 BigInteger z = b.modPow(m, this);
1028 while (!((j == 0 && z.equals(ONE)) || z.equals(thisMinusOne))) {
1029 if (j > 0 && z.equals(ONE) || ++j == a)
1030 return false;
1031 z = z.modPow(TWO, this);
1032 }
1033 }
1034 return true;
1035 }
1036
1037 /**
1038 * This internal constructor differs from its public cousin
1039 * with the arguments reversed in two ways: it assumes that its
1040 * arguments are correct, and it doesn't copy the magnitude array.
1041 */
1042 BigInteger(int[] magnitude, int signum) {
1043 this.signum = (magnitude.length == 0 ? 0 : signum);
1044 this.mag = magnitude;
1045 if (mag.length >= MAX_MAG_LENGTH) {
1046 checkRange();
1047 }
1048 }
1049
1050 /**
1051 * This private constructor is for internal use and assumes that its
1052 * arguments are correct.
1053 */
1054 private BigInteger(byte[] magnitude, int signum) {
1055 this.signum = (magnitude.length == 0 ? 0 : signum);
1056 this.mag = stripLeadingZeroBytes(magnitude);
1057 if (mag.length >= MAX_MAG_LENGTH) {
1058 checkRange();
1059 }
1060 }
1061
1062 /**
1063 * Throws an {@code ArithmeticException} if the {@code BigInteger} would be
1064 * out of the supported range.
1065 *
1066 * @throws ArithmeticException if {@code this} exceeds the supported range.
1067 */
1068 private void checkRange() {
1069 if (mag.length > MAX_MAG_LENGTH || mag.length == MAX_MAG_LENGTH && mag[0] < 0) {
1070 reportOverflow();
1071 }
1072 }
1073
1074 private static void reportOverflow() {
1075 throw new ArithmeticException("BigInteger would overflow supported range");
1076 }
1077
1078 //Static Factory Methods
1079
1080 /**
1081 * Returns a BigInteger whose value is equal to that of the
1082 * specified {@code long}. This "static factory method" is
1083 * provided in preference to a ({@code long}) constructor
1084 * because it allows for reuse of frequently used BigIntegers.
1085 *
1086 * @param val value of the BigInteger to return.
1087 * @return a BigInteger with the specified value.
1088 */
1089 public static BigInteger valueOf(long val) {
1090 // If -MAX_CONSTANT < val < MAX_CONSTANT, return stashed constant
1091 if (val == 0)
1092 return ZERO;
1093 if (val > 0 && val <= MAX_CONSTANT)
1094 return posConst[(int) val];
1095 else if (val < 0 && val >= -MAX_CONSTANT)
1096 return negConst[(int) -val];
1097
1098 return new BigInteger(val);
1099 }
1100
1101 /**
1102 * Constructs a BigInteger with the specified value, which may not be zero.
1103 */
1104 private BigInteger(long val) {
1105 if (val < 0) {
1106 val = -val;
1107 signum = -1;
1108 } else {
1109 signum = 1;
1110 }
1111
1112 int highWord = (int)(val >>> 32);
1113 if (highWord == 0) {
1114 mag = new int[1];
1115 mag[0] = (int)val;
1116 } else {
1117 mag = new int[2];
1118 mag[0] = highWord;
1119 mag[1] = (int)val;
1120 }
1121 }
1122
1123 /**
1124 * Returns a BigInteger with the given two's complement representation.
1125 * Assumes that the input array will not be modified (the returned
1126 * BigInteger will reference the input array if feasible).
1127 */
1128 private static BigInteger valueOf(int val[]) {
1129 return (val[0] > 0 ? new BigInteger(val, 1) : new BigInteger(val));
1130 }
1131
1132 // Constants
1133
1134 /**
1135 * Initialize static constant array when class is loaded.
1136 */
1137 private final static int MAX_CONSTANT = 16;
1138 private static BigInteger posConst[] = new BigInteger[MAX_CONSTANT+1];
1139 private static BigInteger negConst[] = new BigInteger[MAX_CONSTANT+1];
1140
1141 /**
1142 * The cache of powers of each radix. This allows us to not have to
1143 * recalculate powers of radix^(2^n) more than once. This speeds
1144 * Schoenhage recursive base conversion significantly.
1145 */
1146 private static volatile BigInteger[][] powerCache;
1147
1148 /** The cache of logarithms of radices for base conversion. */
1149 private static final double[] logCache;
1150
1151 /** The natural log of 2. This is used in computing cache indices. */
1152 private static final double LOG_TWO = Math.log(2.0);
1153
1154 static {
1155 for (int i = 1; i <= MAX_CONSTANT; i++) {
1156 int[] magnitude = new int[1];
1157 magnitude[0] = i;
1158 posConst[i] = new BigInteger(magnitude, 1);
1159 negConst[i] = new BigInteger(magnitude, -1);
1160 }
1161
1162 /*
1163 * Initialize the cache of radix^(2^x) values used for base conversion
1164 * with just the very first value. Additional values will be created
1165 * on demand.
1166 */
1167 powerCache = new BigInteger[Character.MAX_RADIX+1][];
1168 logCache = new double[Character.MAX_RADIX+1];
1169
1170 for (int i=Character.MIN_RADIX; i <= Character.MAX_RADIX; i++) {
1171 powerCache[i] = new BigInteger[] { BigInteger.valueOf(i) };
1172 logCache[i] = Math.log(i);
1173 }
1174 }
1175
1176 /**
1177 * The BigInteger constant zero.
1178 *
1179 * @since 1.2
1180 */
1181 public static final BigInteger ZERO = new BigInteger(new int[0], 0);
1182
1183 /**
1184 * The BigInteger constant one.
1185 *
1186 * @since 1.2
1187 */
1188 public static final BigInteger ONE = valueOf(1);
1189
1190 /**
1191 * The BigInteger constant two. (Not exported.)
1192 */
1193 private static final BigInteger TWO = valueOf(2);
1194
1195 /**
1196 * The BigInteger constant -1. (Not exported.)
1197 */
1198 private static final BigInteger NEGATIVE_ONE = valueOf(-1);
1199
1200 /**
1201 * The BigInteger constant ten.
1202 *
1203 * @since 1.5
1204 */
1205 public static final BigInteger TEN = valueOf(10);
1206
1207 // Arithmetic Operations
1208
1209 /**
1210 * Returns a BigInteger whose value is {@code (this + val)}.
1211 *
1212 * @param val value to be added to this BigInteger.
1213 * @return {@code this + val}
1214 */
1215 public BigInteger add(BigInteger val) {
1216 if (val.signum == 0)
1217 return this;
1218 if (signum == 0)
1219 return val;
1220 if (val.signum == signum)
1221 return new BigInteger(add(mag, val.mag), signum);
1222
1223 int cmp = compareMagnitude(val);
1224 if (cmp == 0)
1225 return ZERO;
1226 int[] resultMag = (cmp > 0 ? subtract(mag, val.mag)
1227 : subtract(val.mag, mag));
1228 resultMag = trustedStripLeadingZeroInts(resultMag);
1229
1230 return new BigInteger(resultMag, cmp == signum ? 1 : -1);
1231 }
1232
1233 /**
1234 * Package private methods used by BigDecimal code to add a BigInteger
1235 * with a long. Assumes val is not equal to INFLATED.
1236 */
1237 BigInteger add(long val) {
1238 if (val == 0)
1239 return this;
1240 if (signum == 0)
1241 return valueOf(val);
1242 if (Long.signum(val) == signum)
1243 return new BigInteger(add(mag, Math.abs(val)), signum);
1244 int cmp = compareMagnitude(val);
1245 if (cmp == 0)
1246 return ZERO;
1247 int[] resultMag = (cmp > 0 ? subtract(mag, Math.abs(val)) : subtract(Math.abs(val), mag));
1248 resultMag = trustedStripLeadingZeroInts(resultMag);
1249 return new BigInteger(resultMag, cmp == signum ? 1 : -1);
1250 }
1251
1252 /**
1253 * Adds the contents of the int array x and long value val. This
1254 * method allocates a new int array to hold the answer and returns
1255 * a reference to that array. Assumes x.length > 0 and val is
1256 * non-negative
1257 */
1258 private static int[] add(int[] x, long val) {
1259 int[] y;
1260 long sum = 0;
1261 int xIndex = x.length;
1262 int[] result;
1263 int highWord = (int)(val >>> 32);
1264 if (highWord == 0) {
1265 result = new int[xIndex];
1266 sum = (x[--xIndex] & LONG_MASK) + val;
1267 result[xIndex] = (int)sum;
1268 } else {
1269 if (xIndex == 1) {
1270 result = new int[2];
1271 sum = val + (x[0] & LONG_MASK);
1272 result[1] = (int)sum;
1273 result[0] = (int)(sum >>> 32);
1274 return result;
1275 } else {
1276 result = new int[xIndex];
1277 sum = (x[--xIndex] & LONG_MASK) + (val & LONG_MASK);
1278 result[xIndex] = (int)sum;
1279 sum = (x[--xIndex] & LONG_MASK) + (highWord & LONG_MASK) + (sum >>> 32);
1280 result[xIndex] = (int)sum;
1281 }
1282 }
1283 // Copy remainder of longer number while carry propagation is required
1284 boolean carry = (sum >>> 32 != 0);
1285 while (xIndex > 0 && carry)
1286 carry = ((result[--xIndex] = x[xIndex] + 1) == 0);
1287 // Copy remainder of longer number
1288 while (xIndex > 0)
1289 result[--xIndex] = x[xIndex];
1290 // Grow result if necessary
1291 if (carry) {
1292 int bigger[] = new int[result.length + 1];
1293 System.arraycopy(result, 0, bigger, 1, result.length);
1294 bigger[0] = 0x01;
1295 return bigger;
1296 }
1297 return result;
1298 }
1299
1300 /**
1301 * Adds the contents of the int arrays x and y. This method allocates
1302 * a new int array to hold the answer and returns a reference to that
1303 * array.
1304 */
1305 private static int[] add(int[] x, int[] y) {
1306 // If x is shorter, swap the two arrays
1307 if (x.length < y.length) {
1308 int[] tmp = x;
1309 x = y;
1310 y = tmp;
1311 }
1312
1313 int xIndex = x.length;
1314 int yIndex = y.length;
1315 int result[] = new int[xIndex];
1316 long sum = 0;
1317 if (yIndex == 1) {
1318 sum = (x[--xIndex] & LONG_MASK) + (y[0] & LONG_MASK) ;
1319 result[xIndex] = (int)sum;
1320 } else {
1321 // Add common parts of both numbers
1322 while (yIndex > 0) {
1323 sum = (x[--xIndex] & LONG_MASK) +
1324 (y[--yIndex] & LONG_MASK) + (sum >>> 32);
1325 result[xIndex] = (int)sum;
1326 }
1327 }
1328 // Copy remainder of longer number while carry propagation is required
1329 boolean carry = (sum >>> 32 != 0);
1330 while (xIndex > 0 && carry)
1331 carry = ((result[--xIndex] = x[xIndex] + 1) == 0);
1332
1333 // Copy remainder of longer number
1334 while (xIndex > 0)
1335 result[--xIndex] = x[xIndex];
1336
1337 // Grow result if necessary
1338 if (carry) {
1339 int bigger[] = new int[result.length + 1];
1340 System.arraycopy(result, 0, bigger, 1, result.length);
1341 bigger[0] = 0x01;
1342 return bigger;
1343 }
1344 return result;
1345 }
1346
1347 private static int[] subtract(long val, int[] little) {
1348 int highWord = (int)(val >>> 32);
1349 if (highWord == 0) {
1350 int result[] = new int[1];
1351 result[0] = (int)(val - (little[0] & LONG_MASK));
1352 return result;
1353 } else {
1354 int result[] = new int[2];
1355 if (little.length == 1) {
1356 long difference = ((int)val & LONG_MASK) - (little[0] & LONG_MASK);
1357 result[1] = (int)difference;
1358 // Subtract remainder of longer number while borrow propagates
1359 boolean borrow = (difference >> 32 != 0);
1360 if (borrow) {
1361 result[0] = highWord - 1;
1362 } else { // Copy remainder of longer number
1363 result[0] = highWord;
1364 }
1365 return result;
1366 } else { // little.length == 2
1367 long difference = ((int)val & LONG_MASK) - (little[1] & LONG_MASK);
1368 result[1] = (int)difference;
1369 difference = (highWord & LONG_MASK) - (little[0] & LONG_MASK) + (difference >> 32);
1370 result[0] = (int)difference;
1371 return result;
1372 }
1373 }
1374 }
1375
1376 /**
1377 * Subtracts the contents of the second argument (val) from the
1378 * first (big). The first int array (big) must represent a larger number
1379 * than the second. This method allocates the space necessary to hold the
1380 * answer.
1381 * assumes val >= 0
1382 */
1383 private static int[] subtract(int[] big, long val) {
1384 int highWord = (int)(val >>> 32);
1385 int bigIndex = big.length;
1386 int result[] = new int[bigIndex];
1387 long difference = 0;
1388
1389 if (highWord == 0) {
1390 difference = (big[--bigIndex] & LONG_MASK) - val;
1391 result[bigIndex] = (int)difference;
1392 } else {
1393 difference = (big[--bigIndex] & LONG_MASK) - (val & LONG_MASK);
1394 result[bigIndex] = (int)difference;
1395 difference = (big[--bigIndex] & LONG_MASK) - (highWord & LONG_MASK) + (difference >> 32);
1396 result[bigIndex] = (int)difference;
1397 }
1398
1399 // Subtract remainder of longer number while borrow propagates
1400 boolean borrow = (difference >> 32 != 0);
1401 while (bigIndex > 0 && borrow)
1402 borrow = ((result[--bigIndex] = big[bigIndex] - 1) == -1);
1403
1404 // Copy remainder of longer number
1405 while (bigIndex > 0)
1406 result[--bigIndex] = big[bigIndex];
1407
1408 return result;
1409 }
1410
1411 /**
1412 * Returns a BigInteger whose value is {@code (this - val)}.
1413 *
1414 * @param val value to be subtracted from this BigInteger.
1415 * @return {@code this - val}
1416 */
1417 public BigInteger subtract(BigInteger val) {
1418 if (val.signum == 0)
1419 return this;
1420 if (signum == 0)
1421 return val.negate();
1422 if (val.signum != signum)
1423 return new BigInteger(add(mag, val.mag), signum);
1424
1425 int cmp = compareMagnitude(val);
1426 if (cmp == 0)
1427 return ZERO;
1428 int[] resultMag = (cmp > 0 ? subtract(mag, val.mag)
1429 : subtract(val.mag, mag));
1430 resultMag = trustedStripLeadingZeroInts(resultMag);
1431 return new BigInteger(resultMag, cmp == signum ? 1 : -1);
1432 }
1433
1434 /**
1435 * Subtracts the contents of the second int arrays (little) from the
1436 * first (big). The first int array (big) must represent a larger number
1437 * than the second. This method allocates the space necessary to hold the
1438 * answer.
1439 */
1440 private static int[] subtract(int[] big, int[] little) {
1441 int bigIndex = big.length;
1442 int result[] = new int[bigIndex];
1443 int littleIndex = little.length;
1444 long difference = 0;
1445
1446 // Subtract common parts of both numbers
1447 while (littleIndex > 0) {
1448 difference = (big[--bigIndex] & LONG_MASK) -
1449 (little[--littleIndex] & LONG_MASK) +
1450 (difference >> 32);
1451 result[bigIndex] = (int)difference;
1452 }
1453
1454 // Subtract remainder of longer number while borrow propagates
1455 boolean borrow = (difference >> 32 != 0);
1456 while (bigIndex > 0 && borrow)
1457 borrow = ((result[--bigIndex] = big[bigIndex] - 1) == -1);
1458
1459 // Copy remainder of longer number
1460 while (bigIndex > 0)
1461 result[--bigIndex] = big[bigIndex];
1462
1463 return result;
1464 }
1465
1466 /**
1467 * Returns a BigInteger whose value is {@code (this * val)}.
1468 *
1469 * @implNote An implementation may offer better algorithmic
1470 * performance when {@code val == this}.
1471 *
1472 * @param val value to be multiplied by this BigInteger.
1473 * @return {@code this * val}
1474 */
1475 public BigInteger multiply(BigInteger val) {
1476 if (val.signum == 0 || signum == 0)
1477 return ZERO;
1478
1479 int xlen = mag.length;
1480
1481 if (val == this && xlen > MULTIPLY_SQUARE_THRESHOLD) {
1482 return square();
1483 }
1484
1485 int ylen = val.mag.length;
1486
1487 if ((xlen < KARATSUBA_THRESHOLD) || (ylen < KARATSUBA_THRESHOLD)) {
1488 int resultSign = signum == val.signum ? 1 : -1;
1489 if (val.mag.length == 1) {
1490 return multiplyByInt(mag,val.mag[0], resultSign);
1491 }
1492 if (mag.length == 1) {
1493 return multiplyByInt(val.mag,mag[0], resultSign);
1494 }
1495 int[] result = multiplyToLen(mag, xlen,
1496 val.mag, ylen, null);
1497 result = trustedStripLeadingZeroInts(result);
1498 return new BigInteger(result, resultSign);
1499 } else {
1500 if ((xlen < TOOM_COOK_THRESHOLD) && (ylen < TOOM_COOK_THRESHOLD)) {
1501 return multiplyKaratsuba(this, val);
1502 } else {
1503 return multiplyToomCook3(this, val);
1504 }
1505 }
1506 }
1507
1508 private static BigInteger multiplyByInt(int[] x, int y, int sign) {
1509 if (Integer.bitCount(y) == 1) {
1510 return new BigInteger(shiftLeft(x,Integer.numberOfTrailingZeros(y)), sign);
1511 }
1512 int xlen = x.length;
1513 int[] rmag = new int[xlen + 1];
1514 long carry = 0;
1515 long yl = y & LONG_MASK;
1516 int rstart = rmag.length - 1;
1517 for (int i = xlen - 1; i >= 0; i--) {
1518 long product = (x[i] & LONG_MASK) * yl + carry;
1519 rmag[rstart--] = (int)product;
1520 carry = product >>> 32;
1521 }
1522 if (carry == 0L) {
1523 rmag = java.util.Arrays.copyOfRange(rmag, 1, rmag.length);
1524 } else {
1525 rmag[rstart] = (int)carry;
1526 }
1527 return new BigInteger(rmag, sign);
1528 }
1529
1530 /**
1531 * Package private methods used by BigDecimal code to multiply a BigInteger
1532 * with a long. Assumes v is not equal to INFLATED.
1533 */
1534 BigInteger multiply(long v) {
1535 if (v == 0 || signum == 0)
1536 return ZERO;
1537 if (v == BigDecimal.INFLATED)
1538 return multiply(BigInteger.valueOf(v));
1539 int rsign = (v > 0 ? signum : -signum);
1540 if (v < 0)
1541 v = -v;
1542 long dh = v >>> 32; // higher order bits
1543 long dl = v & LONG_MASK; // lower order bits
1544
1545 int xlen = mag.length;
1546 int[] value = mag;
1547 int[] rmag = (dh == 0L) ? (new int[xlen + 1]) : (new int[xlen + 2]);
1548 long carry = 0;
1549 int rstart = rmag.length - 1;
1550 for (int i = xlen - 1; i >= 0; i--) {
1551 long product = (value[i] & LONG_MASK) * dl + carry;
1552 rmag[rstart--] = (int)product;
1553 carry = product >>> 32;
1554 }
1555 rmag[rstart] = (int)carry;
1556 if (dh != 0L) {
1557 carry = 0;
1558 rstart = rmag.length - 2;
1559 for (int i = xlen - 1; i >= 0; i--) {
1560 long product = (value[i] & LONG_MASK) * dh +
1561 (rmag[rstart] & LONG_MASK) + carry;
1562 rmag[rstart--] = (int)product;
1563 carry = product >>> 32;
1564 }
1565 rmag[0] = (int)carry;
1566 }
1567 if (carry == 0L)
1568 rmag = java.util.Arrays.copyOfRange(rmag, 1, rmag.length);
1569 return new BigInteger(rmag, rsign);
1570 }
1571
1572 /**
1573 * Multiplies int arrays x and y to the specified lengths and places
1574 * the result into z. There will be no leading zeros in the resultant array.
1575 */
1576 private int[] multiplyToLen(int[] x, int xlen, int[] y, int ylen, int[] z) {
1577 int xstart = xlen - 1;
1578 int ystart = ylen - 1;
1579
1580 if (z == null || z.length < (xlen+ ylen))
1581 z = new int[xlen+ylen];
1582
1583 long carry = 0;
1584 for (int j=ystart, k=ystart+1+xstart; j >= 0; j--, k--) {
1585 long product = (y[j] & LONG_MASK) *
1586 (x[xstart] & LONG_MASK) + carry;
1587 z[k] = (int)product;
1588 carry = product >>> 32;
1589 }
1590 z[xstart] = (int)carry;
1591
1592 for (int i = xstart-1; i >= 0; i--) {
1593 carry = 0;
1594 for (int j=ystart, k=ystart+1+i; j >= 0; j--, k--) {
1595 long product = (y[j] & LONG_MASK) *
1596 (x[i] & LONG_MASK) +
1597 (z[k] & LONG_MASK) + carry;
1598 z[k] = (int)product;
1599 carry = product >>> 32;
1600 }
1601 z[i] = (int)carry;
1602 }
1603 return z;
1604 }
1605
1606 /**
1607 * Multiplies two BigIntegers using the Karatsuba multiplication
1608 * algorithm. This is a recursive divide-and-conquer algorithm which is
1609 * more efficient for large numbers than what is commonly called the
1610 * "grade-school" algorithm used in multiplyToLen. If the numbers to be
1611 * multiplied have length n, the "grade-school" algorithm has an
1612 * asymptotic complexity of O(n^2). In contrast, the Karatsuba algorithm
1613 * has complexity of O(n^(log2(3))), or O(n^1.585). It achieves this
1614 * increased performance by doing 3 multiplies instead of 4 when
1615 * evaluating the product. As it has some overhead, should be used when
1616 * both numbers are larger than a certain threshold (found
1617 * experimentally).
1618 *
1619 * See: http://en.wikipedia.org/wiki/Karatsuba_algorithm
1620 */
1621 private static BigInteger multiplyKaratsuba(BigInteger x, BigInteger y) {
1622 int xlen = x.mag.length;
1623 int ylen = y.mag.length;
1624
1625 // The number of ints in each half of the number.
1626 int half = (Math.max(xlen, ylen)+1) / 2;
1627
1628 // xl and yl are the lower halves of x and y respectively,
1629 // xh and yh are the upper halves.
1630 BigInteger xl = x.getLower(half);
1631 BigInteger xh = x.getUpper(half);
1632 BigInteger yl = y.getLower(half);
1633 BigInteger yh = y.getUpper(half);
1634
1635 BigInteger p1 = xh.multiply(yh); // p1 = xh*yh
1636 BigInteger p2 = xl.multiply(yl); // p2 = xl*yl
1637
1638 // p3=(xh+xl)*(yh+yl)
1639 BigInteger p3 = xh.add(xl).multiply(yh.add(yl));
1640
1641 // result = p1 * 2^(32*2*half) + (p3 - p1 - p2) * 2^(32*half) + p2
1642 BigInteger result = p1.shiftLeft(32*half).add(p3.subtract(p1).subtract(p2)).shiftLeft(32*half).add(p2);
1643
1644 if (x.signum != y.signum) {
1645 return result.negate();
1646 } else {
1647 return result;
1648 }
1649 }
1650
1651 /**
1652 * Multiplies two BigIntegers using a 3-way Toom-Cook multiplication
1653 * algorithm. This is a recursive divide-and-conquer algorithm which is
1654 * more efficient for large numbers than what is commonly called the
1655 * "grade-school" algorithm used in multiplyToLen. If the numbers to be
1656 * multiplied have length n, the "grade-school" algorithm has an
1657 * asymptotic complexity of O(n^2). In contrast, 3-way Toom-Cook has a
1658 * complexity of about O(n^1.465). It achieves this increased asymptotic
1659 * performance by breaking each number into three parts and by doing 5
1660 * multiplies instead of 9 when evaluating the product. Due to overhead
1661 * (additions, shifts, and one division) in the Toom-Cook algorithm, it
1662 * should only be used when both numbers are larger than a certain
1663 * threshold (found experimentally). This threshold is generally larger
1664 * than that for Karatsuba multiplication, so this algorithm is generally
1665 * only used when numbers become significantly larger.
1666 *
1667 * The algorithm used is the "optimal" 3-way Toom-Cook algorithm outlined
1668 * by Marco Bodrato.
1669 *
1670 * See: http://bodrato.it/toom-cook/
1671 * http://bodrato.it/papers/#WAIFI2007
1672 *
1673 * "Towards Optimal Toom-Cook Multiplication for Univariate and
1674 * Multivariate Polynomials in Characteristic 2 and 0." by Marco BODRATO;
1675 * In C.Carlet and B.Sunar, Eds., "WAIFI'07 proceedings", p. 116-133,
1676 * LNCS #4547. Springer, Madrid, Spain, June 21-22, 2007.
1677 *
1678 */
1679 private static BigInteger multiplyToomCook3(BigInteger a, BigInteger b) {
1680 int alen = a.mag.length;
1681 int blen = b.mag.length;
1682
1683 int largest = Math.max(alen, blen);
1684
1685 // k is the size (in ints) of the lower-order slices.
1686 int k = (largest+2)/3; // Equal to ceil(largest/3)
1687
1688 // r is the size (in ints) of the highest-order slice.
1689 int r = largest - 2*k;
1690
1691 // Obtain slices of the numbers. a2 and b2 are the most significant
1692 // bits of the numbers a and b, and a0 and b0 the least significant.
1693 BigInteger a0, a1, a2, b0, b1, b2;
1694 a2 = a.getToomSlice(k, r, 0, largest);
1695 a1 = a.getToomSlice(k, r, 1, largest);
1696 a0 = a.getToomSlice(k, r, 2, largest);
1697 b2 = b.getToomSlice(k, r, 0, largest);
1698 b1 = b.getToomSlice(k, r, 1, largest);
1699 b0 = b.getToomSlice(k, r, 2, largest);
1700
1701 BigInteger v0, v1, v2, vm1, vinf, t1, t2, tm1, da1, db1;
1702
1703 v0 = a0.multiply(b0);
1704 da1 = a2.add(a0);
1705 db1 = b2.add(b0);
1706 vm1 = da1.subtract(a1).multiply(db1.subtract(b1));
1707 da1 = da1.add(a1);
1708 db1 = db1.add(b1);
1709 v1 = da1.multiply(db1);
1710 v2 = da1.add(a2).shiftLeft(1).subtract(a0).multiply(
1711 db1.add(b2).shiftLeft(1).subtract(b0));
1712 vinf = a2.multiply(b2);
1713
1714 // The algorithm requires two divisions by 2 and one by 3.
1715 // All divisions are known to be exact, that is, they do not produce
1716 // remainders, and all results are positive. The divisions by 2 are
1717 // implemented as right shifts which are relatively efficient, leaving
1718 // only an exact division by 3, which is done by a specialized
1719 // linear-time algorithm.
1720 t2 = v2.subtract(vm1).exactDivideBy3();
1721 tm1 = v1.subtract(vm1).shiftRight(1);
1722 t1 = v1.subtract(v0);
1723 t2 = t2.subtract(t1).shiftRight(1);
1724 t1 = t1.subtract(tm1).subtract(vinf);
1725 t2 = t2.subtract(vinf.shiftLeft(1));
1726 tm1 = tm1.subtract(t2);
1727
1728 // Number of bits to shift left.
1729 int ss = k*32;
1730
1731 BigInteger result = vinf.shiftLeft(ss).add(t2).shiftLeft(ss).add(t1).shiftLeft(ss).add(tm1).shiftLeft(ss).add(v0);
1732
1733 if (a.signum != b.signum) {
1734 return result.negate();
1735 } else {
1736 return result;
1737 }
1738 }
1739
1740
1741 /**
1742 * Returns a slice of a BigInteger for use in Toom-Cook multiplication.
1743 *
1744 * @param lowerSize The size of the lower-order bit slices.
1745 * @param upperSize The size of the higher-order bit slices.
1746 * @param slice The index of which slice is requested, which must be a
1747 * number from 0 to size-1. Slice 0 is the highest-order bits, and slice
1748 * size-1 are the lowest-order bits. Slice 0 may be of different size than
1749 * the other slices.
1750 * @param fullsize The size of the larger integer array, used to align
1751 * slices to the appropriate position when multiplying different-sized
1752 * numbers.
1753 */
1754 private BigInteger getToomSlice(int lowerSize, int upperSize, int slice,
1755 int fullsize) {
1756 int start, end, sliceSize, len, offset;
1757
1758 len = mag.length;
1759 offset = fullsize - len;
1760
1761 if (slice == 0) {
1762 start = 0 - offset;
1763 end = upperSize - 1 - offset;
1764 } else {
1765 start = upperSize + (slice-1)*lowerSize - offset;
1766 end = start + lowerSize - 1;
1767 }
1768
1769 if (start < 0) {
1770 start = 0;
1771 }
1772 if (end < 0) {
1773 return ZERO;
1774 }
1775
1776 sliceSize = (end-start) + 1;
1777
1778 if (sliceSize <= 0) {
1779 return ZERO;
1780 }
1781
1782 // While performing Toom-Cook, all slices are positive and
1783 // the sign is adjusted when the final number is composed.
1784 if (start == 0 && sliceSize >= len) {
1785 return this.abs();
1786 }
1787
1788 int intSlice[] = new int[sliceSize];
1789 System.arraycopy(mag, start, intSlice, 0, sliceSize);
1790
1791 return new BigInteger(trustedStripLeadingZeroInts(intSlice), 1);
1792 }
1793
1794 /**
1795 * Does an exact division (that is, the remainder is known to be zero)
1796 * of the specified number by 3. This is used in Toom-Cook
1797 * multiplication. This is an efficient algorithm that runs in linear
1798 * time. If the argument is not exactly divisible by 3, results are
1799 * undefined. Note that this is expected to be called with positive
1800 * arguments only.
1801 */
1802 private BigInteger exactDivideBy3() {
1803 int len = mag.length;
1804 int[] result = new int[len];
1805 long x, w, q, borrow;
1806 borrow = 0L;
1807 for (int i=len-1; i >= 0; i--) {
1808 x = (mag[i] & LONG_MASK);
1809 w = x - borrow;
1810 if (borrow > x) { // Did we make the number go negative?
1811 borrow = 1L;
1812 } else {
1813 borrow = 0L;
1814 }
1815
1816 // 0xAAAAAAAB is the modular inverse of 3 (mod 2^32). Thus,
1817 // the effect of this is to divide by 3 (mod 2^32).
1818 // This is much faster than division on most architectures.
1819 q = (w * 0xAAAAAAABL) & LONG_MASK;
1820 result[i] = (int) q;
1821
1822 // Now check the borrow. The second check can of course be
1823 // eliminated if the first fails.
1824 if (q >= 0x55555556L) {
1825 borrow++;
1826 if (q >= 0xAAAAAAABL)
1827 borrow++;
1828 }
1829 }
1830 result = trustedStripLeadingZeroInts(result);
1831 return new BigInteger(result, signum);
1832 }
1833
1834 /**
1835 * Returns a new BigInteger representing n lower ints of the number.
1836 * This is used by Karatsuba multiplication and Karatsuba squaring.
1837 */
1838 private BigInteger getLower(int n) {
1839 int len = mag.length;
1840
1841 if (len <= n) {
1842 return abs();
1843 }
1844
1845 int lowerInts[] = new int[n];
1846 System.arraycopy(mag, len-n, lowerInts, 0, n);
1847
1848 return new BigInteger(trustedStripLeadingZeroInts(lowerInts), 1);
1849 }
1850
1851 /**
1852 * Returns a new BigInteger representing mag.length-n upper
1853 * ints of the number. This is used by Karatsuba multiplication and
1854 * Karatsuba squaring.
1855 */
1856 private BigInteger getUpper(int n) {
1857 int len = mag.length;
1858
1859 if (len <= n) {
1860 return ZERO;
1861 }
1862
1863 int upperLen = len - n;
1864 int upperInts[] = new int[upperLen];
1865 System.arraycopy(mag, 0, upperInts, 0, upperLen);
1866
1867 return new BigInteger(trustedStripLeadingZeroInts(upperInts), 1);
1868 }
1869
1870 // Squaring
1871
1872 /**
1873 * Returns a BigInteger whose value is {@code (this<sup>2</sup>)}.
1874 *
1875 * @return {@code this<sup>2</sup>}
1876 */
1877 private BigInteger square() {
1878 if (signum == 0) {
1879 return ZERO;
1880 }
1881 int len = mag.length;
1882
1883 if (len < KARATSUBA_SQUARE_THRESHOLD) {
1884 int[] z = squareToLen(mag, len, null);
1885 return new BigInteger(trustedStripLeadingZeroInts(z), 1);
1886 } else {
1887 if (len < TOOM_COOK_SQUARE_THRESHOLD) {
1888 return squareKaratsuba();
1889 } else {
1890 return squareToomCook3();
1891 }
1892 }
1893 }
1894
1895 /**
1896 * Squares the contents of the int array x. The result is placed into the
1897 * int array z. The contents of x are not changed.
1898 */
1899 private static final int[] squareToLen(int[] x, int len, int[] z) {
1900 /*
1901 * The algorithm used here is adapted from Colin Plumb's C library.
1902 * Technique: Consider the partial products in the multiplication
1903 * of "abcde" by itself:
1904 *
1905 * a b c d e
1906 * * a b c d e
1907 * ==================
1908 * ae be ce de ee
1909 * ad bd cd dd de
1910 * ac bc cc cd ce
1911 * ab bb bc bd be
1912 * aa ab ac ad ae
1913 *
1914 * Note that everything above the main diagonal:
1915 * ae be ce de = (abcd) * e
1916 * ad bd cd = (abc) * d
1917 * ac bc = (ab) * c
1918 * ab = (a) * b
1919 *
1920 * is a copy of everything below the main diagonal:
1921 * de
1922 * cd ce
1923 * bc bd be
1924 * ab ac ad ae
1925 *
1926 * Thus, the sum is 2 * (off the diagonal) + diagonal.
1927 *
1928 * This is accumulated beginning with the diagonal (which
1929 * consist of the squares of the digits of the input), which is then
1930 * divided by two, the off-diagonal added, and multiplied by two
1931 * again. The low bit is simply a copy of the low bit of the
1932 * input, so it doesn't need special care.
1933 */
1934 int zlen = len << 1;
1935 if (z == null || z.length < zlen)
1936 z = new int[zlen];
1937
1938 // Store the squares, right shifted one bit (i.e., divided by 2)
1939 int lastProductLowWord = 0;
1940 for (int j=0, i=0; j < len; j++) {
1941 long piece = (x[j] & LONG_MASK);
1942 long product = piece * piece;
1943 z[i++] = (lastProductLowWord << 31) | (int)(product >>> 33);
1944 z[i++] = (int)(product >>> 1);
1945 lastProductLowWord = (int)product;
1946 }
1947
1948 // Add in off-diagonal sums
1949 for (int i=len, offset=1; i > 0; i--, offset+=2) {
1950 int t = x[i-1];
1951 t = mulAdd(z, x, offset, i-1, t);
1952 addOne(z, offset-1, i, t);
1953 }
1954
1955 // Shift back up and set low bit
1956 primitiveLeftShift(z, zlen, 1);
1957 z[zlen-1] |= x[len-1] & 1;
1958
1959 return z;
1960 }
1961
1962 /**
1963 * Squares a BigInteger using the Karatsuba squaring algorithm. It should
1964 * be used when both numbers are larger than a certain threshold (found
1965 * experimentally). It is a recursive divide-and-conquer algorithm that
1966 * has better asymptotic performance than the algorithm used in
1967 * squareToLen.
1968 */
1969 private BigInteger squareKaratsuba() {
1970 int half = (mag.length+1) / 2;
1971
1972 BigInteger xl = getLower(half);
1973 BigInteger xh = getUpper(half);
1974
1975 BigInteger xhs = xh.square(); // xhs = xh^2
1976 BigInteger xls = xl.square(); // xls = xl^2
1977
1978 // xh^2 << 64 + (((xl+xh)^2 - (xh^2 + xl^2)) << 32) + xl^2
1979 return xhs.shiftLeft(half*32).add(xl.add(xh).square().subtract(xhs.add(xls))).shiftLeft(half*32).add(xls);
1980 }
1981
1982 /**
1983 * Squares a BigInteger using the 3-way Toom-Cook squaring algorithm. It
1984 * should be used when both numbers are larger than a certain threshold
1985 * (found experimentally). It is a recursive divide-and-conquer algorithm
1986 * that has better asymptotic performance than the algorithm used in
1987 * squareToLen or squareKaratsuba.
1988 */
1989 private BigInteger squareToomCook3() {
1990 int len = mag.length;
1991
1992 // k is the size (in ints) of the lower-order slices.
1993 int k = (len+2)/3; // Equal to ceil(largest/3)
1994
1995 // r is the size (in ints) of the highest-order slice.
1996 int r = len - 2*k;
1997
1998 // Obtain slices of the numbers. a2 is the most significant
1999 // bits of the number, and a0 the least significant.
2000 BigInteger a0, a1, a2;
2001 a2 = getToomSlice(k, r, 0, len);
2002 a1 = getToomSlice(k, r, 1, len);
2003 a0 = getToomSlice(k, r, 2, len);
2004 BigInteger v0, v1, v2, vm1, vinf, t1, t2, tm1, da1;
2005
2006 v0 = a0.square();
2007 da1 = a2.add(a0);
2008 vm1 = da1.subtract(a1).square();
2009 da1 = da1.add(a1);
2010 v1 = da1.square();
2011 vinf = a2.square();
2012 v2 = da1.add(a2).shiftLeft(1).subtract(a0).square();
2013
2014 // The algorithm requires two divisions by 2 and one by 3.
2015 // All divisions are known to be exact, that is, they do not produce
2016 // remainders, and all results are positive. The divisions by 2 are
2017 // implemented as right shifts which are relatively efficient, leaving
2018 // only a division by 3.
2019 // The division by 3 is done by an optimized algorithm for this case.
2020 t2 = v2.subtract(vm1).exactDivideBy3();
2021 tm1 = v1.subtract(vm1).shiftRight(1);
2022 t1 = v1.subtract(v0);
2023 t2 = t2.subtract(t1).shiftRight(1);
2024 t1 = t1.subtract(tm1).subtract(vinf);
2025 t2 = t2.subtract(vinf.shiftLeft(1));
2026 tm1 = tm1.subtract(t2);
2027
2028 // Number of bits to shift left.
2029 int ss = k*32;
2030
2031 return vinf.shiftLeft(ss).add(t2).shiftLeft(ss).add(t1).shiftLeft(ss).add(tm1).shiftLeft(ss).add(v0);
2032 }
2033
2034 // Division
2035
2036 /**
2037 * Returns a BigInteger whose value is {@code (this / val)}.
2038 *
2039 * @param val value by which this BigInteger is to be divided.
2040 * @return {@code this / val}
2041 * @throws ArithmeticException if {@code val} is zero.
2042 */
2043 public BigInteger divide(BigInteger val) {
2044 if (val.mag.length < BURNIKEL_ZIEGLER_THRESHOLD ||
2045 mag.length - val.mag.length < BURNIKEL_ZIEGLER_OFFSET) {
2046 return divideKnuth(val);
2047 } else {
2048 return divideBurnikelZiegler(val);
2049 }
2050 }
2051
2052 /**
2053 * Returns a BigInteger whose value is {@code (this / val)} using an O(n^2) algorithm from Knuth.
2054 *
2055 * @param val value by which this BigInteger is to be divided.
2056 * @return {@code this / val}
2057 * @throws ArithmeticException if {@code val} is zero.
2058 * @see MutableBigInteger#divideKnuth(MutableBigInteger, MutableBigInteger, boolean)
2059 */
2060 private BigInteger divideKnuth(BigInteger val) {
2061 MutableBigInteger q = new MutableBigInteger(),
2062 a = new MutableBigInteger(this.mag),
2063 b = new MutableBigInteger(val.mag);
2064
2065 a.divideKnuth(b, q, false);
2066 return q.toBigInteger(this.signum * val.signum);
2067 }
2068
2069 /**
2070 * Returns an array of two BigIntegers containing {@code (this / val)}
2071 * followed by {@code (this % val)}.
2072 *
2073 * @param val value by which this BigInteger is to be divided, and the
2074 * remainder computed.
2075 * @return an array of two BigIntegers: the quotient {@code (this / val)}
2076 * is the initial element, and the remainder {@code (this % val)}
2077 * is the final element.
2078 * @throws ArithmeticException if {@code val} is zero.
2079 */
2080 public BigInteger[] divideAndRemainder(BigInteger val) {
2081 if (val.mag.length < BURNIKEL_ZIEGLER_THRESHOLD ||
2082 mag.length - val.mag.length < BURNIKEL_ZIEGLER_OFFSET) {
2083 return divideAndRemainderKnuth(val);
2084 } else {
2085 return divideAndRemainderBurnikelZiegler(val);
2086 }
2087 }
2088
2089 /** Long division */
2090 private BigInteger[] divideAndRemainderKnuth(BigInteger val) {
2091 BigInteger[] result = new BigInteger[2];
2092 MutableBigInteger q = new MutableBigInteger(),
2093 a = new MutableBigInteger(this.mag),
2094 b = new MutableBigInteger(val.mag);
2095 MutableBigInteger r = a.divideKnuth(b, q);
2096 result[0] = q.toBigInteger(this.signum == val.signum ? 1 : -1);
2097 result[1] = r.toBigInteger(this.signum);
2098 return result;
2099 }
2100
2101 /**
2102 * Returns a BigInteger whose value is {@code (this % val)}.
2103 *
2104 * @param val value by which this BigInteger is to be divided, and the
2105 * remainder computed.
2106 * @return {@code this % val}
2107 * @throws ArithmeticException if {@code val} is zero.
2108 */
2109 public BigInteger remainder(BigInteger val) {
2110 if (val.mag.length < BURNIKEL_ZIEGLER_THRESHOLD ||
2111 mag.length - val.mag.length < BURNIKEL_ZIEGLER_OFFSET) {
2112 return remainderKnuth(val);
2113 } else {
2114 return remainderBurnikelZiegler(val);
2115 }
2116 }
2117
2118 /** Long division */
2119 private BigInteger remainderKnuth(BigInteger val) {
2120 MutableBigInteger q = new MutableBigInteger(),
2121 a = new MutableBigInteger(this.mag),
2122 b = new MutableBigInteger(val.mag);
2123
2124 return a.divideKnuth(b, q).toBigInteger(this.signum);
2125 }
2126
2127 /**
2128 * Calculates {@code this / val} using the Burnikel-Ziegler algorithm.
2129 * @param val the divisor
2130 * @return {@code this / val}
2131 */
2132 private BigInteger divideBurnikelZiegler(BigInteger val) {
2133 return divideAndRemainderBurnikelZiegler(val)[0];
2134 }
2135
2136 /**
2137 * Calculates {@code this % val} using the Burnikel-Ziegler algorithm.
2138 * @param val the divisor
2139 * @return {@code this % val}
2140 */
2141 private BigInteger remainderBurnikelZiegler(BigInteger val) {
2142 return divideAndRemainderBurnikelZiegler(val)[1];
2143 }
2144
2145 /**
2146 * Computes {@code this / val} and {@code this % val} using the
2147 * Burnikel-Ziegler algorithm.
2148 * @param val the divisor
2149 * @return an array containing the quotient and remainder
2150 */
2151 private BigInteger[] divideAndRemainderBurnikelZiegler(BigInteger val) {
2152 MutableBigInteger q = new MutableBigInteger();
2153 MutableBigInteger r = new MutableBigInteger(this).divideAndRemainderBurnikelZiegler(new MutableBigInteger(val), q);
2154 BigInteger qBigInt = q.isZero() ? ZERO : q.toBigInteger(signum*val.signum);
2155 BigInteger rBigInt = r.isZero() ? ZERO : r.toBigInteger(signum);
2156 return new BigInteger[] {qBigInt, rBigInt};
2157 }
2158
2159 /**
2160 * Returns a BigInteger whose value is <tt>(this<sup>exponent</sup>)</tt>.
2161 * Note that {@code exponent} is an integer rather than a BigInteger.
2162 *
2163 * @param exponent exponent to which this BigInteger is to be raised.
2164 * @return <tt>this<sup>exponent</sup></tt>
2165 * @throws ArithmeticException {@code exponent} is negative. (This would
2166 * cause the operation to yield a non-integer value.)
2167 */
2168 public BigInteger pow(int exponent) {
2169 if (exponent < 0) {
2170 throw new ArithmeticException("Negative exponent");
2171 }
2172 if (signum == 0) {
2173 return (exponent == 0 ? ONE : this);
2174 }
2175
2176 BigInteger partToSquare = this.abs();
2177
2178 // Factor out powers of two from the base, as the exponentiation of
2179 // these can be done by left shifts only.
2180 // The remaining part can then be exponentiated faster. The
2181 // powers of two will be multiplied back at the end.
2182 int powersOfTwo = partToSquare.getLowestSetBit();
2183 long bitsToShift = (long)powersOfTwo * exponent;
2184 if (bitsToShift > Integer.MAX_VALUE) {
2185 reportOverflow();
2186 }
2187
2188 int remainingBits;
2189
2190 // Factor the powers of two out quickly by shifting right, if needed.
2191 if (powersOfTwo > 0) {
2192 partToSquare = partToSquare.shiftRight(powersOfTwo);
2193 remainingBits = partToSquare.bitLength();
2194 if (remainingBits == 1) { // Nothing left but +/- 1?
2195 if (signum < 0 && (exponent&1) == 1) {
2196 return NEGATIVE_ONE.shiftLeft(powersOfTwo*exponent);
2197 } else {
2198 return ONE.shiftLeft(powersOfTwo*exponent);
2199 }
2200 }
2201 } else {
2202 remainingBits = partToSquare.bitLength();
2203 if (remainingBits == 1) { // Nothing left but +/- 1?
2204 if (signum < 0 && (exponent&1) == 1) {
2205 return NEGATIVE_ONE;
2206 } else {
2207 return ONE;
2208 }
2209 }
2210 }
2211
2212 // This is a quick way to approximate the size of the result,
2213 // similar to doing log2[n] * exponent. This will give an upper bound
2214 // of how big the result can be, and which algorithm to use.
2215 long scaleFactor = (long)remainingBits * exponent;
2216
2217 // Use slightly different algorithms for small and large operands.
2218 // See if the result will safely fit into a long. (Largest 2^63-1)
2219 if (partToSquare.mag.length == 1 && scaleFactor <= 62) {
2220 // Small number algorithm. Everything fits into a long.
2221 int newSign = (signum <0 && (exponent&1) == 1 ? -1 : 1);
2222 long result = 1;
2223 long baseToPow2 = partToSquare.mag[0] & LONG_MASK;
2224
2225 int workingExponent = exponent;
2226
2227 // Perform exponentiation using repeated squaring trick
2228 while (workingExponent != 0) {
2229 if ((workingExponent & 1) == 1) {
2230 result = result * baseToPow2;
2231 }
2232
2233 if ((workingExponent >>>= 1) != 0) {
2234 baseToPow2 = baseToPow2 * baseToPow2;
2235 }
2236 }
2237
2238 // Multiply back the powers of two (quickly, by shifting left)
2239 if (powersOfTwo > 0) {
2240 if (bitsToShift + scaleFactor <= 62) { // Fits in long?
2241 return valueOf((result << bitsToShift) * newSign);
2242 } else {
2243 return valueOf(result*newSign).shiftLeft((int) bitsToShift);
2244 }
2245 }
2246 else {
2247 return valueOf(result*newSign);
2248 }
2249 } else {
2250 // Large number algorithm. This is basically identical to
2251 // the algorithm above, but calls multiply() and square()
2252 // which may use more efficient algorithms for large numbers.
2253 BigInteger answer = ONE;
2254
2255 int workingExponent = exponent;
2256 // Perform exponentiation using repeated squaring trick
2257 while (workingExponent != 0) {
2258 if ((workingExponent & 1) == 1) {
2259 answer = answer.multiply(partToSquare);
2260 }
2261
2262 if ((workingExponent >>>= 1) != 0) {
2263 partToSquare = partToSquare.square();
2264 }
2265 }
2266 // Multiply back the (exponentiated) powers of two (quickly,
2267 // by shifting left)
2268 if (powersOfTwo > 0) {
2269 answer = answer.shiftLeft(powersOfTwo*exponent);
2270 }
2271
2272 if (signum < 0 && (exponent&1) == 1) {
2273 return answer.negate();
2274 } else {
2275 return answer;
2276 }
2277 }
2278 }
2279
2280 /**
2281 * Returns a BigInteger whose value is the greatest common divisor of
2282 * {@code abs(this)} and {@code abs(val)}. Returns 0 if
2283 * {@code this == 0 && val == 0}.
2284 *
2285 * @param val value with which the GCD is to be computed.
2286 * @return {@code GCD(abs(this), abs(val))}
2287 */
2288 public BigInteger gcd(BigInteger val) {
2289 if (val.signum == 0)
2290 return this.abs();
2291 else if (this.signum == 0)
2292 return val.abs();
2293
2294 MutableBigInteger a = new MutableBigInteger(this);
2295 MutableBigInteger b = new MutableBigInteger(val);
2296
2297 MutableBigInteger result = a.hybridGCD(b);
2298
2299 return result.toBigInteger(1);
2300 }
2301
2302 /**
2303 * Package private method to return bit length for an integer.
2304 */
2305 static int bitLengthForInt(int n) {
2306 return 32 - Integer.numberOfLeadingZeros(n);
2307 }
2308
2309 /**
2310 * Left shift int array a up to len by n bits. Returns the array that
2311 * results from the shift since space may have to be reallocated.
2312 */
2313 private static int[] leftShift(int[] a, int len, int n) {
2314 int nInts = n >>> 5;
2315 int nBits = n&0x1F;
2316 int bitsInHighWord = bitLengthForInt(a[0]);
2317
2318 // If shift can be done without recopy, do so
2319 if (n <= (32-bitsInHighWord)) {
2320 primitiveLeftShift(a, len, nBits);
2321 return a;
2322 } else { // Array must be resized
2323 if (nBits <= (32-bitsInHighWord)) {
2324 int result[] = new int[nInts+len];
2325 System.arraycopy(a, 0, result, 0, len);
2326 primitiveLeftShift(result, result.length, nBits);
2327 return result;
2328 } else {
2329 int result[] = new int[nInts+len+1];
2330 System.arraycopy(a, 0, result, 0, len);
2331 primitiveRightShift(result, result.length, 32 - nBits);
2332 return result;
2333 }
2334 }
2335 }
2336
2337 // shifts a up to len right n bits assumes no leading zeros, 0<n<32
2338 static void primitiveRightShift(int[] a, int len, int n) {
2339 int n2 = 32 - n;
2340 for (int i=len-1, c=a[i]; i > 0; i--) {
2341 int b = c;
2342 c = a[i-1];
2343 a[i] = (c << n2) | (b >>> n);
2344 }
2345 a[0] >>>= n;
2346 }
2347
2348 // shifts a up to len left n bits assumes no leading zeros, 0<=n<32
2349 static void primitiveLeftShift(int[] a, int len, int n) {
2350 if (len == 0 || n == 0)
2351 return;
2352
2353 int n2 = 32 - n;
2354 for (int i=0, c=a[i], m=i+len-1; i < m; i++) {
2355 int b = c;
2356 c = a[i+1];
2357 a[i] = (b << n) | (c >>> n2);
2358 }
2359 a[len-1] <<= n;
2360 }
2361
2362 /**
2363 * Calculate bitlength of contents of the first len elements an int array,
2364 * assuming there are no leading zero ints.
2365 */
2366 private static int bitLength(int[] val, int len) {
2367 if (len == 0)
2368 return 0;
2369 return ((len - 1) << 5) + bitLengthForInt(val[0]);
2370 }
2371
2372 /**
2373 * Returns a BigInteger whose value is the absolute value of this
2374 * BigInteger.
2375 *
2376 * @return {@code abs(this)}
2377 */
2378 public BigInteger abs() {
2379 return (signum >= 0 ? this : this.negate());
2380 }
2381
2382 /**
2383 * Returns a BigInteger whose value is {@code (-this)}.
2384 *
2385 * @return {@code -this}
2386 */
2387 public BigInteger negate() {
2388 return new BigInteger(this.mag, -this.signum);
2389 }
2390
2391 /**
2392 * Returns the signum function of this BigInteger.
2393 *
2394 * @return -1, 0 or 1 as the value of this BigInteger is negative, zero or
2395 * positive.
2396 */
2397 public int signum() {
2398 return this.signum;
2399 }
2400
2401 // Modular Arithmetic Operations
2402
2403 /**
2404 * Returns a BigInteger whose value is {@code (this mod m}). This method
2405 * differs from {@code remainder} in that it always returns a
2406 * <i>non-negative</i> BigInteger.
2407 *
2408 * @param m the modulus.
2409 * @return {@code this mod m}
2410 * @throws ArithmeticException {@code m} ≤ 0
2411 * @see #remainder
2412 */
2413 public BigInteger mod(BigInteger m) {
2414 if (m.signum <= 0)
2415 throw new ArithmeticException("BigInteger: modulus not positive");
2416
2417 BigInteger result = this.remainder(m);
2418 return (result.signum >= 0 ? result : result.add(m));
2419 }
2420
2421 /**
2422 * Returns a BigInteger whose value is
2423 * <tt>(this<sup>exponent</sup> mod m)</tt>. (Unlike {@code pow}, this
2424 * method permits negative exponents.)
2425 *
2426 * @param exponent the exponent.
2427 * @param m the modulus.
2428 * @return <tt>this<sup>exponent</sup> mod m</tt>
2429 * @throws ArithmeticException {@code m} ≤ 0 or the exponent is
2430 * negative and this BigInteger is not <i>relatively
2431 * prime</i> to {@code m}.
2432 * @see #modInverse
2433 */
2434 public BigInteger modPow(BigInteger exponent, BigInteger m) {
2435 if (m.signum <= 0)
2436 throw new ArithmeticException("BigInteger: modulus not positive");
2437
2438 // Trivial cases
2439 if (exponent.signum == 0)
2440 return (m.equals(ONE) ? ZERO : ONE);
2441
2442 if (this.equals(ONE))
2443 return (m.equals(ONE) ? ZERO : ONE);
2444
2445 if (this.equals(ZERO) && exponent.signum >= 0)
2446 return ZERO;
2447
2448 if (this.equals(negConst[1]) && (!exponent.testBit(0)))
2449 return (m.equals(ONE) ? ZERO : ONE);
2450
2451 boolean invertResult;
2452 if ((invertResult = (exponent.signum < 0)))
2453 exponent = exponent.negate();
2454
2455 BigInteger base = (this.signum < 0 || this.compareTo(m) >= 0
2456 ? this.mod(m) : this);
2457 BigInteger result;
2458 if (m.testBit(0)) { // odd modulus
2459 result = base.oddModPow(exponent, m);
2460 } else {
2461 /*
2462 * Even modulus. Tear it into an "odd part" (m1) and power of two
2463 * (m2), exponentiate mod m1, manually exponentiate mod m2, and
2464 * use Chinese Remainder Theorem to combine results.
2465 */
2466
2467 // Tear m apart into odd part (m1) and power of 2 (m2)
2468 int p = m.getLowestSetBit(); // Max pow of 2 that divides m
2469
2470 BigInteger m1 = m.shiftRight(p); // m/2**p
2471 BigInteger m2 = ONE.shiftLeft(p); // 2**p
2472
2473 // Calculate new base from m1
2474 BigInteger base2 = (this.signum < 0 || this.compareTo(m1) >= 0
2475 ? this.mod(m1) : this);
2476
2477 // Caculate (base ** exponent) mod m1.
2478 BigInteger a1 = (m1.equals(ONE) ? ZERO :
2479 base2.oddModPow(exponent, m1));
2480
2481 // Calculate (this ** exponent) mod m2
2482 BigInteger a2 = base.modPow2(exponent, p);
2483
2484 // Combine results using Chinese Remainder Theorem
2485 BigInteger y1 = m2.modInverse(m1);
2486 BigInteger y2 = m1.modInverse(m2);
2487
2488 if (m.mag.length < MAX_MAG_LENGTH / 2) {
2489 result = a1.multiply(m2).multiply(y1).add(a2.multiply(m1).multiply(y2)).mod(m);
2490 } else {
2491 MutableBigInteger t1 = new MutableBigInteger();
2492 new MutableBigInteger(a1.multiply(m2)).multiply(new MutableBigInteger(y1), t1);
2493 MutableBigInteger t2 = new MutableBigInteger();
2494 new MutableBigInteger(a2.multiply(m1)).multiply(new MutableBigInteger(y2), t2);
2495 t1.add(t2);
2496 MutableBigInteger q = new MutableBigInteger();
2497 result = t1.divide(new MutableBigInteger(m), q).toBigInteger();
2498 }
2499 }
2500
2501 return (invertResult ? result.modInverse(m) : result);
2502 }
2503
2504 static int[] bnExpModThreshTable = {7, 25, 81, 241, 673, 1793,
2505 Integer.MAX_VALUE}; // Sentinel
2506
2507 /**
2508 * Returns a BigInteger whose value is x to the power of y mod z.
2509 * Assumes: z is odd && x < z.
2510 */
2511 private BigInteger oddModPow(BigInteger y, BigInteger z) {
2512 /*
2513 * The algorithm is adapted from Colin Plumb's C library.
2514 *
2515 * The window algorithm:
2516 * The idea is to keep a running product of b1 = n^(high-order bits of exp)
2517 * and then keep appending exponent bits to it. The following patterns
2518 * apply to a 3-bit window (k = 3):
2519 * To append 0: square
2520 * To append 1: square, multiply by n^1
2521 * To append 10: square, multiply by n^1, square
2522 * To append 11: square, square, multiply by n^3
2523 * To append 100: square, multiply by n^1, square, square
2524 * To append 101: square, square, square, multiply by n^5
2525 * To append 110: square, square, multiply by n^3, square
2526 * To append 111: square, square, square, multiply by n^7
2527 *
2528 * Since each pattern involves only one multiply, the longer the pattern
2529 * the better, except that a 0 (no multiplies) can be appended directly.
2530 * We precompute a table of odd powers of n, up to 2^k, and can then
2531 * multiply k bits of exponent at a time. Actually, assuming random
2532 * exponents, there is on average one zero bit between needs to
2533 * multiply (1/2 of the time there's none, 1/4 of the time there's 1,
2534 * 1/8 of the time, there's 2, 1/32 of the time, there's 3, etc.), so
2535 * you have to do one multiply per k+1 bits of exponent.
2536 *
2537 * The loop walks down the exponent, squaring the result buffer as
2538 * it goes. There is a wbits+1 bit lookahead buffer, buf, that is
2539 * filled with the upcoming exponent bits. (What is read after the
2540 * end of the exponent is unimportant, but it is filled with zero here.)
2541 * When the most-significant bit of this buffer becomes set, i.e.
2542 * (buf & tblmask) != 0, we have to decide what pattern to multiply
2543 * by, and when to do it. We decide, remember to do it in future
2544 * after a suitable number of squarings have passed (e.g. a pattern
2545 * of "100" in the buffer requires that we multiply by n^1 immediately;
2546 * a pattern of "110" calls for multiplying by n^3 after one more
2547 * squaring), clear the buffer, and continue.
2548 *
2549 * When we start, there is one more optimization: the result buffer
2550 * is implcitly one, so squaring it or multiplying by it can be
2551 * optimized away. Further, if we start with a pattern like "100"
2552 * in the lookahead window, rather than placing n into the buffer
2553 * and then starting to square it, we have already computed n^2
2554 * to compute the odd-powers table, so we can place that into
2555 * the buffer and save a squaring.
2556 *
2557 * This means that if you have a k-bit window, to compute n^z,
2558 * where z is the high k bits of the exponent, 1/2 of the time
2559 * it requires no squarings. 1/4 of the time, it requires 1
2560 * squaring, ... 1/2^(k-1) of the time, it reqires k-2 squarings.
2561 * And the remaining 1/2^(k-1) of the time, the top k bits are a
2562 * 1 followed by k-1 0 bits, so it again only requires k-2
2563 * squarings, not k-1. The average of these is 1. Add that
2564 * to the one squaring we have to do to compute the table,
2565 * and you'll see that a k-bit window saves k-2 squarings
2566 * as well as reducing the multiplies. (It actually doesn't
2567 * hurt in the case k = 1, either.)
2568 */
2569 // Special case for exponent of one
2570 if (y.equals(ONE))
2571 return this;
2572
2573 // Special case for base of zero
2574 if (signum == 0)
2575 return ZERO;
2576
2577 int[] base = mag.clone();
2578 int[] exp = y.mag;
2579 int[] mod = z.mag;
2580 int modLen = mod.length;
2581
2582 // Select an appropriate window size
2583 int wbits = 0;
2584 int ebits = bitLength(exp, exp.length);
2585 // if exponent is 65537 (0x10001), use minimum window size
2586 if ((ebits != 17) || (exp[0] != 65537)) {
2587 while (ebits > bnExpModThreshTable[wbits]) {
2588 wbits++;
2589 }
2590 }
2591
2592 // Calculate appropriate table size
2593 int tblmask = 1 << wbits;
2594
2595 // Allocate table for precomputed odd powers of base in Montgomery form
2596 int[][] table = new int[tblmask][];
2597 for (int i=0; i < tblmask; i++)
2598 table[i] = new int[modLen];
2599
2600 // Compute the modular inverse
2601 int inv = -MutableBigInteger.inverseMod32(mod[modLen-1]);
2602
2603 // Convert base to Montgomery form
2604 int[] a = leftShift(base, base.length, modLen << 5);
2605
2606 MutableBigInteger q = new MutableBigInteger(),
2607 a2 = new MutableBigInteger(a),
2608 b2 = new MutableBigInteger(mod);
2609
2610 MutableBigInteger r= a2.divide(b2, q);
2611 table[0] = r.toIntArray();
2612
2613 // Pad table[0] with leading zeros so its length is at least modLen
2614 if (table[0].length < modLen) {
2615 int offset = modLen - table[0].length;
2616 int[] t2 = new int[modLen];
2617 for (int i=0; i < table[0].length; i++)
2618 t2[i+offset] = table[0][i];
2619 table[0] = t2;
2620 }
2621
2622 // Set b to the square of the base
2623 int[] b = squareToLen(table[0], modLen, null);
2624 b = montReduce(b, mod, modLen, inv);
2625
2626 // Set t to high half of b
2627 int[] t = Arrays.copyOf(b, modLen);
2628
2629 // Fill in the table with odd powers of the base
2630 for (int i=1; i < tblmask; i++) {
2631 int[] prod = multiplyToLen(t, modLen, table[i-1], modLen, null);
2632 table[i] = montReduce(prod, mod, modLen, inv);
2633 }
2634
2635 // Pre load the window that slides over the exponent
2636 int bitpos = 1 << ((ebits-1) & (32-1));
2637
2638 int buf = 0;
2639 int elen = exp.length;
2640 int eIndex = 0;
2641 for (int i = 0; i <= wbits; i++) {
2642 buf = (buf << 1) | (((exp[eIndex] & bitpos) != 0)?1:0);
2643 bitpos >>>= 1;
2644 if (bitpos == 0) {
2645 eIndex++;
2646 bitpos = 1 << (32-1);
2647 elen--;
2648 }
2649 }
2650
2651 int multpos = ebits;
2652
2653 // The first iteration, which is hoisted out of the main loop
2654 ebits--;
2655 boolean isone = true;
2656
2657 multpos = ebits - wbits;
2658 while ((buf & 1) == 0) {
2659 buf >>>= 1;
2660 multpos++;
2661 }
2662
2663 int[] mult = table[buf >>> 1];
2664
2665 buf = 0;
2666 if (multpos == ebits)
2667 isone = false;
2668
2669 // The main loop
2670 while (true) {
2671 ebits--;
2672 // Advance the window
2673 buf <<= 1;
2674
2675 if (elen != 0) {
2676 buf |= ((exp[eIndex] & bitpos) != 0) ? 1 : 0;
2677 bitpos >>>= 1;
2678 if (bitpos == 0) {
2679 eIndex++;
2680 bitpos = 1 << (32-1);
2681 elen--;
2682 }
2683 }
2684
2685 // Examine the window for pending multiplies
2686 if ((buf & tblmask) != 0) {
2687 multpos = ebits - wbits;
2688 while ((buf & 1) == 0) {
2689 buf >>>= 1;
2690 multpos++;
2691 }
2692 mult = table[buf >>> 1];
2693 buf = 0;
2694 }
2695
2696 // Perform multiply
2697 if (ebits == multpos) {
2698 if (isone) {
2699 b = mult.clone();
2700 isone = false;
2701 } else {
2702 t = b;
2703 a = multiplyToLen(t, modLen, mult, modLen, a);
2704 a = montReduce(a, mod, modLen, inv);
2705 t = a; a = b; b = t;
2706 }
2707 }
2708
2709 // Check if done
2710 if (ebits == 0)
2711 break;
2712
2713 // Square the input
2714 if (!isone) {
2715 t = b;
2716 a = squareToLen(t, modLen, a);
2717 a = montReduce(a, mod, modLen, inv);
2718 t = a; a = b; b = t;
2719 }
2720 }
2721
2722 // Convert result out of Montgomery form and return
2723 int[] t2 = new int[2*modLen];
2724 System.arraycopy(b, 0, t2, modLen, modLen);
2725
2726 b = montReduce(t2, mod, modLen, inv);
2727
2728 t2 = Arrays.copyOf(b, modLen);
2729
2730 return new BigInteger(1, t2);
2731 }
2732
2733 /**
2734 * Montgomery reduce n, modulo mod. This reduces modulo mod and divides
2735 * by 2^(32*mlen). Adapted from Colin Plumb's C library.
2736 */
2737 private static int[] montReduce(int[] n, int[] mod, int mlen, int inv) {
2738 int c=0;
2739 int len = mlen;
2740 int offset=0;
2741
2742 do {
2743 int nEnd = n[n.length-1-offset];
2744 int carry = mulAdd(n, mod, offset, mlen, inv * nEnd);
2745 c += addOne(n, offset, mlen, carry);
2746 offset++;
2747 } while (--len > 0);
2748
2749 while (c > 0)
2750 c += subN(n, mod, mlen);
2751
2752 while (intArrayCmpToLen(n, mod, mlen) >= 0)
2753 subN(n, mod, mlen);
2754
2755 return n;
2756 }
2757
2758
2759 /*
2760 * Returns -1, 0 or +1 as big-endian unsigned int array arg1 is less than,
2761 * equal to, or greater than arg2 up to length len.
2762 */
2763 private static int intArrayCmpToLen(int[] arg1, int[] arg2, int len) {
2764 for (int i=0; i < len; i++) {
2765 long b1 = arg1[i] & LONG_MASK;
2766 long b2 = arg2[i] & LONG_MASK;
2767 if (b1 < b2)
2768 return -1;
2769 if (b1 > b2)
2770 return 1;
2771 }
2772 return 0;
2773 }
2774
2775 /**
2776 * Subtracts two numbers of same length, returning borrow.
2777 */
2778 private static int subN(int[] a, int[] b, int len) {
2779 long sum = 0;
2780
2781 while (--len >= 0) {
2782 sum = (a[len] & LONG_MASK) -
2783 (b[len] & LONG_MASK) + (sum >> 32);
2784 a[len] = (int)sum;
2785 }
2786
2787 return (int)(sum >> 32);
2788 }
2789
2790 /**
2791 * Multiply an array by one word k and add to result, return the carry
2792 */
2793 static int mulAdd(int[] out, int[] in, int offset, int len, int k) {
2794 long kLong = k & LONG_MASK;
2795 long carry = 0;
2796
2797 offset = out.length-offset - 1;
2798 for (int j=len-1; j >= 0; j--) {
2799 long product = (in[j] & LONG_MASK) * kLong +
2800 (out[offset] & LONG_MASK) + carry;
2801 out[offset--] = (int)product;
2802 carry = product >>> 32;
2803 }
2804 return (int)carry;
2805 }
2806
2807 /**
2808 * Add one word to the number a mlen words into a. Return the resulting
2809 * carry.
2810 */
2811 static int addOne(int[] a, int offset, int mlen, int carry) {
2812 offset = a.length-1-mlen-offset;
2813 long t = (a[offset] & LONG_MASK) + (carry & LONG_MASK);
2814
2815 a[offset] = (int)t;
2816 if ((t >>> 32) == 0)
2817 return 0;
2818 while (--mlen >= 0) {
2819 if (--offset < 0) { // Carry out of number
2820 return 1;
2821 } else {
2822 a[offset]++;
2823 if (a[offset] != 0)
2824 return 0;
2825 }
2826 }
2827 return 1;
2828 }
2829
2830 /**
2831 * Returns a BigInteger whose value is (this ** exponent) mod (2**p)
2832 */
2833 private BigInteger modPow2(BigInteger exponent, int p) {
2834 /*
2835 * Perform exponentiation using repeated squaring trick, chopping off
2836 * high order bits as indicated by modulus.
2837 */
2838 BigInteger result = ONE;
2839 BigInteger baseToPow2 = this.mod2(p);
2840 int expOffset = 0;
2841
2842 int limit = exponent.bitLength();
2843
2844 if (this.testBit(0))
2845 limit = (p-1) < limit ? (p-1) : limit;
2846
2847 while (expOffset < limit) {
2848 if (exponent.testBit(expOffset))
2849 result = result.multiply(baseToPow2).mod2(p);
2850 expOffset++;
2851 if (expOffset < limit)
2852 baseToPow2 = baseToPow2.square().mod2(p);
2853 }
2854
2855 return result;
2856 }
2857
2858 /**
2859 * Returns a BigInteger whose value is this mod(2**p).
2860 * Assumes that this {@code BigInteger >= 0} and {@code p > 0}.
2861 */
2862 private BigInteger mod2(int p) {
2863 if (bitLength() <= p)
2864 return this;
2865
2866 // Copy remaining ints of mag
2867 int numInts = (p + 31) >>> 5;
2868 int[] mag = new int[numInts];
2869 System.arraycopy(this.mag, (this.mag.length - numInts), mag, 0, numInts);
2870
2871 // Mask out any excess bits
2872 int excessBits = (numInts << 5) - p;
2873 mag[0] &= (1L << (32-excessBits)) - 1;
2874
2875 return (mag[0] == 0 ? new BigInteger(1, mag) : new BigInteger(mag, 1));
2876 }
2877
2878 /**
2879 * Returns a BigInteger whose value is {@code (this}<sup>-1</sup> {@code mod m)}.
2880 *
2881 * @param m the modulus.
2882 * @return {@code this}<sup>-1</sup> {@code mod m}.
2883 * @throws ArithmeticException {@code m} ≤ 0, or this BigInteger
2884 * has no multiplicative inverse mod m (that is, this BigInteger
2885 * is not <i>relatively prime</i> to m).
2886 */
2887 public BigInteger modInverse(BigInteger m) {
2888 if (m.signum != 1)
2889 throw new ArithmeticException("BigInteger: modulus not positive");
2890
2891 if (m.equals(ONE))
2892 return ZERO;
2893
2894 // Calculate (this mod m)
2895 BigInteger modVal = this;
2896 if (signum < 0 || (this.compareMagnitude(m) >= 0))
2897 modVal = this.mod(m);
2898
2899 if (modVal.equals(ONE))
2900 return ONE;
2901
2902 MutableBigInteger a = new MutableBigInteger(modVal);
2903 MutableBigInteger b = new MutableBigInteger(m);
2904
2905 MutableBigInteger result = a.mutableModInverse(b);
2906 return result.toBigInteger(1);
2907 }
2908
2909 // Shift Operations
2910
2911 /**
2912 * Returns a BigInteger whose value is {@code (this << n)}.
2913 * The shift distance, {@code n}, may be negative, in which case
2914 * this method performs a right shift.
2915 * (Computes <tt>floor(this * 2<sup>n</sup>)</tt>.)
2916 *
2917 * @param n shift distance, in bits.
2918 * @return {@code this << n}
2919 * @see #shiftRight
2920 */
2921 public BigInteger shiftLeft(int n) {
2922 if (signum == 0)
2923 return ZERO;
2924 if (n > 0) {
2925 return new BigInteger(shiftLeft(mag, n), signum);
2926 } else if (n == 0) {
2927 return this;
2928 } else {
2929 // Possible int overflow in (-n) is not a trouble,
2930 // because shiftRightImpl considers its argument unsigned
2931 return shiftRightImpl(-n);
2932 }
2933 }
2934
2935 /**
2936 * Returns a magnitude array whose value is {@code (mag << n)}.
2937 * The shift distance, {@code n}, is considered unnsigned.
2938 * (Computes <tt>this * 2<sup>n</sup></tt>.)
2939 *
2940 * @param mag magnitude, the most-significant int ({@code mag[0]}) must be non-zero.
2941 * @param n unsigned shift distance, in bits.
2942 * @return {@code mag << n}
2943 */
2944 private static int[] shiftLeft(int[] mag, int n) {
2945 int nInts = n >>> 5;
2946 int nBits = n & 0x1f;
2947 int magLen = mag.length;
2948 int newMag[] = null;
2949
2950 if (nBits == 0) {
2951 newMag = new int[magLen + nInts];
2952 System.arraycopy(mag, 0, newMag, 0, magLen);
2953 } else {
2954 int i = 0;
2955 int nBits2 = 32 - nBits;
2956 int highBits = mag[0] >>> nBits2;
2957 if (highBits != 0) {
2958 newMag = new int[magLen + nInts + 1];
2959 newMag[i++] = highBits;
2960 } else {
2961 newMag = new int[magLen + nInts];
2962 }
2963 int j=0;
2964 while (j < magLen-1)
2965 newMag[i++] = mag[j++] << nBits | mag[j] >>> nBits2;
2966 newMag[i] = mag[j] << nBits;
2967 }
2968 return newMag;
2969 }
2970
2971 /**
2972 * Returns a BigInteger whose value is {@code (this >> n)}. Sign
2973 * extension is performed. The shift distance, {@code n}, may be
2974 * negative, in which case this method performs a left shift.
2975 * (Computes <tt>floor(this / 2<sup>n</sup>)</tt>.)
2976 *
2977 * @param n shift distance, in bits.
2978 * @return {@code this >> n}
2979 * @see #shiftLeft
2980 */
2981 public BigInteger shiftRight(int n) {
2982 if (signum == 0)
2983 return ZERO;
2984 if (n > 0) {
2985 return shiftRightImpl(n);
2986 } else if (n == 0) {
2987 return this;
2988 } else {
2989 // Possible int overflow in {@code -n} is not a trouble,
2990 // because shiftLeft considers its argument unsigned
2991 return new BigInteger(shiftLeft(mag, -n), signum);
2992 }
2993 }
2994
2995 /**
2996 * Returns a BigInteger whose value is {@code (this >> n)}. The shift
2997 * distance, {@code n}, is considered unsigned.
2998 * (Computes <tt>floor(this * 2<sup>-n</sup>)</tt>.)
2999 *
3000 * @param n unsigned shift distance, in bits.
3001 * @return {@code this >> n}
3002 */
3003 private BigInteger shiftRightImpl(int n) {
3004 int nInts = n >>> 5;
3005 int nBits = n & 0x1f;
3006 int magLen = mag.length;
3007 int newMag[] = null;
3008
3009 // Special case: entire contents shifted off the end
3010 if (nInts >= magLen)
3011 return (signum >= 0 ? ZERO : negConst[1]);
3012
3013 if (nBits == 0) {
3014 int newMagLen = magLen - nInts;
3015 newMag = Arrays.copyOf(mag, newMagLen);
3016 } else {
3017 int i = 0;
3018 int highBits = mag[0] >>> nBits;
3019 if (highBits != 0) {
3020 newMag = new int[magLen - nInts];
3021 newMag[i++] = highBits;
3022 } else {
3023 newMag = new int[magLen - nInts -1];
3024 }
3025
3026 int nBits2 = 32 - nBits;
3027 int j=0;
3028 while (j < magLen - nInts - 1)
3029 newMag[i++] = (mag[j++] << nBits2) | (mag[j] >>> nBits);
3030 }
3031
3032 if (signum < 0) {
3033 // Find out whether any one-bits were shifted off the end.
3034 boolean onesLost = false;
3035 for (int i=magLen-1, j=magLen-nInts; i >= j && !onesLost; i--)
3036 onesLost = (mag[i] != 0);
3037 if (!onesLost && nBits != 0)
3038 onesLost = (mag[magLen - nInts - 1] << (32 - nBits) != 0);
3039
3040 if (onesLost)
3041 newMag = javaIncrement(newMag);
3042 }
3043
3044 return new BigInteger(newMag, signum);
3045 }
3046
3047 int[] javaIncrement(int[] val) {
3048 int lastSum = 0;
3049 for (int i=val.length-1; i >= 0 && lastSum == 0; i--)
3050 lastSum = (val[i] += 1);
3051 if (lastSum == 0) {
3052 val = new int[val.length+1];
3053 val[0] = 1;
3054 }
3055 return val;
3056 }
3057
3058 // Bitwise Operations
3059
3060 /**
3061 * Returns a BigInteger whose value is {@code (this & val)}. (This
3062 * method returns a negative BigInteger if and only if this and val are
3063 * both negative.)
3064 *
3065 * @param val value to be AND'ed with this BigInteger.
3066 * @return {@code this & val}
3067 */
3068 public BigInteger and(BigInteger val) {
3069 int[] result = new int[Math.max(intLength(), val.intLength())];
3070 for (int i=0; i < result.length; i++)
3071 result[i] = (getInt(result.length-i-1)
3072 & val.getInt(result.length-i-1));
3073
3074 return valueOf(result);
3075 }
3076
3077 /**
3078 * Returns a BigInteger whose value is {@code (this | val)}. (This method
3079 * returns a negative BigInteger if and only if either this or val is
3080 * negative.)
3081 *
3082 * @param val value to be OR'ed with this BigInteger.
3083 * @return {@code this | val}
3084 */
3085 public BigInteger or(BigInteger val) {
3086 int[] result = new int[Math.max(intLength(), val.intLength())];
3087 for (int i=0; i < result.length; i++)
3088 result[i] = (getInt(result.length-i-1)
3089 | val.getInt(result.length-i-1));
3090
3091 return valueOf(result);
3092 }
3093
3094 /**
3095 * Returns a BigInteger whose value is {@code (this ^ val)}. (This method
3096 * returns a negative BigInteger if and only if exactly one of this and
3097 * val are negative.)
3098 *
3099 * @param val value to be XOR'ed with this BigInteger.
3100 * @return {@code this ^ val}
3101 */
3102 public BigInteger xor(BigInteger val) {
3103 int[] result = new int[Math.max(intLength(), val.intLength())];
3104 for (int i=0; i < result.length; i++)
3105 result[i] = (getInt(result.length-i-1)
3106 ^ val.getInt(result.length-i-1));
3107
3108 return valueOf(result);
3109 }
3110
3111 /**
3112 * Returns a BigInteger whose value is {@code (~this)}. (This method
3113 * returns a negative value if and only if this BigInteger is
3114 * non-negative.)
3115 *
3116 * @return {@code ~this}
3117 */
3118 public BigInteger not() {
3119 int[] result = new int[intLength()];
3120 for (int i=0; i < result.length; i++)
3121 result[i] = ~getInt(result.length-i-1);
3122
3123 return valueOf(result);
3124 }
3125
3126 /**
3127 * Returns a BigInteger whose value is {@code (this & ~val)}. This
3128 * method, which is equivalent to {@code and(val.not())}, is provided as
3129 * a convenience for masking operations. (This method returns a negative
3130 * BigInteger if and only if {@code this} is negative and {@code val} is
3131 * positive.)
3132 *
3133 * @param val value to be complemented and AND'ed with this BigInteger.
3134 * @return {@code this & ~val}
3135 */
3136 public BigInteger andNot(BigInteger val) {
3137 int[] result = new int[Math.max(intLength(), val.intLength())];
3138 for (int i=0; i < result.length; i++)
3139 result[i] = (getInt(result.length-i-1)
3140 & ~val.getInt(result.length-i-1));
3141
3142 return valueOf(result);
3143 }
3144
3145
3146 // Single Bit Operations
3147
3148 /**
3149 * Returns {@code true} if and only if the designated bit is set.
3150 * (Computes {@code ((this & (1<<n)) != 0)}.)
3151 *
3152 * @param n index of bit to test.
3153 * @return {@code true} if and only if the designated bit is set.
3154 * @throws ArithmeticException {@code n} is negative.
3155 */
3156 public boolean testBit(int n) {
3157 if (n < 0)
3158 throw new ArithmeticException("Negative bit address");
3159
3160 return (getInt(n >>> 5) & (1 << (n & 31))) != 0;
3161 }
3162
3163 /**
3164 * Returns a BigInteger whose value is equivalent to this BigInteger
3165 * with the designated bit set. (Computes {@code (this | (1<<n))}.)
3166 *
3167 * @param n index of bit to set.
3168 * @return {@code this | (1<<n)}
3169 * @throws ArithmeticException {@code n} is negative.
3170 */
3171 public BigInteger setBit(int n) {
3172 if (n < 0)
3173 throw new ArithmeticException("Negative bit address");
3174
3175 int intNum = n >>> 5;
3176 int[] result = new int[Math.max(intLength(), intNum+2)];
3177
3178 for (int i=0; i < result.length; i++)
3179 result[result.length-i-1] = getInt(i);
3180
3181 result[result.length-intNum-1] |= (1 << (n & 31));
3182
3183 return valueOf(result);
3184 }
3185
3186 /**
3187 * Returns a BigInteger whose value is equivalent to this BigInteger
3188 * with the designated bit cleared.
3189 * (Computes {@code (this & ~(1<<n))}.)
3190 *
3191 * @param n index of bit to clear.
3192 * @return {@code this & ~(1<<n)}
3193 * @throws ArithmeticException {@code n} is negative.
3194 */
3195 public BigInteger clearBit(int n) {
3196 if (n < 0)
3197 throw new ArithmeticException("Negative bit address");
3198
3199 int intNum = n >>> 5;
3200 int[] result = new int[Math.max(intLength(), ((n + 1) >>> 5) + 1)];
3201
3202 for (int i=0; i < result.length; i++)
3203 result[result.length-i-1] = getInt(i);
3204
3205 result[result.length-intNum-1] &= ~(1 << (n & 31));
3206
3207 return valueOf(result);
3208 }
3209
3210 /**
3211 * Returns a BigInteger whose value is equivalent to this BigInteger
3212 * with the designated bit flipped.
3213 * (Computes {@code (this ^ (1<<n))}.)
3214 *
3215 * @param n index of bit to flip.
3216 * @return {@code this ^ (1<<n)}
3217 * @throws ArithmeticException {@code n} is negative.
3218 */
3219 public BigInteger flipBit(int n) {
3220 if (n < 0)
3221 throw new ArithmeticException("Negative bit address");
3222
3223 int intNum = n >>> 5;
3224 int[] result = new int[Math.max(intLength(), intNum+2)];
3225
3226 for (int i=0; i < result.length; i++)
3227 result[result.length-i-1] = getInt(i);
3228
3229 result[result.length-intNum-1] ^= (1 << (n & 31));
3230
3231 return valueOf(result);
3232 }
3233
3234 /**
3235 * Returns the index of the rightmost (lowest-order) one bit in this
3236 * BigInteger (the number of zero bits to the right of the rightmost
3237 * one bit). Returns -1 if this BigInteger contains no one bits.
3238 * (Computes {@code (this == 0? -1 : log2(this & -this))}.)
3239 *
3240 * @return index of the rightmost one bit in this BigInteger.
3241 */
3242 public int getLowestSetBit() {
3243 @SuppressWarnings("deprecation") int lsb = lowestSetBit - 2;
3244 if (lsb == -2) { // lowestSetBit not initialized yet
3245 lsb = 0;
3246 if (signum == 0) {
3247 lsb -= 1;
3248 } else {
3249 // Search for lowest order nonzero int
3250 int i,b;
3251 for (i=0; (b = getInt(i)) == 0; i++)
3252 ;
3253 lsb += (i << 5) + Integer.numberOfTrailingZeros(b);
3254 }
3255 lowestSetBit = lsb + 2;
3256 }
3257 return lsb;
3258 }
3259
3260
3261 // Miscellaneous Bit Operations
3262
3263 /**
3264 * Returns the number of bits in the minimal two's-complement
3265 * representation of this BigInteger, <i>excluding</i> a sign bit.
3266 * For positive BigIntegers, this is equivalent to the number of bits in
3267 * the ordinary binary representation. (Computes
3268 * {@code (ceil(log2(this < 0 ? -this : this+1)))}.)
3269 *
3270 * @return number of bits in the minimal two's-complement
3271 * representation of this BigInteger, <i>excluding</i> a sign bit.
3272 */
3273 public int bitLength() {
3274 @SuppressWarnings("deprecation") int n = bitLength - 1;
3275 if (n == -1) { // bitLength not initialized yet
3276 int[] m = mag;
3277 int len = m.length;
3278 if (len == 0) {
3279 n = 0; // offset by one to initialize
3280 } else {
3281 // Calculate the bit length of the magnitude
3282 int magBitLength = ((len - 1) << 5) + bitLengthForInt(mag[0]);
3283 if (signum < 0) {
3284 // Check if magnitude is a power of two
3285 boolean pow2 = (Integer.bitCount(mag[0]) == 1);
3286 for (int i=1; i< len && pow2; i++)
3287 pow2 = (mag[i] == 0);
3288
3289 n = (pow2 ? magBitLength -1 : magBitLength);
3290 } else {
3291 n = magBitLength;
3292 }
3293 }
3294 bitLength = n + 1;
3295 }
3296 return n;
3297 }
3298
3299 /**
3300 * Returns the number of bits in the two's complement representation
3301 * of this BigInteger that differ from its sign bit. This method is
3302 * useful when implementing bit-vector style sets atop BigIntegers.
3303 *
3304 * @return number of bits in the two's complement representation
3305 * of this BigInteger that differ from its sign bit.
3306 */
3307 public int bitCount() {
3308 @SuppressWarnings("deprecation") int bc = bitCount - 1;
3309 if (bc == -1) { // bitCount not initialized yet
3310 bc = 0; // offset by one to initialize
3311 // Count the bits in the magnitude
3312 for (int i=0; i < mag.length; i++)
3313 bc += Integer.bitCount(mag[i]);
3314 if (signum < 0) {
3315 // Count the trailing zeros in the magnitude
3316 int magTrailingZeroCount = 0, j;
3317 for (j=mag.length-1; mag[j] == 0; j--)
3318 magTrailingZeroCount += 32;
3319 magTrailingZeroCount += Integer.numberOfTrailingZeros(mag[j]);
3320 bc += magTrailingZeroCount - 1;
3321 }
3322 bitCount = bc + 1;
3323 }
3324 return bc;
3325 }
3326
3327 // Primality Testing
3328
3329 /**
3330 * Returns {@code true} if this BigInteger is probably prime,
3331 * {@code false} if it's definitely composite. If
3332 * {@code certainty} is ≤ 0, {@code true} is
3333 * returned.
3334 *
3335 * @param certainty a measure of the uncertainty that the caller is
3336 * willing to tolerate: if the call returns {@code true}
3337 * the probability that this BigInteger is prime exceeds
3338 * (1 - 1/2<sup>{@code certainty}</sup>). The execution time of
3339 * this method is proportional to the value of this parameter.
3340 * @return {@code true} if this BigInteger is probably prime,
3341 * {@code false} if it's definitely composite.
3342 */
3343 public boolean isProbablePrime(int certainty) {
3344 if (certainty <= 0)
3345 return true;
3346 BigInteger w = this.abs();
3347 if (w.equals(TWO))
3348 return true;
3349 if (!w.testBit(0) || w.equals(ONE))
3350 return false;
3351
3352 return w.primeToCertainty(certainty, null);
3353 }
3354
3355 // Comparison Operations
3356
3357 /**
3358 * Compares this BigInteger with the specified BigInteger. This
3359 * method is provided in preference to individual methods for each
3360 * of the six boolean comparison operators ({@literal <}, ==,
3361 * {@literal >}, {@literal >=}, !=, {@literal <=}). The suggested
3362 * idiom for performing these comparisons is: {@code
3363 * (x.compareTo(y)} <<i>op</i>> {@code 0)}, where
3364 * <<i>op</i>> is one of the six comparison operators.
3365 *
3366 * @param val BigInteger to which this BigInteger is to be compared.
3367 * @return -1, 0 or 1 as this BigInteger is numerically less than, equal
3368 * to, or greater than {@code val}.
3369 */
3370 public int compareTo(BigInteger val) {
3371 if (signum == val.signum) {
3372 switch (signum) {
3373 case 1:
3374 return compareMagnitude(val);
3375 case -1:
3376 return val.compareMagnitude(this);
3377 default:
3378 return 0;
3379 }
3380 }
3381 return signum > val.signum ? 1 : -1;
3382 }
3383
3384 /**
3385 * Compares the magnitude array of this BigInteger with the specified
3386 * BigInteger's. This is the version of compareTo ignoring sign.
3387 *
3388 * @param val BigInteger whose magnitude array to be compared.
3389 * @return -1, 0 or 1 as this magnitude array is less than, equal to or
3390 * greater than the magnitude aray for the specified BigInteger's.
3391 */
3392 final int compareMagnitude(BigInteger val) {
3393 int[] m1 = mag;
3394 int len1 = m1.length;
3395 int[] m2 = val.mag;
3396 int len2 = m2.length;
3397 if (len1 < len2)
3398 return -1;
3399 if (len1 > len2)
3400 return 1;
3401 for (int i = 0; i < len1; i++) {
3402 int a = m1[i];
3403 int b = m2[i];
3404 if (a != b)
3405 return ((a & LONG_MASK) < (b & LONG_MASK)) ? -1 : 1;
3406 }
3407 return 0;
3408 }
3409
3410 /**
3411 * Version of compareMagnitude that compares magnitude with long value.
3412 * val can't be Long.MIN_VALUE.
3413 */
3414 final int compareMagnitude(long val) {
3415 assert val != Long.MIN_VALUE;
3416 int[] m1 = mag;
3417 int len = m1.length;
3418 if (len > 2) {
3419 return 1;
3420 }
3421 if (val < 0) {
3422 val = -val;
3423 }
3424 int highWord = (int)(val >>> 32);
3425 if (highWord == 0) {
3426 if (len < 1)
3427 return -1;
3428 if (len > 1)
3429 return 1;
3430 int a = m1[0];
3431 int b = (int)val;
3432 if (a != b) {
3433 return ((a & LONG_MASK) < (b & LONG_MASK))? -1 : 1;
3434 }
3435 return 0;
3436 } else {
3437 if (len < 2)
3438 return -1;
3439 int a = m1[0];
3440 int b = highWord;
3441 if (a != b) {
3442 return ((a & LONG_MASK) < (b & LONG_MASK))? -1 : 1;
3443 }
3444 a = m1[1];
3445 b = (int)val;
3446 if (a != b) {
3447 return ((a & LONG_MASK) < (b & LONG_MASK))? -1 : 1;
3448 }
3449 return 0;
3450 }
3451 }
3452
3453 /**
3454 * Compares this BigInteger with the specified Object for equality.
3455 *
3456 * @param x Object to which this BigInteger is to be compared.
3457 * @return {@code true} if and only if the specified Object is a
3458 * BigInteger whose value is numerically equal to this BigInteger.
3459 */
3460 public boolean equals(Object x) {
3461 // This test is just an optimization, which may or may not help
3462 if (x == this)
3463 return true;
3464
3465 if (!(x instanceof BigInteger))
3466 return false;
3467
3468 BigInteger xInt = (BigInteger) x;
3469 if (xInt.signum != signum)
3470 return false;
3471
3472 int[] m = mag;
3473 int len = m.length;
3474 int[] xm = xInt.mag;
3475 if (len != xm.length)
3476 return false;
3477
3478 for (int i = 0; i < len; i++)
3479 if (xm[i] != m[i])
3480 return false;
3481
3482 return true;
3483 }
3484
3485 /**
3486 * Returns the minimum of this BigInteger and {@code val}.
3487 *
3488 * @param val value with which the minimum is to be computed.
3489 * @return the BigInteger whose value is the lesser of this BigInteger and
3490 * {@code val}. If they are equal, either may be returned.
3491 */
3492 public BigInteger min(BigInteger val) {
3493 return (compareTo(val) < 0 ? this : val);
3494 }
3495
3496 /**
3497 * Returns the maximum of this BigInteger and {@code val}.
3498 *
3499 * @param val value with which the maximum is to be computed.
3500 * @return the BigInteger whose value is the greater of this and
3501 * {@code val}. If they are equal, either may be returned.
3502 */
3503 public BigInteger max(BigInteger val) {
3504 return (compareTo(val) > 0 ? this : val);
3505 }
3506
3507
3508 // Hash Function
3509
3510 /**
3511 * Returns the hash code for this BigInteger.
3512 *
3513 * @return hash code for this BigInteger.
3514 */
3515 public int hashCode() {
3516 int hashCode = 0;
3517
3518 for (int i=0; i < mag.length; i++)
3519 hashCode = (int)(31*hashCode + (mag[i] & LONG_MASK));
3520
3521 return hashCode * signum;
3522 }
3523
3524 /**
3525 * Returns the String representation of this BigInteger in the
3526 * given radix. If the radix is outside the range from {@link
3527 * Character#MIN_RADIX} to {@link Character#MAX_RADIX} inclusive,
3528 * it will default to 10 (as is the case for
3529 * {@code Integer.toString}). The digit-to-character mapping
3530 * provided by {@code Character.forDigit} is used, and a minus
3531 * sign is prepended if appropriate. (This representation is
3532 * compatible with the {@link #BigInteger(String, int) (String,
3533 * int)} constructor.)
3534 *
3535 * @param radix radix of the String representation.
3536 * @return String representation of this BigInteger in the given radix.
3537 * @see Integer#toString
3538 * @see Character#forDigit
3539 * @see #BigInteger(java.lang.String, int)
3540 */
3541 public String toString(int radix) {
3542 if (signum == 0)
3543 return "0";
3544 if (radix < Character.MIN_RADIX || radix > Character.MAX_RADIX)
3545 radix = 10;
3546
3547 // If it's small enough, use smallToString.
3548 if (mag.length <= SCHOENHAGE_BASE_CONVERSION_THRESHOLD)
3549 return smallToString(radix);
3550
3551 // Otherwise use recursive toString, which requires positive arguments.
3552 // The results will be concatenated into this StringBuilder
3553 StringBuilder sb = new StringBuilder();
3554 if (signum < 0) {
3555 toString(this.negate(), sb, radix, 0);
3556 sb.insert(0, '-');
3557 }
3558 else
3559 toString(this, sb, radix, 0);
3560
3561 return sb.toString();
3562 }
3563
3564 /** This method is used to perform toString when arguments are small. */
3565 private String smallToString(int radix) {
3566 if (signum == 0) {
3567 return "0";
3568 }
3569
3570 // Compute upper bound on number of digit groups and allocate space
3571 int maxNumDigitGroups = (4*mag.length + 6)/7;
3572 String digitGroup[] = new String[maxNumDigitGroups];
3573
3574 // Translate number to string, a digit group at a time
3575 BigInteger tmp = this.abs();
3576 int numGroups = 0;
3577 while (tmp.signum != 0) {
3578 BigInteger d = longRadix[radix];
3579
3580 MutableBigInteger q = new MutableBigInteger(),
3581 a = new MutableBigInteger(tmp.mag),
3582 b = new MutableBigInteger(d.mag);
3583 MutableBigInteger r = a.divide(b, q);
3584 BigInteger q2 = q.toBigInteger(tmp.signum * d.signum);
3585 BigInteger r2 = r.toBigInteger(tmp.signum * d.signum);
3586
3587 digitGroup[numGroups++] = Long.toString(r2.longValue(), radix);
3588 tmp = q2;
3589 }
3590
3591 // Put sign (if any) and first digit group into result buffer
3592 StringBuilder buf = new StringBuilder(numGroups*digitsPerLong[radix]+1);
3593 if (signum < 0) {
3594 buf.append('-');
3595 }
3596 buf.append(digitGroup[numGroups-1]);
3597
3598 // Append remaining digit groups padded with leading zeros
3599 for (int i=numGroups-2; i >= 0; i--) {
3600 // Prepend (any) leading zeros for this digit group
3601 int numLeadingZeros = digitsPerLong[radix]-digitGroup[i].length();
3602 if (numLeadingZeros != 0) {
3603 buf.append(zeros[numLeadingZeros]);
3604 }
3605 buf.append(digitGroup[i]);
3606 }
3607 return buf.toString();
3608 }
3609
3610 /**
3611 * Converts the specified BigInteger to a string and appends to
3612 * {@code sb}. This implements the recursive Schoenhage algorithm
3613 * for base conversions.
3614 * <p/>
3615 * See Knuth, Donald, _The Art of Computer Programming_, Vol. 2,
3616 * Answers to Exercises (4.4) Question 14.
3617 *
3618 * @param u The number to convert to a string.
3619 * @param sb The StringBuilder that will be appended to in place.
3620 * @param radix The base to convert to.
3621 * @param digits The minimum number of digits to pad to.
3622 */
3623 private static void toString(BigInteger u, StringBuilder sb, int radix,
3624 int digits) {
3625 /* If we're smaller than a certain threshold, use the smallToString
3626 method, padding with leading zeroes when necessary. */
3627 if (u.mag.length <= SCHOENHAGE_BASE_CONVERSION_THRESHOLD) {
3628 String s = u.smallToString(radix);
3629
3630 // Pad with internal zeros if necessary.
3631 // Don't pad if we're at the beginning of the string.
3632 if ((s.length() < digits) && (sb.length() > 0)) {
3633 for (int i=s.length(); i < digits; i++) { // May be a faster way to
3634 sb.append('0'); // do this?
3635 }
3636 }
3637
3638 sb.append(s);
3639 return;
3640 }
3641
3642 int b, n;
3643 b = u.bitLength();
3644
3645 // Calculate a value for n in the equation radix^(2^n) = u
3646 // and subtract 1 from that value. This is used to find the
3647 // cache index that contains the best value to divide u.
3648 n = (int) Math.round(Math.log(b * LOG_TWO / logCache[radix]) / LOG_TWO - 1.0);
3649 BigInteger v = getRadixConversionCache(radix, n);
3650 BigInteger[] results;
3651 results = u.divideAndRemainder(v);
3652
3653 int expectedDigits = 1 << n;
3654
3655 // Now recursively build the two halves of each number.
3656 toString(results[0], sb, radix, digits-expectedDigits);
3657 toString(results[1], sb, radix, expectedDigits);
3658 }
3659
3660 /**
3661 * Returns the value radix^(2^exponent) from the cache.
3662 * If this value doesn't already exist in the cache, it is added.
3663 * <p/>
3664 * This could be changed to a more complicated caching method using
3665 * {@code Future}.
3666 */
3667 private static BigInteger getRadixConversionCache(int radix, int exponent) {
3668 BigInteger[] cacheLine = powerCache[radix]; // volatile read
3669 if (exponent < cacheLine.length) {
3670 return cacheLine[exponent];
3671 }
3672
3673 int oldLength = cacheLine.length;
3674 cacheLine = Arrays.copyOf(cacheLine, exponent + 1);
3675 for (int i = oldLength; i <= exponent; i++) {
3676 cacheLine[i] = cacheLine[i - 1].pow(2);
3677 }
3678
3679 BigInteger[][] pc = powerCache; // volatile read again
3680 if (exponent >= pc[radix].length) {
3681 pc = pc.clone();
3682 pc[radix] = cacheLine;
3683 powerCache = pc; // volatile write, publish
3684 }
3685 return cacheLine[exponent];
3686 }
3687
3688 /* zero[i] is a string of i consecutive zeros. */
3689 private static String zeros[] = new String[64];
3690 static {
3691 zeros[63] =
3692 "000000000000000000000000000000000000000000000000000000000000000";
3693 for (int i=0; i < 63; i++)
3694 zeros[i] = zeros[63].substring(0, i);
3695 }
3696
3697 /**
3698 * Returns the decimal String representation of this BigInteger.
3699 * The digit-to-character mapping provided by
3700 * {@code Character.forDigit} is used, and a minus sign is
3701 * prepended if appropriate. (This representation is compatible
3702 * with the {@link #BigInteger(String) (String)} constructor, and
3703 * allows for String concatenation with Java's + operator.)
3704 *
3705 * @return decimal String representation of this BigInteger.
3706 * @see Character#forDigit
3707 * @see #BigInteger(java.lang.String)
3708 */
3709 public String toString() {
3710 return toString(10);
3711 }
3712
3713 /**
3714 * Returns a byte array containing the two's-complement
3715 * representation of this BigInteger. The byte array will be in
3716 * <i>big-endian</i> byte-order: the most significant byte is in
3717 * the zeroth element. The array will contain the minimum number
3718 * of bytes required to represent this BigInteger, including at
3719 * least one sign bit, which is {@code (ceil((this.bitLength() +
3720 * 1)/8))}. (This representation is compatible with the
3721 * {@link #BigInteger(byte[]) (byte[])} constructor.)
3722 *
3723 * @return a byte array containing the two's-complement representation of
3724 * this BigInteger.
3725 * @see #BigInteger(byte[])
3726 */
3727 public byte[] toByteArray() {
3728 int byteLen = bitLength()/8 + 1;
3729 byte[] byteArray = new byte[byteLen];
3730
3731 for (int i=byteLen-1, bytesCopied=4, nextInt=0, intIndex=0; i >= 0; i--) {
3732 if (bytesCopied == 4) {
3733 nextInt = getInt(intIndex++);
3734 bytesCopied = 1;
3735 } else {
3736 nextInt >>>= 8;
3737 bytesCopied++;
3738 }
3739 byteArray[i] = (byte)nextInt;
3740 }
3741 return byteArray;
3742 }
3743
3744 /**
3745 * Converts this BigInteger to an {@code int}. This
3746 * conversion is analogous to a
3747 * <i>narrowing primitive conversion</i> from {@code long} to
3748 * {@code int} as defined in section 5.1.3 of
3749 * <cite>The Java™ Language Specification</cite>:
3750 * if this BigInteger is too big to fit in an
3751 * {@code int}, only the low-order 32 bits are returned.
3752 * Note that this conversion can lose information about the
3753 * overall magnitude of the BigInteger value as well as return a
3754 * result with the opposite sign.
3755 *
3756 * @return this BigInteger converted to an {@code int}.
3757 * @see #intValueExact()
3758 */
3759 public int intValue() {
3760 int result = 0;
3761 result = getInt(0);
3762 return result;
3763 }
3764
3765 /**
3766 * Converts this BigInteger to a {@code long}. This
3767 * conversion is analogous to a
3768 * <i>narrowing primitive conversion</i> from {@code long} to
3769 * {@code int} as defined in section 5.1.3 of
3770 * <cite>The Java™ Language Specification</cite>:
3771 * if this BigInteger is too big to fit in a
3772 * {@code long}, only the low-order 64 bits are returned.
3773 * Note that this conversion can lose information about the
3774 * overall magnitude of the BigInteger value as well as return a
3775 * result with the opposite sign.
3776 *
3777 * @return this BigInteger converted to a {@code long}.
3778 * @see #longValueExact()
3779 */
3780 public long longValue() {
3781 long result = 0;
3782
3783 for (int i=1; i >= 0; i--)
3784 result = (result << 32) + (getInt(i) & LONG_MASK);
3785 return result;
3786 }
3787
3788 /**
3789 * Converts this BigInteger to a {@code float}. This
3790 * conversion is similar to the
3791 * <i>narrowing primitive conversion</i> from {@code double} to
3792 * {@code float} as defined in section 5.1.3 of
3793 * <cite>The Java™ Language Specification</cite>:
3794 * if this BigInteger has too great a magnitude
3795 * to represent as a {@code float}, it will be converted to
3796 * {@link Float#NEGATIVE_INFINITY} or {@link
3797 * Float#POSITIVE_INFINITY} as appropriate. Note that even when
3798 * the return value is finite, this conversion can lose
3799 * information about the precision of the BigInteger value.
3800 *
3801 * @return this BigInteger converted to a {@code float}.
3802 */
3803 public float floatValue() {
3804 if (signum == 0) {
3805 return 0.0f;
3806 }
3807
3808 int exponent = ((mag.length - 1) << 5) + bitLengthForInt(mag[0]) - 1;
3809
3810 // exponent == floor(log2(abs(this)))
3811 if (exponent < Long.SIZE - 1) {
3812 return longValue();
3813 } else if (exponent > Float.MAX_EXPONENT) {
3814 return signum > 0 ? Float.POSITIVE_INFINITY : Float.NEGATIVE_INFINITY;
3815 }
3816
3817 /*
3818 * We need the top SIGNIFICAND_WIDTH bits, including the "implicit"
3819 * one bit. To make rounding easier, we pick out the top
3820 * SIGNIFICAND_WIDTH + 1 bits, so we have one to help us round up or
3821 * down. twiceSignifFloor will contain the top SIGNIFICAND_WIDTH + 1
3822 * bits, and signifFloor the top SIGNIFICAND_WIDTH.
3823 *
3824 * It helps to consider the real number signif = abs(this) *
3825 * 2^(SIGNIFICAND_WIDTH - 1 - exponent).
3826 */
3827 int shift = exponent - FloatConsts.SIGNIFICAND_WIDTH;
3828
3829 int twiceSignifFloor;
3830 // twiceSignifFloor will be == abs().shiftRight(shift).intValue()
3831 // We do the shift into an int directly to improve performance.
3832
3833 int nBits = shift & 0x1f;
3834 int nBits2 = 32 - nBits;
3835
3836 if (nBits == 0) {
3837 twiceSignifFloor = mag[0];
3838 } else {
3839 twiceSignifFloor = mag[0] >>> nBits;
3840 if (twiceSignifFloor == 0) {
3841 twiceSignifFloor = (mag[0] << nBits2) | (mag[1] >>> nBits);
3842 }
3843 }
3844
3845 int signifFloor = twiceSignifFloor >> 1;
3846 signifFloor &= FloatConsts.SIGNIF_BIT_MASK; // remove the implied bit
3847
3848 /*
3849 * We round up if either the fractional part of signif is strictly
3850 * greater than 0.5 (which is true if the 0.5 bit is set and any lower
3851 * bit is set), or if the fractional part of signif is >= 0.5 and
3852 * signifFloor is odd (which is true if both the 0.5 bit and the 1 bit
3853 * are set). This is equivalent to the desired HALF_EVEN rounding.
3854 */
3855 boolean increment = (twiceSignifFloor & 1) != 0
3856 && ((signifFloor & 1) != 0 || abs().getLowestSetBit() < shift);
3857 int signifRounded = increment ? signifFloor + 1 : signifFloor;
3858 int bits = ((exponent + FloatConsts.EXP_BIAS))
3859 << (FloatConsts.SIGNIFICAND_WIDTH - 1);
3860 bits += signifRounded;
3861 /*
3862 * If signifRounded == 2^24, we'd need to set all of the significand
3863 * bits to zero and add 1 to the exponent. This is exactly the behavior
3864 * we get from just adding signifRounded to bits directly. If the
3865 * exponent is Float.MAX_EXPONENT, we round up (correctly) to
3866 * Float.POSITIVE_INFINITY.
3867 */
3868 bits |= signum & FloatConsts.SIGN_BIT_MASK;
3869 return Float.intBitsToFloat(bits);
3870 }
3871
3872 /**
3873 * Converts this BigInteger to a {@code double}. This
3874 * conversion is similar to the
3875 * <i>narrowing primitive conversion</i> from {@code double} to
3876 * {@code float} as defined in section 5.1.3 of
3877 * <cite>The Java™ Language Specification</cite>:
3878 * if this BigInteger has too great a magnitude
3879 * to represent as a {@code double}, it will be converted to
3880 * {@link Double#NEGATIVE_INFINITY} or {@link
3881 * Double#POSITIVE_INFINITY} as appropriate. Note that even when
3882 * the return value is finite, this conversion can lose
3883 * information about the precision of the BigInteger value.
3884 *
3885 * @return this BigInteger converted to a {@code double}.
3886 */
3887 public double doubleValue() {
3888 if (signum == 0) {
3889 return 0.0;
3890 }
3891
3892 int exponent = ((mag.length - 1) << 5) + bitLengthForInt(mag[0]) - 1;
3893
3894 // exponent == floor(log2(abs(this))Double)
3895 if (exponent < Long.SIZE - 1) {
3896 return longValue();
3897 } else if (exponent > Double.MAX_EXPONENT) {
3898 return signum > 0 ? Double.POSITIVE_INFINITY : Double.NEGATIVE_INFINITY;
3899 }
3900
3901 /*
3902 * We need the top SIGNIFICAND_WIDTH bits, including the "implicit"
3903 * one bit. To make rounding easier, we pick out the top
3904 * SIGNIFICAND_WIDTH + 1 bits, so we have one to help us round up or
3905 * down. twiceSignifFloor will contain the top SIGNIFICAND_WIDTH + 1
3906 * bits, and signifFloor the top SIGNIFICAND_WIDTH.
3907 *
3908 * It helps to consider the real number signif = abs(this) *
3909 * 2^(SIGNIFICAND_WIDTH - 1 - exponent).
3910 */
3911 int shift = exponent - DoubleConsts.SIGNIFICAND_WIDTH;
3912
3913 long twiceSignifFloor;
3914 // twiceSignifFloor will be == abs().shiftRight(shift).longValue()
3915 // We do the shift into a long directly to improve performance.
3916
3917 int nBits = shift & 0x1f;
3918 int nBits2 = 32 - nBits;
3919
3920 int highBits;
3921 int lowBits;
3922 if (nBits == 0) {
3923 highBits = mag[0];
3924 lowBits = mag[1];
3925 } else {
3926 highBits = mag[0] >>> nBits;
3927 lowBits = (mag[0] << nBits2) | (mag[1] >>> nBits);
3928 if (highBits == 0) {
3929 highBits = lowBits;
3930 lowBits = (mag[1] << nBits2) | (mag[2] >>> nBits);
3931 }
3932 }
3933
3934 twiceSignifFloor = ((highBits & LONG_MASK) << 32)
3935 | (lowBits & LONG_MASK);
3936
3937 long signifFloor = twiceSignifFloor >> 1;
3938 signifFloor &= DoubleConsts.SIGNIF_BIT_MASK; // remove the implied bit
3939
3940 /*
3941 * We round up if either the fractional part of signif is strictly
3942 * greater than 0.5 (which is true if the 0.5 bit is set and any lower
3943 * bit is set), or if the fractional part of signif is >= 0.5 and
3944 * signifFloor is odd (which is true if both the 0.5 bit and the 1 bit
3945 * are set). This is equivalent to the desired HALF_EVEN rounding.
3946 */
3947 boolean increment = (twiceSignifFloor & 1) != 0
3948 && ((signifFloor & 1) != 0 || abs().getLowestSetBit() < shift);
3949 long signifRounded = increment ? signifFloor + 1 : signifFloor;
3950 long bits = (long) ((exponent + DoubleConsts.EXP_BIAS))
3951 << (DoubleConsts.SIGNIFICAND_WIDTH - 1);
3952 bits += signifRounded;
3953 /*
3954 * If signifRounded == 2^53, we'd need to set all of the significand
3955 * bits to zero and add 1 to the exponent. This is exactly the behavior
3956 * we get from just adding signifRounded to bits directly. If the
3957 * exponent is Double.MAX_EXPONENT, we round up (correctly) to
3958 * Double.POSITIVE_INFINITY.
3959 */
3960 bits |= signum & DoubleConsts.SIGN_BIT_MASK;
3961 return Double.longBitsToDouble(bits);
3962 }
3963
3964 /**
3965 * Returns a copy of the input array stripped of any leading zero bytes.
3966 */
3967 private static int[] stripLeadingZeroInts(int val[]) {
3968 int vlen = val.length;
3969 int keep;
3970
3971 // Find first nonzero byte
3972 for (keep = 0; keep < vlen && val[keep] == 0; keep++)
3973 ;
3974 return java.util.Arrays.copyOfRange(val, keep, vlen);
3975 }
3976
3977 /**
3978 * Returns the input array stripped of any leading zero bytes.
3979 * Since the source is trusted the copying may be skipped.
3980 */
3981 private static int[] trustedStripLeadingZeroInts(int val[]) {
3982 int vlen = val.length;
3983 int keep;
3984
3985 // Find first nonzero byte
3986 for (keep = 0; keep < vlen && val[keep] == 0; keep++)
3987 ;
3988 return keep == 0 ? val : java.util.Arrays.copyOfRange(val, keep, vlen);
3989 }
3990
3991 /**
3992 * Returns a copy of the input array stripped of any leading zero bytes.
3993 */
3994 private static int[] stripLeadingZeroBytes(byte a[]) {
3995 int byteLength = a.length;
3996 int keep;
3997
3998 // Find first nonzero byte
3999 for (keep = 0; keep < byteLength && a[keep] == 0; keep++)
4000 ;
4001
4002 // Allocate new array and copy relevant part of input array
4003 int intLength = ((byteLength - keep) + 3) >>> 2;
4004 int[] result = new int[intLength];
4005 int b = byteLength - 1;
4006 for (int i = intLength-1; i >= 0; i--) {
4007 result[i] = a[b--] & 0xff;
4008 int bytesRemaining = b - keep + 1;
4009 int bytesToTransfer = Math.min(3, bytesRemaining);
4010 for (int j=8; j <= (bytesToTransfer << 3); j += 8)
4011 result[i] |= ((a[b--] & 0xff) << j);
4012 }
4013 return result;
4014 }
4015
4016 /**
4017 * Takes an array a representing a negative 2's-complement number and
4018 * returns the minimal (no leading zero bytes) unsigned whose value is -a.
4019 */
4020 private static int[] makePositive(byte a[]) {
4021 int keep, k;
4022 int byteLength = a.length;
4023
4024 // Find first non-sign (0xff) byte of input
4025 for (keep=0; keep < byteLength && a[keep] == -1; keep++)
4026 ;
4027
4028
4029 /* Allocate output array. If all non-sign bytes are 0x00, we must
4030 * allocate space for one extra output byte. */
4031 for (k=keep; k < byteLength && a[k] == 0; k++)
4032 ;
4033
4034 int extraByte = (k == byteLength) ? 1 : 0;
4035 int intLength = ((byteLength - keep + extraByte) + 3) >>> 2;
4036 int result[] = new int[intLength];
4037
4038 /* Copy one's complement of input into output, leaving extra
4039 * byte (if it exists) == 0x00 */
4040 int b = byteLength - 1;
4041 for (int i = intLength-1; i >= 0; i--) {
4042 result[i] = a[b--] & 0xff;
4043 int numBytesToTransfer = Math.min(3, b-keep+1);
4044 if (numBytesToTransfer < 0)
4045 numBytesToTransfer = 0;
4046 for (int j=8; j <= 8*numBytesToTransfer; j += 8)
4047 result[i] |= ((a[b--] & 0xff) << j);
4048
4049 // Mask indicates which bits must be complemented
4050 int mask = -1 >>> (8*(3-numBytesToTransfer));
4051 result[i] = ~result[i] & mask;
4052 }
4053
4054 // Add one to one's complement to generate two's complement
4055 for (int i=result.length-1; i >= 0; i--) {
4056 result[i] = (int)((result[i] & LONG_MASK) + 1);
4057 if (result[i] != 0)
4058 break;
4059 }
4060
4061 return result;
4062 }
4063
4064 /**
4065 * Takes an array a representing a negative 2's-complement number and
4066 * returns the minimal (no leading zero ints) unsigned whose value is -a.
4067 */
4068 private static int[] makePositive(int a[]) {
4069 int keep, j;
4070
4071 // Find first non-sign (0xffffffff) int of input
4072 for (keep=0; keep < a.length && a[keep] == -1; keep++)
4073 ;
4074
4075 /* Allocate output array. If all non-sign ints are 0x00, we must
4076 * allocate space for one extra output int. */
4077 for (j=keep; j < a.length && a[j] == 0; j++)
4078 ;
4079 int extraInt = (j == a.length ? 1 : 0);
4080 int result[] = new int[a.length - keep + extraInt];
4081
4082 /* Copy one's complement of input into output, leaving extra
4083 * int (if it exists) == 0x00 */
4084 for (int i = keep; i < a.length; i++)
4085 result[i - keep + extraInt] = ~a[i];
4086
4087 // Add one to one's complement to generate two's complement
4088 for (int i=result.length-1; ++result[i] == 0; i--)
4089 ;
4090
4091 return result;
4092 }
4093
4094 /*
4095 * The following two arrays are used for fast String conversions. Both
4096 * are indexed by radix. The first is the number of digits of the given
4097 * radix that can fit in a Java long without "going negative", i.e., the
4098 * highest integer n such that radix**n < 2**63. The second is the
4099 * "long radix" that tears each number into "long digits", each of which
4100 * consists of the number of digits in the corresponding element in
4101 * digitsPerLong (longRadix[i] = i**digitPerLong[i]). Both arrays have
4102 * nonsense values in their 0 and 1 elements, as radixes 0 and 1 are not
4103 * used.
4104 */
4105 private static int digitsPerLong[] = {0, 0,
4106 62, 39, 31, 27, 24, 22, 20, 19, 18, 18, 17, 17, 16, 16, 15, 15, 15, 14,
4107 14, 14, 14, 13, 13, 13, 13, 13, 13, 12, 12, 12, 12, 12, 12, 12, 12};
4108
4109 private static BigInteger longRadix[] = {null, null,
4110 valueOf(0x4000000000000000L), valueOf(0x383d9170b85ff80bL),
4111 valueOf(0x4000000000000000L), valueOf(0x6765c793fa10079dL),
4112 valueOf(0x41c21cb8e1000000L), valueOf(0x3642798750226111L),
4113 valueOf(0x1000000000000000L), valueOf(0x12bf307ae81ffd59L),
4114 valueOf( 0xde0b6b3a7640000L), valueOf(0x4d28cb56c33fa539L),
4115 valueOf(0x1eca170c00000000L), valueOf(0x780c7372621bd74dL),
4116 valueOf(0x1e39a5057d810000L), valueOf(0x5b27ac993df97701L),
4117 valueOf(0x1000000000000000L), valueOf(0x27b95e997e21d9f1L),
4118 valueOf(0x5da0e1e53c5c8000L), valueOf( 0xb16a458ef403f19L),
4119 valueOf(0x16bcc41e90000000L), valueOf(0x2d04b7fdd9c0ef49L),
4120 valueOf(0x5658597bcaa24000L), valueOf( 0x6feb266931a75b7L),
4121 valueOf( 0xc29e98000000000L), valueOf(0x14adf4b7320334b9L),
4122 valueOf(0x226ed36478bfa000L), valueOf(0x383d9170b85ff80bL),
4123 valueOf(0x5a3c23e39c000000L), valueOf( 0x4e900abb53e6b71L),
4124 valueOf( 0x7600ec618141000L), valueOf( 0xaee5720ee830681L),
4125 valueOf(0x1000000000000000L), valueOf(0x172588ad4f5f0981L),
4126 valueOf(0x211e44f7d02c1000L), valueOf(0x2ee56725f06e5c71L),
4127 valueOf(0x41c21cb8e1000000L)};
4128
4129 /*
4130 * These two arrays are the integer analogue of above.
4131 */
4132 private static int digitsPerInt[] = {0, 0, 30, 19, 15, 13, 11,
4133 11, 10, 9, 9, 8, 8, 8, 8, 7, 7, 7, 7, 7, 7, 7, 6, 6, 6, 6,
4134 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 5};
4135
4136 private static int intRadix[] = {0, 0,
4137 0x40000000, 0x4546b3db, 0x40000000, 0x48c27395, 0x159fd800,
4138 0x75db9c97, 0x40000000, 0x17179149, 0x3b9aca00, 0xcc6db61,
4139 0x19a10000, 0x309f1021, 0x57f6c100, 0xa2f1b6f, 0x10000000,
4140 0x18754571, 0x247dbc80, 0x3547667b, 0x4c4b4000, 0x6b5a6e1d,
4141 0x6c20a40, 0x8d2d931, 0xb640000, 0xe8d4a51, 0x1269ae40,
4142 0x17179149, 0x1cb91000, 0x23744899, 0x2b73a840, 0x34e63b41,
4143 0x40000000, 0x4cfa3cc1, 0x5c13d840, 0x6d91b519, 0x39aa400
4144 };
4145
4146 /**
4147 * These routines provide access to the two's complement representation
4148 * of BigIntegers.
4149 */
4150
4151 /**
4152 * Returns the length of the two's complement representation in ints,
4153 * including space for at least one sign bit.
4154 */
4155 private int intLength() {
4156 return (bitLength() >>> 5) + 1;
4157 }
4158
4159 /* Returns sign bit */
4160 private int signBit() {
4161 return signum < 0 ? 1 : 0;
4162 }
4163
4164 /* Returns an int of sign bits */
4165 private int signInt() {
4166 return signum < 0 ? -1 : 0;
4167 }
4168
4169 /**
4170 * Returns the specified int of the little-endian two's complement
4171 * representation (int 0 is the least significant). The int number can
4172 * be arbitrarily high (values are logically preceded by infinitely many
4173 * sign ints).
4174 */
4175 private int getInt(int n) {
4176 if (n < 0)
4177 return 0;
4178 if (n >= mag.length)
4179 return signInt();
4180
4181 int magInt = mag[mag.length-n-1];
4182
4183 return (signum >= 0 ? magInt :
4184 (n <= firstNonzeroIntNum() ? -magInt : ~magInt));
4185 }
4186
4187 /**
4188 * Returns the index of the int that contains the first nonzero int in the
4189 * little-endian binary representation of the magnitude (int 0 is the
4190 * least significant). If the magnitude is zero, return value is undefined.
4191 */
4192 private int firstNonzeroIntNum() {
4193 int fn = firstNonzeroIntNum - 2;
4194 if (fn == -2) { // firstNonzeroIntNum not initialized yet
4195 fn = 0;
4196
4197 // Search for the first nonzero int
4198 int i;
4199 int mlen = mag.length;
4200 for (i = mlen - 1; i >= 0 && mag[i] == 0; i--)
4201 ;
4202 fn = mlen - i - 1;
4203 firstNonzeroIntNum = fn + 2; // offset by two to initialize
4204 }
4205 return fn;
4206 }
4207
4208 /** use serialVersionUID from JDK 1.1. for interoperability */
4209 private static final long serialVersionUID = -8287574255936472291L;
4210
4211 /**
4212 * Serializable fields for BigInteger.
4213 *
4214 * @serialField signum int
4215 * signum of this BigInteger.
4216 * @serialField magnitude int[]
4217 * magnitude array of this BigInteger.
4218 * @serialField bitCount int
4219 * number of bits in this BigInteger
4220 * @serialField bitLength int
4221 * the number of bits in the minimal two's-complement
4222 * representation of this BigInteger
4223 * @serialField lowestSetBit int
4224 * lowest set bit in the twos complement representation
4225 */
4226 private static final ObjectStreamField[] serialPersistentFields = {
4227 new ObjectStreamField("signum", Integer.TYPE),
4228 new ObjectStreamField("magnitude", byte[].class),
4229 new ObjectStreamField("bitCount", Integer.TYPE),
4230 new ObjectStreamField("bitLength", Integer.TYPE),
4231 new ObjectStreamField("firstNonzeroByteNum", Integer.TYPE),
4232 new ObjectStreamField("lowestSetBit", Integer.TYPE)
4233 };
4234
4235 /**
4236 * Reconstitute the {@code BigInteger} instance from a stream (that is,
4237 * deserialize it). The magnitude is read in as an array of bytes
4238 * for historical reasons, but it is converted to an array of ints
4239 * and the byte array is discarded.
4240 * Note:
4241 * The current convention is to initialize the cache fields, bitCount,
4242 * bitLength and lowestSetBit, to 0 rather than some other marker value.
4243 * Therefore, no explicit action to set these fields needs to be taken in
4244 * readObject because those fields already have a 0 value be default since
4245 * defaultReadObject is not being used.
4246 */
4247 private void readObject(java.io.ObjectInputStream s)
4248 throws java.io.IOException, ClassNotFoundException {
4249 /*
4250 * In order to maintain compatibility with previous serialized forms,
4251 * the magnitude of a BigInteger is serialized as an array of bytes.
4252 * The magnitude field is used as a temporary store for the byte array
4253 * that is deserialized. The cached computation fields should be
4254 * transient but are serialized for compatibility reasons.
4255 */
4256
4257 // prepare to read the alternate persistent fields
4258 ObjectInputStream.GetField fields = s.readFields();
4259
4260 // Read the alternate persistent fields that we care about
4261 int sign = fields.get("signum", -2);
4262 byte[] magnitude = (byte[])fields.get("magnitude", null);
4263
4264 // Validate signum
4265 if (sign < -1 || sign > 1) {
4266 String message = "BigInteger: Invalid signum value";
4267 if (fields.defaulted("signum"))
4268 message = "BigInteger: Signum not present in stream";
4269 throw new java.io.StreamCorruptedException(message);
4270 }
4271 int[] mag = stripLeadingZeroBytes(magnitude);
4272 if ((mag.length == 0) != (sign == 0)) {
4273 String message = "BigInteger: signum-magnitude mismatch";
4274 if (fields.defaulted("magnitude"))
4275 message = "BigInteger: Magnitude not present in stream";
4276 throw new java.io.StreamCorruptedException(message);
4277 }
4278
4279 // Commit final fields via Unsafe
4280 UnsafeHolder.putSign(this, sign);
4281
4282 // Calculate mag field from magnitude and discard magnitude
4283 UnsafeHolder.putMag(this, mag);
4284 if (mag.length >= MAX_MAG_LENGTH) {
4285 try {
4286 checkRange();
4287 } catch (ArithmeticException e) {
4288 throw new java.io.StreamCorruptedException("BigInteger: Out of the supported range");
4289 }
4290 }
4291 }
4292
4293 // Support for resetting final fields while deserializing
4294 private static class UnsafeHolder {
4295 private static final sun.misc.Unsafe unsafe;
4296 private static final long signumOffset;
4297 private static final long magOffset;
4298 static {
4299 try {
4300 unsafe = sun.misc.Unsafe.getUnsafe();
4301 signumOffset = unsafe.objectFieldOffset
4302 (BigInteger.class.getDeclaredField("signum"));
4303 magOffset = unsafe.objectFieldOffset
4304 (BigInteger.class.getDeclaredField("mag"));
4305 } catch (Exception ex) {
4306 throw new ExceptionInInitializerError(ex);
4307 }
4308 }
4309
4310 static void putSign(BigInteger bi, int sign) {
4311 unsafe.putIntVolatile(bi, signumOffset, sign);
4312 }
4313
4314 static void putMag(BigInteger bi, int[] magnitude) {
4315 unsafe.putObjectVolatile(bi, magOffset, magnitude);
4316 }
4317 }
4318
4319 /**
4320 * Save the {@code BigInteger} instance to a stream.
4321 * The magnitude of a BigInteger is serialized as a byte array for
4322 * historical reasons.
4323 *
4324 * @serialData two necessary fields are written as well as obsolete
4325 * fields for compatibility with older versions.
4326 */
4327 private void writeObject(ObjectOutputStream s) throws IOException {
4328 // set the values of the Serializable fields
4329 ObjectOutputStream.PutField fields = s.putFields();
4330 fields.put("signum", signum);
4331 fields.put("magnitude", magSerializedForm());
4332 // The values written for cached fields are compatible with older
4333 // versions, but are ignored in readObject so don't otherwise matter.
4334 fields.put("bitCount", -1);
4335 fields.put("bitLength", -1);
4336 fields.put("lowestSetBit", -2);
4337 fields.put("firstNonzeroByteNum", -2);
4338
4339 // save them
4340 s.writeFields();
4341 }
4342
4343 /**
4344 * Returns the mag array as an array of bytes.
4345 */
4346 private byte[] magSerializedForm() {
4347 int len = mag.length;
4348
4349 int bitLen = (len == 0 ? 0 : ((len - 1) << 5) + bitLengthForInt(mag[0]));
4350 int byteLen = (bitLen + 7) >>> 3;
4351 byte[] result = new byte[byteLen];
4352
4353 for (int i = byteLen - 1, bytesCopied = 4, intIndex = len - 1, nextInt = 0;
4354 i >= 0; i--) {
4355 if (bytesCopied == 4) {
4356 nextInt = mag[intIndex--];
4357 bytesCopied = 1;
4358 } else {
4359 nextInt >>>= 8;
4360 bytesCopied++;
4361 }
4362 result[i] = (byte)nextInt;
4363 }
4364 return result;
4365 }
4366
4367 /**
4368 * Converts this {@code BigInteger} to a {@code long}, checking
4369 * for lost information. If the value of this {@code BigInteger}
4370 * is out of the range of the {@code long} type, then an
4371 * {@code ArithmeticException} is thrown.
4372 *
4373 * @return this {@code BigInteger} converted to a {@code long}.
4374 * @throws ArithmeticException if the value of {@code this} will
4375 * not exactly fit in a {@code long}.
4376 * @see BigInteger#longValue
4377 * @since 1.8
4378 */
4379 public long longValueExact() {
4380 if (mag.length <= 2 && bitLength() <= 63)
4381 return longValue();
4382 else
4383 throw new ArithmeticException("BigInteger out of long range");
4384 }
4385
4386 /**
4387 * Converts this {@code BigInteger} to an {@code int}, checking
4388 * for lost information. If the value of this {@code BigInteger}
4389 * is out of the range of the {@code int} type, then an
4390 * {@code ArithmeticException} is thrown.
4391 *
4392 * @return this {@code BigInteger} converted to an {@code int}.
4393 * @throws ArithmeticException if the value of {@code this} will
4394 * not exactly fit in a {@code int}.
4395 * @see BigInteger#intValue
4396 * @since 1.8
4397 */
4398 public int intValueExact() {
4399 if (mag.length <= 1 && bitLength() <= 31)
4400 return intValue();
4401 else
4402 throw new ArithmeticException("BigInteger out of int range");
4403 }
4404
4405 /**
4406 * Converts this {@code BigInteger} to a {@code short}, checking
4407 * for lost information. If the value of this {@code BigInteger}
4408 * is out of the range of the {@code short} type, then an
4409 * {@code ArithmeticException} is thrown.
4410 *
4411 * @return this {@code BigInteger} converted to a {@code short}.
4412 * @throws ArithmeticException if the value of {@code this} will
4413 * not exactly fit in a {@code short}.
4414 * @see BigInteger#shortValue
4415 * @since 1.8
4416 */
4417 public short shortValueExact() {
4418 if (mag.length <= 1 && bitLength() <= 31) {
4419 int value = intValue();
4420 if (value >= Short.MIN_VALUE && value <= Short.MAX_VALUE)
4421 return shortValue();
4422 }
4423 throw new ArithmeticException("BigInteger out of short range");
4424 }
4425
4426 /**
4427 * Converts this {@code BigInteger} to a {@code byte}, checking
4428 * for lost information. If the value of this {@code BigInteger}
4429 * is out of the range of the {@code byte} type, then an
4430 * {@code ArithmeticException} is thrown.
4431 *
4432 * @return this {@code BigInteger} converted to a {@code byte}.
4433 * @throws ArithmeticException if the value of {@code this} will
4434 * not exactly fit in a {@code byte}.
4435 * @see BigInteger#byteValue
4436 * @since 1.8
4437 */
4438 public byte byteValueExact() {
4439 if (mag.length <= 1 && bitLength() <= 31) {
4440 int value = intValue();
4441 if (value >= Byte.MIN_VALUE && value <= Byte.MAX_VALUE)
4442 return byteValue();
4443 }
4444 throw new ArithmeticException("BigInteger out of byte range");
4445 }
4446 }