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 sun.misc.DoubleConsts;
39 import sun.misc.FloatConsts;
40
41 /**
42 * Immutable arbitrary-precision integers. All operations behave as if
43 * BigIntegers were represented in two's-complement notation (like Java's
44 * primitive integer types). BigInteger provides analogues to all of Java's
45 * primitive integer operators, and all relevant methods from java.lang.Math.
46 * Additionally, BigInteger provides operations for modular arithmetic, GCD
47 * calculation, primality testing, prime generation, bit manipulation,
48 * and a few other miscellaneous operations.
49 *
50 * <p>Semantics of arithmetic operations exactly mimic those of Java's integer
51 * arithmetic operators, as defined in <i>The Java Language Specification</i>.
52 * For example, division by zero throws an {@code ArithmeticException}, and
53 * division of a negative by a positive yields a negative (or zero) remainder.
54 * All of the details in the Spec concerning overflow are ignored, as
55 * BigIntegers are made as large as necessary to accommodate the results of an
56 * operation.
57 *
58 * <p>Semantics of shift operations extend those of Java's shift operators
59 * to allow for negative shift distances. A right-shift with a negative
60 * shift distance results in a left shift, and vice-versa. The unsigned
61 * right shift operator ({@code >>>}) is omitted, as this operation makes
62 * little sense in combination with the "infinite word size" abstraction
63 * provided by this class.
64 *
65 * <p>Semantics of bitwise logical operations exactly mimic those of Java's
66 * bitwise integer operators. The binary operators ({@code and},
67 * {@code or}, {@code xor}) implicitly perform sign extension on the shorter
68 * of the two operands prior to performing the operation.
69 *
70 * <p>Comparison operations perform signed integer comparisons, analogous to
71 * those performed by Java's relational and equality operators.
72 *
73 * <p>Modular arithmetic operations are provided to compute residues, perform
74 * exponentiation, and compute multiplicative inverses. These methods always
75 * return a non-negative result, between {@code 0} and {@code (modulus - 1)},
76 * inclusive.
77 *
78 * <p>Bit operations operate on a single bit of the two's-complement
79 * representation of their operand. If necessary, the operand is sign-
80 * extended so that it contains the designated bit. None of the single-bit
81 * operations can produce a BigInteger with a different sign from the
82 * BigInteger being operated on, as they affect only a single bit, and the
83 * "infinite word size" abstraction provided by this class ensures that there
84 * are infinitely many "virtual sign bits" preceding each BigInteger.
85 *
86 * <p>For the sake of brevity and clarity, pseudo-code is used throughout the
87 * descriptions of BigInteger methods. The pseudo-code expression
88 * {@code (i + j)} is shorthand for "a BigInteger whose value is
89 * that of the BigInteger {@code i} plus that of the BigInteger {@code j}."
90 * The pseudo-code expression {@code (i == j)} is shorthand for
91 * "{@code true} if and only if the BigInteger {@code i} represents the same
92 * value as the BigInteger {@code j}." Other pseudo-code expressions are
93 * interpreted similarly.
94 *
95 * <p>All methods and constructors in this class throw
96 * {@code NullPointerException} when passed
97 * a null object reference for any input parameter.
98 *
99 * @see BigDecimal
100 * @author Josh Bloch
101 * @author Michael McCloskey
102 * @author Alan Eliasen
103 * @author Timothy Buktu
104 * @since JDK1.1
105 */
106
107 public class BigInteger extends Number implements Comparable<BigInteger> {
108 /**
109 * The signum of this BigInteger: -1 for negative, 0 for zero, or
110 * 1 for positive. Note that the BigInteger zero <i>must</i> have
111 * a signum of 0. This is necessary to ensures that there is exactly one
112 * representation for each BigInteger value.
113 *
114 * @serial
115 */
116 final int signum;
117
118 /**
119 * The magnitude of this BigInteger, in <i>big-endian</i> order: the
120 * zeroth element of this array is the most-significant int of the
121 * magnitude. The magnitude must be "minimal" in that the most-significant
122 * int ({@code mag[0]}) must be non-zero. This is necessary to
123 * ensure that there is exactly one representation for each BigInteger
124 * value. Note that this implies that the BigInteger zero has a
125 * zero-length mag array.
126 */
127 final int[] mag;
128
129 // These "redundant fields" are initialized with recognizable nonsense
130 // values, and cached the first time they are needed (or never, if they
131 // aren't needed).
132
133 /**
134 * One plus the bitCount of this BigInteger. Zeros means unitialized.
135 *
136 * @serial
137 * @see #bitCount
138 * @deprecated Deprecated since logical value is offset from stored
139 * value and correction factor is applied in accessor method.
140 */
141 @Deprecated
142 private int bitCount;
143
144 /**
145 * One plus the bitLength of this BigInteger. Zeros means unitialized.
146 * (either value is acceptable).
147 *
148 * @serial
149 * @see #bitLength()
150 * @deprecated Deprecated since logical value is offset from stored
151 * value and correction factor is applied in accessor method.
152 */
153 @Deprecated
154 private int bitLength;
155
156 /**
157 * Two plus the lowest set bit of this BigInteger, as returned by
158 * getLowestSetBit().
159 *
160 * @serial
161 * @see #getLowestSetBit
162 * @deprecated Deprecated since logical value is offset from stored
163 * value and correction factor is applied in accessor method.
164 */
165 @Deprecated
166 private int lowestSetBit;
167
168 /**
169 * Two plus the index of the lowest-order int in the magnitude of this
170 * BigInteger that contains a nonzero int, or -2 (either value is acceptable).
171 * The least significant int has int-number 0, the next int in order of
172 * increasing significance has int-number 1, and so forth.
173 * @deprecated Deprecated since logical value is offset from stored
174 * value and correction factor is applied in accessor method.
175 */
176 @Deprecated
177 private int firstNonzeroIntNum;
178
179 /**
180 * This mask is used to obtain the value of an int as if it were unsigned.
181 */
182 final static long LONG_MASK = 0xffffffffL;
183
184 /**
185 * The threshold value for using Karatsuba multiplication. If the number
186 * of ints in both mag arrays are greater than this number, then
187 * Karatsuba multiplication will be used. This value is found
188 * experimentally to work well.
189 */
190 private static final int KARATSUBA_THRESHOLD = 50;
191
192 /**
193 * The threshold value for using 3-way Toom-Cook multiplication.
194 * If the number of ints in each mag array is greater than the
195 * Karatsuba threshold, and the number of ints in at least one of
196 * the mag arrays is greater than this threshold, then Toom-Cook
197 * multiplication will be used.
198 */
199 private static final int TOOM_COOK_THRESHOLD = 75;
200
201 /**
202 * The threshold value for using Karatsuba squaring. If the number
203 * of ints in the number are larger than this value,
204 * Karatsuba squaring will be used. This value is found
205 * experimentally to work well.
206 */
207 private static final int KARATSUBA_SQUARE_THRESHOLD = 90;
208
209 /**
210 * The threshold value for using Toom-Cook squaring. If the number
211 * of ints in the number are larger than this value,
212 * Toom-Cook squaring will be used. This value is found
213 * experimentally to work well.
214 */
215 private static final int TOOM_COOK_SQUARE_THRESHOLD = 140;
216
217 /**
218 * The threshold value for using Burnikel-Ziegler division. If the number
219 * of ints in the number are larger than this value,
220 * Burnikel-Ziegler division will be used. This value is found
221 * experimentally to work well.
222 */
223 static final int BURNIKEL_ZIEGLER_THRESHOLD = 50;
224
225 /**
226 * The threshold value for using Schoenhage recursive base conversion. If
227 * the number of ints in the number are larger than this value,
228 * the Schoenhage algorithm will be used. In practice, it appears that the
229 * Schoenhage routine is faster for any threshold down to 2, and is
230 * relatively flat for thresholds between 2-25, so this choice may be
231 * varied within this range for very small effect.
232 */
233 private static final int SCHOENHAGE_BASE_CONVERSION_THRESHOLD = 8;
234
235 //Constructors
236
237 /**
238 * Translates a byte array containing the two's-complement binary
239 * representation of a BigInteger into a BigInteger. The input array is
240 * assumed to be in <i>big-endian</i> byte-order: the most significant
241 * byte is in the zeroth element.
242 *
243 * @param val big-endian two's-complement binary representation of
244 * BigInteger.
245 * @throws NumberFormatException {@code val} is zero bytes long.
246 */
247 public BigInteger(byte[] val) {
248 if (val.length == 0)
249 throw new NumberFormatException("Zero length BigInteger");
250
251 if (val[0] < 0) {
252 mag = makePositive(val);
253 signum = -1;
254 } else {
255 mag = stripLeadingZeroBytes(val);
256 signum = (mag.length == 0 ? 0 : 1);
257 }
258 }
259
260 /**
261 * This private constructor translates an int array containing the
262 * two's-complement binary representation of a BigInteger into a
263 * BigInteger. The input array is assumed to be in <i>big-endian</i>
264 * int-order: the most significant int is in the zeroth element.
265 */
266 private BigInteger(int[] val) {
267 if (val.length == 0)
268 throw new NumberFormatException("Zero length BigInteger");
269
270 if (val[0] < 0) {
271 mag = makePositive(val);
272 signum = -1;
273 } else {
274 mag = trustedStripLeadingZeroInts(val);
275 signum = (mag.length == 0 ? 0 : 1);
276 }
277 }
278
279 /**
280 * Translates the sign-magnitude representation of a BigInteger into a
281 * BigInteger. The sign is represented as an integer signum value: -1 for
282 * negative, 0 for zero, or 1 for positive. The magnitude is a byte array
283 * in <i>big-endian</i> byte-order: the most significant byte is in the
284 * zeroth element. A zero-length magnitude array is permissible, and will
285 * result in a BigInteger value of 0, whether signum is -1, 0 or 1.
286 *
287 * @param signum signum of the number (-1 for negative, 0 for zero, 1
288 * for positive).
289 * @param magnitude big-endian binary representation of the magnitude of
290 * the number.
291 * @throws NumberFormatException {@code signum} is not one of the three
292 * legal values (-1, 0, and 1), or {@code signum} is 0 and
293 * {@code magnitude} contains one or more non-zero bytes.
294 */
295 public BigInteger(int signum, byte[] magnitude) {
296 this.mag = stripLeadingZeroBytes(magnitude);
297
298 if (signum < -1 || signum > 1)
299 throw(new NumberFormatException("Invalid signum value"));
300
301 if (this.mag.length == 0) {
302 this.signum = 0;
303 } else {
304 if (signum == 0)
305 throw(new NumberFormatException("signum-magnitude mismatch"));
306 this.signum = signum;
307 }
308 }
309
310 /**
311 * A constructor for internal use that translates the sign-magnitude
312 * representation of a BigInteger into a BigInteger. It checks the
313 * arguments and copies the magnitude so this constructor would be
314 * safe for external use.
315 */
316 private BigInteger(int signum, int[] magnitude) {
317 this.mag = stripLeadingZeroInts(magnitude);
318
319 if (signum < -1 || signum > 1)
320 throw(new NumberFormatException("Invalid signum value"));
321
322 if (this.mag.length == 0) {
323 this.signum = 0;
324 } else {
325 if (signum == 0)
326 throw(new NumberFormatException("signum-magnitude mismatch"));
327 this.signum = signum;
328 }
329 }
330
331 /**
332 * Translates the String representation of a BigInteger in the
333 * specified radix into a BigInteger. The String representation
334 * consists of an optional minus or plus sign followed by a
335 * sequence of one or more digits in the specified radix. The
336 * character-to-digit mapping is provided by {@code
337 * Character.digit}. The String may not contain any extraneous
338 * characters (whitespace, for example).
339 *
340 * @param val String representation of BigInteger.
341 * @param radix radix to be used in interpreting {@code val}.
342 * @throws NumberFormatException {@code val} is not a valid representation
343 * of a BigInteger in the specified radix, or {@code radix} is
344 * outside the range from {@link Character#MIN_RADIX} to
345 * {@link Character#MAX_RADIX}, inclusive.
346 * @see Character#digit
347 */
348 public BigInteger(String val, int radix) {
349 int cursor = 0, numDigits;
350 final int len = val.length();
351
352 if (radix < Character.MIN_RADIX || radix > Character.MAX_RADIX)
353 throw new NumberFormatException("Radix out of range");
354 if (len == 0)
355 throw new NumberFormatException("Zero length BigInteger");
356
357 // Check for at most one leading sign
358 int sign = 1;
359 int index1 = val.lastIndexOf('-');
360 int index2 = val.lastIndexOf('+');
361 if ((index1 + index2) <= -1) {
362 // No leading sign character or at most one leading sign character
363 if (index1 == 0 || index2 == 0) {
364 cursor = 1;
365 if (len == 1)
366 throw new NumberFormatException("Zero length BigInteger");
367 }
368 if (index1 == 0)
369 sign = -1;
370 } else
371 throw new NumberFormatException("Illegal embedded sign character");
372
373 // Skip leading zeros and compute number of digits in magnitude
374 while (cursor < len &&
375 Character.digit(val.charAt(cursor), radix) == 0) {
376 cursor++;
377 }
378
379 if (cursor == len) {
380 signum = 0;
381 mag = ZERO.mag;
382 return;
383 }
384
385 numDigits = len - cursor;
386 signum = sign;
387
388 // Pre-allocate array of expected size. May be too large but can
389 // never be too small. Typically exact.
390 int numBits = (int)(((numDigits * bitsPerDigit[radix]) >>> 10) + 1);
391 int numWords = (numBits + 31) >>> 5;
392 int[] magnitude = new int[numWords];
393
394 // Process first (potentially short) digit group
395 int firstGroupLen = numDigits % digitsPerInt[radix];
396 if (firstGroupLen == 0)
397 firstGroupLen = digitsPerInt[radix];
398 String group = val.substring(cursor, cursor += firstGroupLen);
399 magnitude[numWords - 1] = Integer.parseInt(group, radix);
400 if (magnitude[numWords - 1] < 0)
401 throw new NumberFormatException("Illegal digit");
402
403 // Process remaining digit groups
404 int superRadix = intRadix[radix];
405 int groupVal = 0;
406 while (cursor < len) {
407 group = val.substring(cursor, cursor += digitsPerInt[radix]);
408 groupVal = Integer.parseInt(group, radix);
409 if (groupVal < 0)
410 throw new NumberFormatException("Illegal digit");
411 destructiveMulAdd(magnitude, superRadix, groupVal);
412 }
413 // Required for cases where the array was overallocated.
414 mag = trustedStripLeadingZeroInts(magnitude);
415 }
416
417 /*
418 * Constructs a new BigInteger using a char array with radix=10.
419 * Sign is precalculated outside and not allowed in the val.
420 */
421 BigInteger(char[] val, int sign, int len) {
422 int cursor = 0, numDigits;
423
424 // Skip leading zeros and compute number of digits in magnitude
425 while (cursor < len && Character.digit(val[cursor], 10) == 0) {
426 cursor++;
427 }
428 if (cursor == len) {
429 signum = 0;
430 mag = ZERO.mag;
431 return;
432 }
433
434 numDigits = len - cursor;
435 signum = sign;
436 // Pre-allocate array of expected size
437 int numWords;
438 if (len < 10) {
439 numWords = 1;
440 } else {
441 int numBits = (int)(((numDigits * bitsPerDigit[10]) >>> 10) + 1);
442 numWords = (numBits + 31) >>> 5;
443 }
444 int[] magnitude = new int[numWords];
445
446 // Process first (potentially short) digit group
447 int firstGroupLen = numDigits % digitsPerInt[10];
448 if (firstGroupLen == 0)
449 firstGroupLen = digitsPerInt[10];
450 magnitude[numWords - 1] = parseInt(val, cursor, cursor += firstGroupLen);
451
452 // Process remaining digit groups
453 while (cursor < len) {
454 int groupVal = parseInt(val, cursor, cursor += digitsPerInt[10]);
455 destructiveMulAdd(magnitude, intRadix[10], groupVal);
456 }
457 mag = trustedStripLeadingZeroInts(magnitude);
458 }
459
460 // Create an integer with the digits between the two indexes
461 // Assumes start < end. The result may be negative, but it
462 // is to be treated as an unsigned value.
463 private int parseInt(char[] source, int start, int end) {
464 int result = Character.digit(source[start++], 10);
465 if (result == -1)
466 throw new NumberFormatException(new String(source));
467
468 for (int index = start; index < end; index++) {
469 int nextVal = Character.digit(source[index], 10);
470 if (nextVal == -1)
471 throw new NumberFormatException(new String(source));
472 result = 10*result + nextVal;
473 }
474
475 return result;
476 }
477
478 // bitsPerDigit in the given radix times 1024
479 // Rounded up to avoid underallocation.
480 private static long bitsPerDigit[] = { 0, 0,
481 1024, 1624, 2048, 2378, 2648, 2875, 3072, 3247, 3402, 3543, 3672,
482 3790, 3899, 4001, 4096, 4186, 4271, 4350, 4426, 4498, 4567, 4633,
483 4696, 4756, 4814, 4870, 4923, 4975, 5025, 5074, 5120, 5166, 5210,
484 5253, 5295};
485
486 // Multiply x array times word y in place, and add word z
487 private static void destructiveMulAdd(int[] x, int y, int z) {
488 // Perform the multiplication word by word
489 long ylong = y & LONG_MASK;
490 long zlong = z & LONG_MASK;
491 int len = x.length;
492
493 long product = 0;
494 long carry = 0;
495 for (int i = len-1; i >= 0; i--) {
496 product = ylong * (x[i] & LONG_MASK) + carry;
497 x[i] = (int)product;
498 carry = product >>> 32;
499 }
500
501 // Perform the addition
502 long sum = (x[len-1] & LONG_MASK) + zlong;
503 x[len-1] = (int)sum;
504 carry = sum >>> 32;
505 for (int i = len-2; i >= 0; i--) {
506 sum = (x[i] & LONG_MASK) + carry;
507 x[i] = (int)sum;
508 carry = sum >>> 32;
509 }
510 }
511
512 /**
513 * Translates the decimal String representation of a BigInteger into a
514 * BigInteger. The String representation consists of an optional minus
515 * sign followed by a sequence of one or more decimal digits. The
516 * character-to-digit mapping is provided by {@code Character.digit}.
517 * The String may not contain any extraneous characters (whitespace, for
518 * example).
519 *
520 * @param val decimal String representation of BigInteger.
521 * @throws NumberFormatException {@code val} is not a valid representation
522 * of a BigInteger.
523 * @see Character#digit
524 */
525 public BigInteger(String val) {
526 this(val, 10);
527 }
528
529 /**
530 * Constructs a randomly generated BigInteger, uniformly distributed over
531 * the range 0 to (2<sup>{@code numBits}</sup> - 1), inclusive.
532 * The uniformity of the distribution assumes that a fair source of random
533 * bits is provided in {@code rnd}. Note that this constructor always
534 * constructs a non-negative BigInteger.
535 *
536 * @param numBits maximum bitLength of the new BigInteger.
537 * @param rnd source of randomness to be used in computing the new
538 * BigInteger.
539 * @throws IllegalArgumentException {@code numBits} is negative.
540 * @see #bitLength()
541 */
542 public BigInteger(int numBits, Random rnd) {
543 this(1, randomBits(numBits, rnd));
544 }
545
546 private static byte[] randomBits(int numBits, Random rnd) {
547 if (numBits < 0)
548 throw new IllegalArgumentException("numBits must be non-negative");
549 int numBytes = (int)(((long)numBits+7)/8); // avoid overflow
550 byte[] randomBits = new byte[numBytes];
551
552 // Generate random bytes and mask out any excess bits
553 if (numBytes > 0) {
554 rnd.nextBytes(randomBits);
555 int excessBits = 8*numBytes - numBits;
556 randomBits[0] &= (1 << (8-excessBits)) - 1;
557 }
558 return randomBits;
559 }
560
561 /**
562 * Constructs a randomly generated positive BigInteger that is probably
563 * prime, with the specified bitLength.
564 *
565 * <p>It is recommended that the {@link #probablePrime probablePrime}
566 * method be used in preference to this constructor unless there
567 * is a compelling need to specify a certainty.
568 *
569 * @param bitLength bitLength of the returned BigInteger.
570 * @param certainty a measure of the uncertainty that the caller is
571 * willing to tolerate. The probability that the new BigInteger
572 * represents a prime number will exceed
573 * (1 - 1/2<sup>{@code certainty}</sup>). The execution time of
574 * this constructor is proportional to the value of this parameter.
575 * @param rnd source of random bits used to select candidates to be
576 * tested for primality.
577 * @throws ArithmeticException {@code bitLength < 2}.
578 * @see #bitLength()
579 */
580 public BigInteger(int bitLength, int certainty, Random rnd) {
581 BigInteger prime;
582
583 if (bitLength < 2)
584 throw new ArithmeticException("bitLength < 2");
585 prime = (bitLength < SMALL_PRIME_THRESHOLD
586 ? smallPrime(bitLength, certainty, rnd)
587 : largePrime(bitLength, certainty, rnd));
588 signum = 1;
589 mag = prime.mag;
590 }
591
592 // Minimum size in bits that the requested prime number has
593 // before we use the large prime number generating algorithms.
594 // The cutoff of 95 was chosen empirically for best performance.
595 private static final int SMALL_PRIME_THRESHOLD = 95;
596
597 // Certainty required to meet the spec of probablePrime
598 private static final int DEFAULT_PRIME_CERTAINTY = 100;
599
600 /**
601 * Returns a positive BigInteger that is probably prime, with the
602 * specified bitLength. The probability that a BigInteger returned
603 * by this method is composite does not exceed 2<sup>-100</sup>.
604 *
605 * @param bitLength bitLength of the returned BigInteger.
606 * @param rnd source of random bits used to select candidates to be
607 * tested for primality.
608 * @return a BigInteger of {@code bitLength} bits that is probably prime
609 * @throws ArithmeticException {@code bitLength < 2}.
610 * @see #bitLength()
611 * @since 1.4
612 */
613 public static BigInteger probablePrime(int bitLength, Random rnd) {
614 if (bitLength < 2)
615 throw new ArithmeticException("bitLength < 2");
616
617 return (bitLength < SMALL_PRIME_THRESHOLD ?
618 smallPrime(bitLength, DEFAULT_PRIME_CERTAINTY, rnd) :
619 largePrime(bitLength, DEFAULT_PRIME_CERTAINTY, rnd));
620 }
621
622 /**
623 * Find a random number of the specified bitLength that is probably prime.
624 * This method is used for smaller primes, its performance degrades on
625 * larger bitlengths.
626 *
627 * This method assumes bitLength > 1.
628 */
629 private static BigInteger smallPrime(int bitLength, int certainty, Random rnd) {
630 int magLen = (bitLength + 31) >>> 5;
631 int temp[] = new int[magLen];
632 int highBit = 1 << ((bitLength+31) & 0x1f); // High bit of high int
633 int highMask = (highBit << 1) - 1; // Bits to keep in high int
634
635 while (true) {
636 // Construct a candidate
637 for (int i=0; i < magLen; i++)
638 temp[i] = rnd.nextInt();
639 temp[0] = (temp[0] & highMask) | highBit; // Ensure exact length
640 if (bitLength > 2)
641 temp[magLen-1] |= 1; // Make odd if bitlen > 2
642
643 BigInteger p = new BigInteger(temp, 1);
644
645 // Do cheap "pre-test" if applicable
646 if (bitLength > 6) {
647 long r = p.remainder(SMALL_PRIME_PRODUCT).longValue();
648 if ((r%3==0) || (r%5==0) || (r%7==0) || (r%11==0) ||
649 (r%13==0) || (r%17==0) || (r%19==0) || (r%23==0) ||
650 (r%29==0) || (r%31==0) || (r%37==0) || (r%41==0))
651 continue; // Candidate is composite; try another
652 }
653
654 // All candidates of bitLength 2 and 3 are prime by this point
655 if (bitLength < 4)
656 return p;
657
658 // Do expensive test if we survive pre-test (or it's inapplicable)
659 if (p.primeToCertainty(certainty, rnd))
660 return p;
661 }
662 }
663
664 private static final BigInteger SMALL_PRIME_PRODUCT
665 = valueOf(3L*5*7*11*13*17*19*23*29*31*37*41);
666
667 /**
668 * Find a random number of the specified bitLength that is probably prime.
669 * This method is more appropriate for larger bitlengths since it uses
670 * a sieve to eliminate most composites before using a more expensive
671 * test.
672 */
673 private static BigInteger largePrime(int bitLength, int certainty, Random rnd) {
674 BigInteger p;
675 p = new BigInteger(bitLength, rnd).setBit(bitLength-1);
676 p.mag[p.mag.length-1] &= 0xfffffffe;
677
678 // Use a sieve length likely to contain the next prime number
679 int searchLen = (bitLength / 20) * 64;
680 BitSieve searchSieve = new BitSieve(p, searchLen);
681 BigInteger candidate = searchSieve.retrieve(p, certainty, rnd);
682
683 while ((candidate == null) || (candidate.bitLength() != bitLength)) {
684 p = p.add(BigInteger.valueOf(2*searchLen));
685 if (p.bitLength() != bitLength)
686 p = new BigInteger(bitLength, rnd).setBit(bitLength-1);
687 p.mag[p.mag.length-1] &= 0xfffffffe;
688 searchSieve = new BitSieve(p, searchLen);
689 candidate = searchSieve.retrieve(p, certainty, rnd);
690 }
691 return candidate;
692 }
693
694 /**
695 * Returns the first integer greater than this {@code BigInteger} that
696 * is probably prime. The probability that the number returned by this
697 * method is composite does not exceed 2<sup>-100</sup>. This method will
698 * never skip over a prime when searching: if it returns {@code p}, there
699 * is no prime {@code q} such that {@code this < q < p}.
700 *
701 * @return the first integer greater than this {@code BigInteger} that
702 * is probably prime.
703 * @throws ArithmeticException {@code this < 0}.
704 * @since 1.5
705 */
706 public BigInteger nextProbablePrime() {
707 if (this.signum < 0)
708 throw new ArithmeticException("start < 0: " + this);
709
710 // Handle trivial cases
711 if ((this.signum == 0) || this.equals(ONE))
712 return TWO;
713
714 BigInteger result = this.add(ONE);
715
716 // Fastpath for small numbers
717 if (result.bitLength() < SMALL_PRIME_THRESHOLD) {
718
719 // Ensure an odd number
720 if (!result.testBit(0))
721 result = result.add(ONE);
722
723 while (true) {
724 // Do cheap "pre-test" if applicable
725 if (result.bitLength() > 6) {
726 long r = result.remainder(SMALL_PRIME_PRODUCT).longValue();
727 if ((r%3==0) || (r%5==0) || (r%7==0) || (r%11==0) ||
728 (r%13==0) || (r%17==0) || (r%19==0) || (r%23==0) ||
729 (r%29==0) || (r%31==0) || (r%37==0) || (r%41==0)) {
730 result = result.add(TWO);
731 continue; // Candidate is composite; try another
732 }
733 }
734
735 // All candidates of bitLength 2 and 3 are prime by this point
736 if (result.bitLength() < 4)
737 return result;
738
739 // The expensive test
740 if (result.primeToCertainty(DEFAULT_PRIME_CERTAINTY, null))
741 return result;
742
743 result = result.add(TWO);
744 }
745 }
746
747 // Start at previous even number
748 if (result.testBit(0))
749 result = result.subtract(ONE);
750
751 // Looking for the next large prime
752 int searchLen = (result.bitLength() / 20) * 64;
753
754 while (true) {
755 BitSieve searchSieve = new BitSieve(result, searchLen);
756 BigInteger candidate = searchSieve.retrieve(result,
757 DEFAULT_PRIME_CERTAINTY, null);
758 if (candidate != null)
759 return candidate;
760 result = result.add(BigInteger.valueOf(2 * searchLen));
761 }
762 }
763
764 /**
765 * Returns {@code true} if this BigInteger is probably prime,
766 * {@code false} if it's definitely composite.
767 *
768 * This method assumes bitLength > 2.
769 *
770 * @param certainty a measure of the uncertainty that the caller is
771 * willing to tolerate: if the call returns {@code true}
772 * the probability that this BigInteger is prime exceeds
773 * {@code (1 - 1/2<sup>certainty</sup>)}. The execution time of
774 * this method is proportional to the value of this parameter.
775 * @return {@code true} if this BigInteger is probably prime,
776 * {@code false} if it's definitely composite.
777 */
778 boolean primeToCertainty(int certainty, Random random) {
779 int rounds = 0;
780 int n = (Math.min(certainty, Integer.MAX_VALUE-1)+1)/2;
781
782 // The relationship between the certainty and the number of rounds
783 // we perform is given in the draft standard ANSI X9.80, "PRIME
784 // NUMBER GENERATION, PRIMALITY TESTING, AND PRIMALITY CERTIFICATES".
785 int sizeInBits = this.bitLength();
786 if (sizeInBits < 100) {
787 rounds = 50;
788 rounds = n < rounds ? n : rounds;
789 return passesMillerRabin(rounds, random);
790 }
791
792 if (sizeInBits < 256) {
793 rounds = 27;
794 } else if (sizeInBits < 512) {
795 rounds = 15;
796 } else if (sizeInBits < 768) {
797 rounds = 8;
798 } else if (sizeInBits < 1024) {
799 rounds = 4;
800 } else {
801 rounds = 2;
802 }
803 rounds = n < rounds ? n : rounds;
804
805 return passesMillerRabin(rounds, random) && passesLucasLehmer();
806 }
807
808 /**
809 * Returns true iff this BigInteger is a Lucas-Lehmer probable prime.
810 *
811 * The following assumptions are made:
812 * This BigInteger is a positive, odd number.
813 */
814 private boolean passesLucasLehmer() {
815 BigInteger thisPlusOne = this.add(ONE);
816
817 // Step 1
818 int d = 5;
819 while (jacobiSymbol(d, this) != -1) {
820 // 5, -7, 9, -11, ...
821 d = (d < 0) ? Math.abs(d)+2 : -(d+2);
822 }
823
824 // Step 2
825 BigInteger u = lucasLehmerSequence(d, thisPlusOne, this);
826
827 // Step 3
828 return u.mod(this).equals(ZERO);
829 }
830
831 /**
832 * Computes Jacobi(p,n).
833 * Assumes n positive, odd, n>=3.
834 */
835 private static int jacobiSymbol(int p, BigInteger n) {
836 if (p == 0)
837 return 0;
838
839 // Algorithm and comments adapted from Colin Plumb's C library.
840 int j = 1;
841 int u = n.mag[n.mag.length-1];
842
843 // Make p positive
844 if (p < 0) {
845 p = -p;
846 int n8 = u & 7;
847 if ((n8 == 3) || (n8 == 7))
848 j = -j; // 3 (011) or 7 (111) mod 8
849 }
850
851 // Get rid of factors of 2 in p
852 while ((p & 3) == 0)
853 p >>= 2;
854 if ((p & 1) == 0) {
855 p >>= 1;
856 if (((u ^ (u>>1)) & 2) != 0)
857 j = -j; // 3 (011) or 5 (101) mod 8
858 }
859 if (p == 1)
860 return j;
861 // Then, apply quadratic reciprocity
862 if ((p & u & 2) != 0) // p = u = 3 (mod 4)?
863 j = -j;
864 // And reduce u mod p
865 u = n.mod(BigInteger.valueOf(p)).intValue();
866
867 // Now compute Jacobi(u,p), u < p
868 while (u != 0) {
869 while ((u & 3) == 0)
870 u >>= 2;
871 if ((u & 1) == 0) {
872 u >>= 1;
873 if (((p ^ (p>>1)) & 2) != 0)
874 j = -j; // 3 (011) or 5 (101) mod 8
875 }
876 if (u == 1)
877 return j;
878 // Now both u and p are odd, so use quadratic reciprocity
879 assert (u < p);
880 int t = u; u = p; p = t;
881 if ((u & p & 2) != 0) // u = p = 3 (mod 4)?
882 j = -j;
883 // Now u >= p, so it can be reduced
884 u %= p;
885 }
886 return 0;
887 }
888
889 private static BigInteger lucasLehmerSequence(int z, BigInteger k, BigInteger n) {
890 BigInteger d = BigInteger.valueOf(z);
891 BigInteger u = ONE; BigInteger u2;
892 BigInteger v = ONE; BigInteger v2;
893
894 for (int i=k.bitLength()-2; i >= 0; i--) {
895 u2 = u.multiply(v).mod(n);
896
897 v2 = v.square().add(d.multiply(u.square())).mod(n);
898 if (v2.testBit(0))
899 v2 = v2.subtract(n);
900
901 v2 = v2.shiftRight(1);
902
903 u = u2; v = v2;
904 if (k.testBit(i)) {
905 u2 = u.add(v).mod(n);
906 if (u2.testBit(0))
907 u2 = u2.subtract(n);
908
909 u2 = u2.shiftRight(1);
910 v2 = v.add(d.multiply(u)).mod(n);
911 if (v2.testBit(0))
912 v2 = v2.subtract(n);
913 v2 = v2.shiftRight(1);
914
915 u = u2; v = v2;
916 }
917 }
918 return u;
919 }
920
921 private static volatile Random staticRandom;
922
923 private static Random getSecureRandom() {
924 if (staticRandom == null) {
925 staticRandom = new java.security.SecureRandom();
926 }
927 return staticRandom;
928 }
929
930 /**
931 * Returns true iff this BigInteger passes the specified number of
932 * Miller-Rabin tests. This test is taken from the DSA spec (NIST FIPS
933 * 186-2).
934 *
935 * The following assumptions are made:
936 * This BigInteger is a positive, odd number greater than 2.
937 * iterations<=50.
938 */
939 private boolean passesMillerRabin(int iterations, Random rnd) {
940 // Find a and m such that m is odd and this == 1 + 2**a * m
941 BigInteger thisMinusOne = this.subtract(ONE);
942 BigInteger m = thisMinusOne;
943 int a = m.getLowestSetBit();
944 m = m.shiftRight(a);
945
946 // Do the tests
947 if (rnd == null) {
948 rnd = getSecureRandom();
949 }
950 for (int i=0; i < iterations; i++) {
951 // Generate a uniform random on (1, this)
952 BigInteger b;
953 do {
954 b = new BigInteger(this.bitLength(), rnd);
955 } while (b.compareTo(ONE) <= 0 || b.compareTo(this) >= 0);
956
957 int j = 0;
958 BigInteger z = b.modPow(m, this);
959 while (!((j == 0 && z.equals(ONE)) || z.equals(thisMinusOne))) {
960 if (j > 0 && z.equals(ONE) || ++j == a)
961 return false;
962 z = z.modPow(TWO, this);
963 }
964 }
965 return true;
966 }
967
968 /**
969 * This internal constructor differs from its public cousin
970 * with the arguments reversed in two ways: it assumes that its
971 * arguments are correct, and it doesn't copy the magnitude array.
972 */
973 BigInteger(int[] magnitude, int signum) {
974 this.signum = (magnitude.length == 0 ? 0 : signum);
975 this.mag = magnitude;
976 }
977
978 /**
979 * This private constructor is for internal use and assumes that its
980 * arguments are correct.
981 */
982 private BigInteger(byte[] magnitude, int signum) {
983 this.signum = (magnitude.length == 0 ? 0 : signum);
984 this.mag = stripLeadingZeroBytes(magnitude);
985 }
986
987 //Static Factory Methods
988
989 /**
990 * Returns a BigInteger whose value is equal to that of the
991 * specified {@code long}. This "static factory method" is
992 * provided in preference to a ({@code long}) constructor
993 * because it allows for reuse of frequently used BigIntegers.
994 *
995 * @param val value of the BigInteger to return.
996 * @return a BigInteger with the specified value.
997 */
998 public static BigInteger valueOf(long val) {
999 // If -MAX_CONSTANT < val < MAX_CONSTANT, return stashed constant
1000 if (val == 0)
1001 return ZERO;
1002 if (val > 0 && val <= MAX_CONSTANT)
1003 return posConst[(int) val];
1004 else if (val < 0 && val >= -MAX_CONSTANT)
1005 return negConst[(int) -val];
1006
1007 return new BigInteger(val);
1008 }
1009
1010 /**
1011 * Constructs a BigInteger with the specified value, which may not be zero.
1012 */
1013 private BigInteger(long val) {
1014 if (val < 0) {
1015 val = -val;
1016 signum = -1;
1017 } else {
1018 signum = 1;
1019 }
1020
1021 int highWord = (int)(val >>> 32);
1022 if (highWord == 0) {
1023 mag = new int[1];
1024 mag[0] = (int)val;
1025 } else {
1026 mag = new int[2];
1027 mag[0] = highWord;
1028 mag[1] = (int)val;
1029 }
1030 }
1031
1032 /**
1033 * Returns a BigInteger with the given two's complement representation.
1034 * Assumes that the input array will not be modified (the returned
1035 * BigInteger will reference the input array if feasible).
1036 */
1037 private static BigInteger valueOf(int val[]) {
1038 return (val[0] > 0 ? new BigInteger(val, 1) : new BigInteger(val));
1039 }
1040
1041 // Constants
1042
1043 /**
1044 * Initialize static constant array when class is loaded.
1045 */
1046 private final static int MAX_CONSTANT = 16;
1047 private static BigInteger posConst[] = new BigInteger[MAX_CONSTANT+1];
1048 private static BigInteger negConst[] = new BigInteger[MAX_CONSTANT+1];
1049
1050 /**
1051 * The cache of powers of each radix. This allows us to not have to
1052 * recalculate powers of radix^(2^n) more than once. This speeds
1053 * Schoenhage recursive base conversion significantly.
1054 */
1055 private static volatile BigInteger[][] powerCache;
1056
1057 /** The cache of logarithms of radices for base conversion. */
1058 private static final double[] logCache;
1059
1060 /** The natural log of 2. This is used in computing cache indices. */
1061 private static final double LOG_TWO = Math.log(2.0);
1062
1063 static {
1064 for (int i = 1; i <= MAX_CONSTANT; i++) {
1065 int[] magnitude = new int[1];
1066 magnitude[0] = i;
1067 posConst[i] = new BigInteger(magnitude, 1);
1068 negConst[i] = new BigInteger(magnitude, -1);
1069 }
1070
1071 /*
1072 * Initialize the cache of radix^(2^x) values used for base conversion
1073 * with just the very first value. Additional values will be created
1074 * on demand.
1075 */
1076 powerCache = new BigInteger[Character.MAX_RADIX+1][];
1077 logCache = new double[Character.MAX_RADIX+1];
1078
1079 for (int i=Character.MIN_RADIX; i <= Character.MAX_RADIX; i++) {
1080 powerCache[i] = new BigInteger[] { BigInteger.valueOf(i) };
1081 logCache[i] = Math.log(i);
1082 }
1083 }
1084
1085 /**
1086 * The BigInteger constant zero.
1087 *
1088 * @since 1.2
1089 */
1090 public static final BigInteger ZERO = new BigInteger(new int[0], 0);
1091
1092 /**
1093 * The BigInteger constant one.
1094 *
1095 * @since 1.2
1096 */
1097 public static final BigInteger ONE = valueOf(1);
1098
1099 /**
1100 * The BigInteger constant two. (Not exported.)
1101 */
1102 private static final BigInteger TWO = valueOf(2);
1103
1104 /**
1105 * The BigInteger constant -1. (Not exported.)
1106 */
1107 private static final BigInteger NEGATIVE_ONE = valueOf(-1);
1108
1109 /**
1110 * The BigInteger constant ten.
1111 *
1112 * @since 1.5
1113 */
1114 public static final BigInteger TEN = valueOf(10);
1115
1116 // Arithmetic Operations
1117
1118 /**
1119 * Returns a BigInteger whose value is {@code (this + val)}.
1120 *
1121 * @param val value to be added to this BigInteger.
1122 * @return {@code this + val}
1123 */
1124 public BigInteger add(BigInteger val) {
1125 if (val.signum == 0)
1126 return this;
1127 if (signum == 0)
1128 return val;
1129 if (val.signum == signum)
1130 return new BigInteger(add(mag, val.mag), signum);
1131
1132 int cmp = compareMagnitude(val);
1133 if (cmp == 0)
1134 return ZERO;
1135 int[] resultMag = (cmp > 0 ? subtract(mag, val.mag)
1136 : subtract(val.mag, mag));
1137 resultMag = trustedStripLeadingZeroInts(resultMag);
1138
1139 return new BigInteger(resultMag, cmp == signum ? 1 : -1);
1140 }
1141
1142 /**
1143 * Package private methods used by BigDecimal code to add a BigInteger
1144 * with a long. Assumes val is not equal to INFLATED.
1145 */
1146 BigInteger add(long val) {
1147 if (val == 0)
1148 return this;
1149 if (signum == 0)
1150 return valueOf(val);
1151 if (Long.signum(val) == signum)
1152 return new BigInteger(add(mag, Math.abs(val)), signum);
1153 int cmp = compareMagnitude(val);
1154 if (cmp == 0)
1155 return ZERO;
1156 int[] resultMag = (cmp > 0 ? subtract(mag, Math.abs(val)) : subtract(Math.abs(val), mag));
1157 resultMag = trustedStripLeadingZeroInts(resultMag);
1158 return new BigInteger(resultMag, cmp == signum ? 1 : -1);
1159 }
1160
1161 /**
1162 * Adds the contents of the int array x and long value val. This
1163 * method allocates a new int array to hold the answer and returns
1164 * a reference to that array. Assumes x.length > 0 and val is
1165 * non-negative
1166 */
1167 private static int[] add(int[] x, long val) {
1168 int[] y;
1169 long sum = 0;
1170 int xIndex = x.length;
1171 int[] result;
1172 int highWord = (int)(val >>> 32);
1173 if (highWord == 0) {
1174 result = new int[xIndex];
1175 sum = (x[--xIndex] & LONG_MASK) + val;
1176 result[xIndex] = (int)sum;
1177 } else {
1178 if (xIndex == 1) {
1179 result = new int[2];
1180 sum = val + (x[0] & LONG_MASK);
1181 result[1] = (int)sum;
1182 result[0] = (int)(sum >>> 32);
1183 return result;
1184 } else {
1185 result = new int[xIndex];
1186 sum = (x[--xIndex] & LONG_MASK) + (val & LONG_MASK);
1187 result[xIndex] = (int)sum;
1188 sum = (x[--xIndex] & LONG_MASK) + (highWord & LONG_MASK) + (sum >>> 32);
1189 result[xIndex] = (int)sum;
1190 }
1191 }
1192 // Copy remainder of longer number while carry propagation is required
1193 boolean carry = (sum >>> 32 != 0);
1194 while (xIndex > 0 && carry)
1195 carry = ((result[--xIndex] = x[xIndex] + 1) == 0);
1196 // Copy remainder of longer number
1197 while (xIndex > 0)
1198 result[--xIndex] = x[xIndex];
1199 // Grow result if necessary
1200 if (carry) {
1201 int bigger[] = new int[result.length + 1];
1202 System.arraycopy(result, 0, bigger, 1, result.length);
1203 bigger[0] = 0x01;
1204 return bigger;
1205 }
1206 return result;
1207 }
1208
1209 /**
1210 * Adds the contents of the int arrays x and y. This method allocates
1211 * a new int array to hold the answer and returns a reference to that
1212 * array.
1213 */
1214 private static int[] add(int[] x, int[] y) {
1215 // If x is shorter, swap the two arrays
1216 if (x.length < y.length) {
1217 int[] tmp = x;
1218 x = y;
1219 y = tmp;
1220 }
1221
1222 int xIndex = x.length;
1223 int yIndex = y.length;
1224 int result[] = new int[xIndex];
1225 long sum = 0;
1226 if (yIndex == 1) {
1227 sum = (x[--xIndex] & LONG_MASK) + (y[0] & LONG_MASK) ;
1228 result[xIndex] = (int)sum;
1229 } else {
1230 // Add common parts of both numbers
1231 while (yIndex > 0) {
1232 sum = (x[--xIndex] & LONG_MASK) +
1233 (y[--yIndex] & LONG_MASK) + (sum >>> 32);
1234 result[xIndex] = (int)sum;
1235 }
1236 }
1237 // Copy remainder of longer number while carry propagation is required
1238 boolean carry = (sum >>> 32 != 0);
1239 while (xIndex > 0 && carry)
1240 carry = ((result[--xIndex] = x[xIndex] + 1) == 0);
1241
1242 // Copy remainder of longer number
1243 while (xIndex > 0)
1244 result[--xIndex] = x[xIndex];
1245
1246 // Grow result if necessary
1247 if (carry) {
1248 int bigger[] = new int[result.length + 1];
1249 System.arraycopy(result, 0, bigger, 1, result.length);
1250 bigger[0] = 0x01;
1251 return bigger;
1252 }
1253 return result;
1254 }
1255
1256 private static int[] subtract(long val, int[] little) {
1257 int highWord = (int)(val >>> 32);
1258 if (highWord == 0) {
1259 int result[] = new int[1];
1260 result[0] = (int)(val - (little[0] & LONG_MASK));
1261 return result;
1262 } else {
1263 int result[] = new int[2];
1264 if (little.length == 1) {
1265 long difference = ((int)val & LONG_MASK) - (little[0] & LONG_MASK);
1266 result[1] = (int)difference;
1267 // Subtract remainder of longer number while borrow propagates
1268 boolean borrow = (difference >> 32 != 0);
1269 if (borrow) {
1270 result[0] = highWord - 1;
1271 } else { // Copy remainder of longer number
1272 result[0] = highWord;
1273 }
1274 return result;
1275 } else { // little.length == 2
1276 long difference = ((int)val & LONG_MASK) - (little[1] & LONG_MASK);
1277 result[1] = (int)difference;
1278 difference = (highWord & LONG_MASK) - (little[0] & LONG_MASK) + (difference >> 32);
1279 result[0] = (int)difference;
1280 return result;
1281 }
1282 }
1283 }
1284
1285 /**
1286 * Subtracts the contents of the second argument (val) from the
1287 * first (big). The first int array (big) must represent a larger number
1288 * than the second. This method allocates the space necessary to hold the
1289 * answer.
1290 * assumes val >= 0
1291 */
1292 private static int[] subtract(int[] big, long val) {
1293 int highWord = (int)(val >>> 32);
1294 int bigIndex = big.length;
1295 int result[] = new int[bigIndex];
1296 long difference = 0;
1297
1298 if (highWord == 0) {
1299 difference = (big[--bigIndex] & LONG_MASK) - val;
1300 result[bigIndex] = (int)difference;
1301 } else {
1302 difference = (big[--bigIndex] & LONG_MASK) - (val & LONG_MASK);
1303 result[bigIndex] = (int)difference;
1304 difference = (big[--bigIndex] & LONG_MASK) - (highWord & LONG_MASK) + (difference >> 32);
1305 result[bigIndex] = (int)difference;
1306 }
1307
1308 // Subtract remainder of longer number while borrow propagates
1309 boolean borrow = (difference >> 32 != 0);
1310 while (bigIndex > 0 && borrow)
1311 borrow = ((result[--bigIndex] = big[bigIndex] - 1) == -1);
1312
1313 // Copy remainder of longer number
1314 while (bigIndex > 0)
1315 result[--bigIndex] = big[bigIndex];
1316
1317 return result;
1318 }
1319
1320 /**
1321 * Returns a BigInteger whose value is {@code (this - val)}.
1322 *
1323 * @param val value to be subtracted from this BigInteger.
1324 * @return {@code this - val}
1325 */
1326 public BigInteger subtract(BigInteger val) {
1327 if (val.signum == 0)
1328 return this;
1329 if (signum == 0)
1330 return val.negate();
1331 if (val.signum != signum)
1332 return new BigInteger(add(mag, val.mag), signum);
1333
1334 int cmp = compareMagnitude(val);
1335 if (cmp == 0)
1336 return ZERO;
1337 int[] resultMag = (cmp > 0 ? subtract(mag, val.mag)
1338 : subtract(val.mag, mag));
1339 resultMag = trustedStripLeadingZeroInts(resultMag);
1340 return new BigInteger(resultMag, cmp == signum ? 1 : -1);
1341 }
1342
1343 /**
1344 * Subtracts the contents of the second int arrays (little) from the
1345 * first (big). The first int array (big) must represent a larger number
1346 * than the second. This method allocates the space necessary to hold the
1347 * answer.
1348 */
1349 private static int[] subtract(int[] big, int[] little) {
1350 int bigIndex = big.length;
1351 int result[] = new int[bigIndex];
1352 int littleIndex = little.length;
1353 long difference = 0;
1354
1355 // Subtract common parts of both numbers
1356 while (littleIndex > 0) {
1357 difference = (big[--bigIndex] & LONG_MASK) -
1358 (little[--littleIndex] & LONG_MASK) +
1359 (difference >> 32);
1360 result[bigIndex] = (int)difference;
1361 }
1362
1363 // Subtract remainder of longer number while borrow propagates
1364 boolean borrow = (difference >> 32 != 0);
1365 while (bigIndex > 0 && borrow)
1366 borrow = ((result[--bigIndex] = big[bigIndex] - 1) == -1);
1367
1368 // Copy remainder of longer number
1369 while (bigIndex > 0)
1370 result[--bigIndex] = big[bigIndex];
1371
1372 return result;
1373 }
1374
1375 /**
1376 * Returns a BigInteger whose value is {@code (this * val)}.
1377 *
1378 * @param val value to be multiplied by this BigInteger.
1379 * @return {@code this * val}
1380 */
1381 public BigInteger multiply(BigInteger val) {
1382 if (val.signum == 0 || signum == 0)
1383 return ZERO;
1384
1385 int xlen = mag.length;
1386 int ylen = val.mag.length;
1387
1388 if ((xlen < KARATSUBA_THRESHOLD) || (ylen < KARATSUBA_THRESHOLD)) {
1389 int resultSign = signum == val.signum ? 1 : -1;
1390 if (val.mag.length == 1) {
1391 return multiplyByInt(mag,val.mag[0], resultSign);
1392 }
1393 if (mag.length == 1) {
1394 return multiplyByInt(val.mag,mag[0], resultSign);
1395 }
1396 int[] result = multiplyToLen(mag, xlen,
1397 val.mag, ylen, null);
1398 result = trustedStripLeadingZeroInts(result);
1399 return new BigInteger(result, resultSign);
1400 } else {
1401 if ((xlen < TOOM_COOK_THRESHOLD) && (ylen < TOOM_COOK_THRESHOLD)) {
1402 return multiplyKaratsuba(this, val);
1403 } else {
1404 return multiplyToomCook3(this, val);
1405 }
1406 }
1407 }
1408
1409 private static BigInteger multiplyByInt(int[] x, int y, int sign) {
1410 if (Integer.bitCount(y) == 1) {
1411 return new BigInteger(shiftLeft(x,Integer.numberOfTrailingZeros(y)), sign);
1412 }
1413 int xlen = x.length;
1414 int[] rmag = new int[xlen + 1];
1415 long carry = 0;
1416 long yl = y & LONG_MASK;
1417 int rstart = rmag.length - 1;
1418 for (int i = xlen - 1; i >= 0; i--) {
1419 long product = (x[i] & LONG_MASK) * yl + carry;
1420 rmag[rstart--] = (int)product;
1421 carry = product >>> 32;
1422 }
1423 if (carry == 0L) {
1424 rmag = java.util.Arrays.copyOfRange(rmag, 1, rmag.length);
1425 } else {
1426 rmag[rstart] = (int)carry;
1427 }
1428 return new BigInteger(rmag, sign);
1429 }
1430
1431 /**
1432 * Package private methods used by BigDecimal code to multiply a BigInteger
1433 * with a long. Assumes v is not equal to INFLATED.
1434 */
1435 BigInteger multiply(long v) {
1436 if (v == 0 || signum == 0)
1437 return ZERO;
1438 if (v == BigDecimal.INFLATED)
1439 return multiply(BigInteger.valueOf(v));
1440 int rsign = (v > 0 ? signum : -signum);
1441 if (v < 0)
1442 v = -v;
1443 long dh = v >>> 32; // higher order bits
1444 long dl = v & LONG_MASK; // lower order bits
1445
1446 int xlen = mag.length;
1447 int[] value = mag;
1448 int[] rmag = (dh == 0L) ? (new int[xlen + 1]) : (new int[xlen + 2]);
1449 long carry = 0;
1450 int rstart = rmag.length - 1;
1451 for (int i = xlen - 1; i >= 0; i--) {
1452 long product = (value[i] & LONG_MASK) * dl + carry;
1453 rmag[rstart--] = (int)product;
1454 carry = product >>> 32;
1455 }
1456 rmag[rstart] = (int)carry;
1457 if (dh != 0L) {
1458 carry = 0;
1459 rstart = rmag.length - 2;
1460 for (int i = xlen - 1; i >= 0; i--) {
1461 long product = (value[i] & LONG_MASK) * dh +
1462 (rmag[rstart] & LONG_MASK) + carry;
1463 rmag[rstart--] = (int)product;
1464 carry = product >>> 32;
1465 }
1466 rmag[0] = (int)carry;
1467 }
1468 if (carry == 0L)
1469 rmag = java.util.Arrays.copyOfRange(rmag, 1, rmag.length);
1470 return new BigInteger(rmag, rsign);
1471 }
1472
1473 /**
1474 * Multiplies int arrays x and y to the specified lengths and places
1475 * the result into z. There will be no leading zeros in the resultant array.
1476 */
1477 private int[] multiplyToLen(int[] x, int xlen, int[] y, int ylen, int[] z) {
1478 int xstart = xlen - 1;
1479 int ystart = ylen - 1;
1480
1481 if (z == null || z.length < (xlen+ ylen))
1482 z = new int[xlen+ylen];
1483
1484 long carry = 0;
1485 for (int j=ystart, k=ystart+1+xstart; j >= 0; j--, k--) {
1486 long product = (y[j] & LONG_MASK) *
1487 (x[xstart] & LONG_MASK) + carry;
1488 z[k] = (int)product;
1489 carry = product >>> 32;
1490 }
1491 z[xstart] = (int)carry;
1492
1493 for (int i = xstart-1; i >= 0; i--) {
1494 carry = 0;
1495 for (int j=ystart, k=ystart+1+i; j >= 0; j--, k--) {
1496 long product = (y[j] & LONG_MASK) *
1497 (x[i] & LONG_MASK) +
1498 (z[k] & LONG_MASK) + carry;
1499 z[k] = (int)product;
1500 carry = product >>> 32;
1501 }
1502 z[i] = (int)carry;
1503 }
1504 return z;
1505 }
1506
1507 /**
1508 * Multiplies two BigIntegers using the Karatsuba multiplication
1509 * algorithm. This is a recursive divide-and-conquer algorithm which is
1510 * more efficient for large numbers than what is commonly called the
1511 * "grade-school" algorithm used in multiplyToLen. If the numbers to be
1512 * multiplied have length n, the "grade-school" algorithm has an
1513 * asymptotic complexity of O(n^2). In contrast, the Karatsuba algorithm
1514 * has complexity of O(n^(log2(3))), or O(n^1.585). It achieves this
1515 * increased performance by doing 3 multiplies instead of 4 when
1516 * evaluating the product. As it has some overhead, should be used when
1517 * both numbers are larger than a certain threshold (found
1518 * experimentally).
1519 *
1520 * See: http://en.wikipedia.org/wiki/Karatsuba_algorithm
1521 */
1522 private static BigInteger multiplyKaratsuba(BigInteger x, BigInteger y) {
1523 int xlen = x.mag.length;
1524 int ylen = y.mag.length;
1525
1526 // The number of ints in each half of the number.
1527 int half = (Math.max(xlen, ylen)+1) / 2;
1528
1529 // xl and yl are the lower halves of x and y respectively,
1530 // xh and yh are the upper halves.
1531 BigInteger xl = x.getLower(half);
1532 BigInteger xh = x.getUpper(half);
1533 BigInteger yl = y.getLower(half);
1534 BigInteger yh = y.getUpper(half);
1535
1536 BigInteger p1 = xh.multiply(yh); // p1 = xh*yh
1537 BigInteger p2 = xl.multiply(yl); // p2 = xl*yl
1538
1539 // p3=(xh+xl)*(yh+yl)
1540 BigInteger p3 = xh.add(xl).multiply(yh.add(yl));
1541
1542 // result = p1 * 2^(32*2*half) + (p3 - p1 - p2) * 2^(32*half) + p2
1543 BigInteger result = p1.shiftLeft(32*half).add(p3.subtract(p1).subtract(p2)).shiftLeft(32*half).add(p2);
1544
1545 if (x.signum != y.signum) {
1546 return result.negate();
1547 } else {
1548 return result;
1549 }
1550 }
1551
1552 /**
1553 * Multiplies two BigIntegers using a 3-way Toom-Cook multiplication
1554 * algorithm. This is a recursive divide-and-conquer algorithm which is
1555 * more efficient for large numbers than what is commonly called the
1556 * "grade-school" algorithm used in multiplyToLen. If the numbers to be
1557 * multiplied have length n, the "grade-school" algorithm has an
1558 * asymptotic complexity of O(n^2). In contrast, 3-way Toom-Cook has a
1559 * complexity of about O(n^1.465). It achieves this increased asymptotic
1560 * performance by breaking each number into three parts and by doing 5
1561 * multiplies instead of 9 when evaluating the product. Due to overhead
1562 * (additions, shifts, and one division) in the Toom-Cook algorithm, it
1563 * should only be used when both numbers are larger than a certain
1564 * threshold (found experimentally). This threshold is generally larger
1565 * than that for Karatsuba multiplication, so this algorithm is generally
1566 * only used when numbers become significantly larger.
1567 *
1568 * The algorithm used is the "optimal" 3-way Toom-Cook algorithm outlined
1569 * by Marco Bodrato.
1570 *
1571 * See: http://bodrato.it/toom-cook/
1572 * http://bodrato.it/papers/#WAIFI2007
1573 *
1574 * "Towards Optimal Toom-Cook Multiplication for Univariate and
1575 * Multivariate Polynomials in Characteristic 2 and 0." by Marco BODRATO;
1576 * In C.Carlet and B.Sunar, Eds., "WAIFI'07 proceedings", p. 116-133,
1577 * LNCS #4547. Springer, Madrid, Spain, June 21-22, 2007.
1578 *
1579 */
1580 private static BigInteger multiplyToomCook3(BigInteger a, BigInteger b) {
1581 int alen = a.mag.length;
1582 int blen = b.mag.length;
1583
1584 int largest = Math.max(alen, blen);
1585
1586 // k is the size (in ints) of the lower-order slices.
1587 int k = (largest+2)/3; // Equal to ceil(largest/3)
1588
1589 // r is the size (in ints) of the highest-order slice.
1590 int r = largest - 2*k;
1591
1592 // Obtain slices of the numbers. a2 and b2 are the most significant
1593 // bits of the numbers a and b, and a0 and b0 the least significant.
1594 BigInteger a0, a1, a2, b0, b1, b2;
1595 a2 = a.getToomSlice(k, r, 0, largest);
1596 a1 = a.getToomSlice(k, r, 1, largest);
1597 a0 = a.getToomSlice(k, r, 2, largest);
1598 b2 = b.getToomSlice(k, r, 0, largest);
1599 b1 = b.getToomSlice(k, r, 1, largest);
1600 b0 = b.getToomSlice(k, r, 2, largest);
1601
1602 BigInteger v0, v1, v2, vm1, vinf, t1, t2, tm1, da1, db1;
1603
1604 v0 = a0.multiply(b0);
1605 da1 = a2.add(a0);
1606 db1 = b2.add(b0);
1607 vm1 = da1.subtract(a1).multiply(db1.subtract(b1));
1608 da1 = da1.add(a1);
1609 db1 = db1.add(b1);
1610 v1 = da1.multiply(db1);
1611 v2 = da1.add(a2).shiftLeft(1).subtract(a0).multiply(
1612 db1.add(b2).shiftLeft(1).subtract(b0));
1613 vinf = a2.multiply(b2);
1614
1615 // The algorithm requires two divisions by 2 and one by 3.
1616 // All divisions are known to be exact, that is, they do not produce
1617 // remainders, and all results are positive. The divisions by 2 are
1618 // implemented as right shifts which are relatively efficient, leaving
1619 // only an exact division by 3, which is done by a specialized
1620 // linear-time algorithm.
1621 t2 = v2.subtract(vm1).exactDivideBy3();
1622 tm1 = v1.subtract(vm1).shiftRight(1);
1623 t1 = v1.subtract(v0);
1624 t2 = t2.subtract(t1).shiftRight(1);
1625 t1 = t1.subtract(tm1).subtract(vinf);
1626 t2 = t2.subtract(vinf.shiftLeft(1));
1627 tm1 = tm1.subtract(t2);
1628
1629 // Number of bits to shift left.
1630 int ss = k*32;
1631
1632 BigInteger result = vinf.shiftLeft(ss).add(t2).shiftLeft(ss).add(t1).shiftLeft(ss).add(tm1).shiftLeft(ss).add(v0);
1633
1634 if (a.signum != b.signum) {
1635 return result.negate();
1636 } else {
1637 return result;
1638 }
1639 }
1640
1641
1642 /**
1643 * Returns a slice of a BigInteger for use in Toom-Cook multiplication.
1644 *
1645 * @param lowerSize The size of the lower-order bit slices.
1646 * @param upperSize The size of the higher-order bit slices.
1647 * @param slice The index of which slice is requested, which must be a
1648 * number from 0 to size-1. Slice 0 is the highest-order bits, and slice
1649 * size-1 are the lowest-order bits. Slice 0 may be of different size than
1650 * the other slices.
1651 * @param fullsize The size of the larger integer array, used to align
1652 * slices to the appropriate position when multiplying different-sized
1653 * numbers.
1654 */
1655 private BigInteger getToomSlice(int lowerSize, int upperSize, int slice,
1656 int fullsize) {
1657 int start, end, sliceSize, len, offset;
1658
1659 len = mag.length;
1660 offset = fullsize - len;
1661
1662 if (slice == 0) {
1663 start = 0 - offset;
1664 end = upperSize - 1 - offset;
1665 } else {
1666 start = upperSize + (slice-1)*lowerSize - offset;
1667 end = start + lowerSize - 1;
1668 }
1669
1670 if (start < 0) {
1671 start = 0;
1672 }
1673 if (end < 0) {
1674 return ZERO;
1675 }
1676
1677 sliceSize = (end-start) + 1;
1678
1679 if (sliceSize <= 0) {
1680 return ZERO;
1681 }
1682
1683 // While performing Toom-Cook, all slices are positive and
1684 // the sign is adjusted when the final number is composed.
1685 if (start == 0 && sliceSize >= len) {
1686 return this.abs();
1687 }
1688
1689 int intSlice[] = new int[sliceSize];
1690 System.arraycopy(mag, start, intSlice, 0, sliceSize);
1691
1692 return new BigInteger(trustedStripLeadingZeroInts(intSlice), 1);
1693 }
1694
1695 /**
1696 * Does an exact division (that is, the remainder is known to be zero)
1697 * of the specified number by 3. This is used in Toom-Cook
1698 * multiplication. This is an efficient algorithm that runs in linear
1699 * time. If the argument is not exactly divisible by 3, results are
1700 * undefined. Note that this is expected to be called with positive
1701 * arguments only.
1702 */
1703 private BigInteger exactDivideBy3() {
1704 int len = mag.length;
1705 int[] result = new int[len];
1706 long x, w, q, borrow;
1707 borrow = 0L;
1708 for (int i=len-1; i >= 0; i--) {
1709 x = (mag[i] & LONG_MASK);
1710 w = x - borrow;
1711 if (borrow > x) { // Did we make the number go negative?
1712 borrow = 1L;
1713 } else {
1714 borrow = 0L;
1715 }
1716
1717 // 0xAAAAAAAB is the modular inverse of 3 (mod 2^32). Thus,
1718 // the effect of this is to divide by 3 (mod 2^32).
1719 // This is much faster than division on most architectures.
1720 q = (w * 0xAAAAAAABL) & LONG_MASK;
1721 result[i] = (int) q;
1722
1723 // Now check the borrow. The second check can of course be
1724 // eliminated if the first fails.
1725 if (q >= 0x55555556L) {
1726 borrow++;
1727 if (q >= 0xAAAAAAABL)
1728 borrow++;
1729 }
1730 }
1731 result = trustedStripLeadingZeroInts(result);
1732 return new BigInteger(result, signum);
1733 }
1734
1735 /**
1736 * Returns a new BigInteger representing n lower ints of the number.
1737 * This is used by Karatsuba multiplication and Karatsuba squaring.
1738 */
1739 private BigInteger getLower(int n) {
1740 int len = mag.length;
1741
1742 if (len <= n) {
1743 return this;
1744 }
1745
1746 int lowerInts[] = new int[n];
1747 System.arraycopy(mag, len-n, lowerInts, 0, n);
1748
1749 return new BigInteger(trustedStripLeadingZeroInts(lowerInts), 1);
1750 }
1751
1752 /**
1753 * Returns a new BigInteger representing mag.length-n upper
1754 * ints of the number. This is used by Karatsuba multiplication and
1755 * Karatsuba squaring.
1756 */
1757 private BigInteger getUpper(int n) {
1758 int len = mag.length;
1759
1760 if (len <= n) {
1761 return ZERO;
1762 }
1763
1764 int upperLen = len - n;
1765 int upperInts[] = new int[upperLen];
1766 System.arraycopy(mag, 0, upperInts, 0, upperLen);
1767
1768 return new BigInteger(trustedStripLeadingZeroInts(upperInts), 1);
1769 }
1770
1771 // Squaring
1772
1773 /**
1774 * Returns a BigInteger whose value is {@code (this<sup>2</sup>)}.
1775 *
1776 * @return {@code this<sup>2</sup>}
1777 */
1778 private BigInteger square() {
1779 if (signum == 0) {
1780 return ZERO;
1781 }
1782 int len = mag.length;
1783
1784 if (len < KARATSUBA_SQUARE_THRESHOLD) {
1785 int[] z = squareToLen(mag, len, null);
1786 return new BigInteger(trustedStripLeadingZeroInts(z), 1);
1787 } else {
1788 if (len < TOOM_COOK_SQUARE_THRESHOLD) {
1789 return squareKaratsuba();
1790 } else {
1791 return squareToomCook3();
1792 }
1793 }
1794 }
1795
1796 /**
1797 * Squares the contents of the int array x. The result is placed into the
1798 * int array z. The contents of x are not changed.
1799 */
1800 private static final int[] squareToLen(int[] x, int len, int[] z) {
1801 /*
1802 * The algorithm used here is adapted from Colin Plumb's C library.
1803 * Technique: Consider the partial products in the multiplication
1804 * of "abcde" by itself:
1805 *
1806 * a b c d e
1807 * * a b c d e
1808 * ==================
1809 * ae be ce de ee
1810 * ad bd cd dd de
1811 * ac bc cc cd ce
1812 * ab bb bc bd be
1813 * aa ab ac ad ae
1814 *
1815 * Note that everything above the main diagonal:
1816 * ae be ce de = (abcd) * e
1817 * ad bd cd = (abc) * d
1818 * ac bc = (ab) * c
1819 * ab = (a) * b
1820 *
1821 * is a copy of everything below the main diagonal:
1822 * de
1823 * cd ce
1824 * bc bd be
1825 * ab ac ad ae
1826 *
1827 * Thus, the sum is 2 * (off the diagonal) + diagonal.
1828 *
1829 * This is accumulated beginning with the diagonal (which
1830 * consist of the squares of the digits of the input), which is then
1831 * divided by two, the off-diagonal added, and multiplied by two
1832 * again. The low bit is simply a copy of the low bit of the
1833 * input, so it doesn't need special care.
1834 */
1835 int zlen = len << 1;
1836 if (z == null || z.length < zlen)
1837 z = new int[zlen];
1838
1839 // Store the squares, right shifted one bit (i.e., divided by 2)
1840 int lastProductLowWord = 0;
1841 for (int j=0, i=0; j < len; j++) {
1842 long piece = (x[j] & LONG_MASK);
1843 long product = piece * piece;
1844 z[i++] = (lastProductLowWord << 31) | (int)(product >>> 33);
1845 z[i++] = (int)(product >>> 1);
1846 lastProductLowWord = (int)product;
1847 }
1848
1849 // Add in off-diagonal sums
1850 for (int i=len, offset=1; i > 0; i--, offset+=2) {
1851 int t = x[i-1];
1852 t = mulAdd(z, x, offset, i-1, t);
1853 addOne(z, offset-1, i, t);
1854 }
1855
1856 // Shift back up and set low bit
1857 primitiveLeftShift(z, zlen, 1);
1858 z[zlen-1] |= x[len-1] & 1;
1859
1860 return z;
1861 }
1862
1863 /**
1864 * Squares a BigInteger using the Karatsuba squaring algorithm. It should
1865 * be used when both numbers are larger than a certain threshold (found
1866 * experimentally). It is a recursive divide-and-conquer algorithm that
1867 * has better asymptotic performance than the algorithm used in
1868 * squareToLen.
1869 */
1870 private BigInteger squareKaratsuba() {
1871 int half = (mag.length+1) / 2;
1872
1873 BigInteger xl = getLower(half);
1874 BigInteger xh = getUpper(half);
1875
1876 BigInteger xhs = xh.square(); // xhs = xh^2
1877 BigInteger xls = xl.square(); // xls = xl^2
1878
1879 // xh^2 << 64 + (((xl+xh)^2 - (xh^2 + xl^2)) << 32) + xl^2
1880 return xhs.shiftLeft(half*32).add(xl.add(xh).square().subtract(xhs.add(xls))).shiftLeft(half*32).add(xls);
1881 }
1882
1883 /**
1884 * Squares a BigInteger using the 3-way Toom-Cook squaring algorithm. It
1885 * should be used when both numbers are larger than a certain threshold
1886 * (found experimentally). It is a recursive divide-and-conquer algorithm
1887 * that has better asymptotic performance than the algorithm used in
1888 * squareToLen or squareKaratsuba.
1889 */
1890 private BigInteger squareToomCook3() {
1891 int len = mag.length;
1892
1893 // k is the size (in ints) of the lower-order slices.
1894 int k = (len+2)/3; // Equal to ceil(largest/3)
1895
1896 // r is the size (in ints) of the highest-order slice.
1897 int r = len - 2*k;
1898
1899 // Obtain slices of the numbers. a2 is the most significant
1900 // bits of the number, and a0 the least significant.
1901 BigInteger a0, a1, a2;
1902 a2 = getToomSlice(k, r, 0, len);
1903 a1 = getToomSlice(k, r, 1, len);
1904 a0 = getToomSlice(k, r, 2, len);
1905 BigInteger v0, v1, v2, vm1, vinf, t1, t2, tm1, da1;
1906
1907 v0 = a0.square();
1908 da1 = a2.add(a0);
1909 vm1 = da1.subtract(a1).square();
1910 da1 = da1.add(a1);
1911 v1 = da1.square();
1912 vinf = a2.square();
1913 v2 = da1.add(a2).shiftLeft(1).subtract(a0).square();
1914
1915 // The algorithm requires two divisions by 2 and one by 3.
1916 // All divisions are known to be exact, that is, they do not produce
1917 // remainders, and all results are positive. The divisions by 2 are
1918 // implemented as right shifts which are relatively efficient, leaving
1919 // only a division by 3.
1920 // The division by 3 is done by an optimized algorithm for this case.
1921 t2 = v2.subtract(vm1).exactDivideBy3();
1922 tm1 = v1.subtract(vm1).shiftRight(1);
1923 t1 = v1.subtract(v0);
1924 t2 = t2.subtract(t1).shiftRight(1);
1925 t1 = t1.subtract(tm1).subtract(vinf);
1926 t2 = t2.subtract(vinf.shiftLeft(1));
1927 tm1 = tm1.subtract(t2);
1928
1929 // Number of bits to shift left.
1930 int ss = k*32;
1931
1932 return vinf.shiftLeft(ss).add(t2).shiftLeft(ss).add(t1).shiftLeft(ss).add(tm1).shiftLeft(ss).add(v0);
1933 }
1934
1935 // Division
1936
1937 /**
1938 * Returns a BigInteger whose value is {@code (this / val)}.
1939 *
1940 * @param val value by which this BigInteger is to be divided.
1941 * @return {@code this / val}
1942 * @throws ArithmeticException if {@code val} is zero.
1943 */
1944 public BigInteger divide(BigInteger val) {
1945 if (mag.length < BURNIKEL_ZIEGLER_THRESHOLD ||
1946 val.mag.length < BURNIKEL_ZIEGLER_THRESHOLD) {
1947 return divideKnuth(val);
1948 } else {
1949 return divideBurnikelZiegler(val);
1950 }
1951 }
1952
1953 /**
1954 * Returns a BigInteger whose value is {@code (this / val)} using an O(n^2) algorithm from Knuth.
1955 *
1956 * @param val value by which this BigInteger is to be divided.
1957 * @return {@code this / val}
1958 * @throws ArithmeticException if {@code val} is zero.
1959 * @see MutableBigInteger#divideKnuth(MutableBigInteger, MutableBigInteger, boolean)
1960 */
1961 private BigInteger divideKnuth(BigInteger val) {
1962 MutableBigInteger q = new MutableBigInteger(),
1963 a = new MutableBigInteger(this.mag),
1964 b = new MutableBigInteger(val.mag);
1965
1966 a.divideKnuth(b, q, false);
1967 return q.toBigInteger(this.signum * val.signum);
1968 }
1969
1970 /**
1971 * Returns an array of two BigIntegers containing {@code (this / val)}
1972 * followed by {@code (this % val)}.
1973 *
1974 * @param val value by which this BigInteger is to be divided, and the
1975 * remainder computed.
1976 * @return an array of two BigIntegers: the quotient {@code (this / val)}
1977 * is the initial element, and the remainder {@code (this % val)}
1978 * is the final element.
1979 * @throws ArithmeticException if {@code val} is zero.
1980 */
1981 public BigInteger[] divideAndRemainder(BigInteger val) {
1982 if (mag.length < BURNIKEL_ZIEGLER_THRESHOLD ||
1983 val.mag.length < BURNIKEL_ZIEGLER_THRESHOLD) {
1984 return divideAndRemainderKnuth(val);
1985 } else {
1986 return divideAndRemainderBurnikelZiegler(val);
1987 }
1988 }
1989
1990 /** Long division */
1991 private BigInteger[] divideAndRemainderKnuth(BigInteger val) {
1992 BigInteger[] result = new BigInteger[2];
1993 MutableBigInteger q = new MutableBigInteger(),
1994 a = new MutableBigInteger(this.mag),
1995 b = new MutableBigInteger(val.mag);
1996 MutableBigInteger r = a.divideKnuth(b, q);
1997 result[0] = q.toBigInteger(this.signum == val.signum ? 1 : -1);
1998 result[1] = r.toBigInteger(this.signum);
1999 return result;
2000 }
2001
2002 /**
2003 * Returns a BigInteger whose value is {@code (this % val)}.
2004 *
2005 * @param val value by which this BigInteger is to be divided, and the
2006 * remainder computed.
2007 * @return {@code this % val}
2008 * @throws ArithmeticException if {@code val} is zero.
2009 */
2010 public BigInteger remainder(BigInteger val) {
2011 if (mag.length < BURNIKEL_ZIEGLER_THRESHOLD ||
2012 val.mag.length < BURNIKEL_ZIEGLER_THRESHOLD) {
2013 return remainderKnuth(val);
2014 } else {
2015 return remainderBurnikelZiegler(val);
2016 }
2017 }
2018
2019 /** Long division */
2020 private BigInteger remainderKnuth(BigInteger val) {
2021 MutableBigInteger q = new MutableBigInteger(),
2022 a = new MutableBigInteger(this.mag),
2023 b = new MutableBigInteger(val.mag);
2024
2025 return a.divideKnuth(b, q).toBigInteger(this.signum);
2026 }
2027
2028 /**
2029 * Calculates {@code this / val} using the Burnikel-Ziegler algorithm.
2030 * @param val the divisor
2031 * @return {@code this / val}
2032 */
2033 private BigInteger divideBurnikelZiegler(BigInteger val) {
2034 return divideAndRemainderBurnikelZiegler(val)[0];
2035 }
2036
2037 /**
2038 * Calculates {@code this % val} using the Burnikel-Ziegler algorithm.
2039 * @param val the divisor
2040 * @return {@code this % val}
2041 */
2042 private BigInteger remainderBurnikelZiegler(BigInteger val) {
2043 return divideAndRemainderBurnikelZiegler(val)[1];
2044 }
2045
2046 /**
2047 * Computes {@code this / val} and {@code this % val} using the
2048 * Burnikel-Ziegler algorithm.
2049 * @param val the divisor
2050 * @return an array containing the quotient and remainder
2051 */
2052 private BigInteger[] divideAndRemainderBurnikelZiegler(BigInteger val) {
2053 MutableBigInteger q = new MutableBigInteger();
2054 MutableBigInteger r = new MutableBigInteger(this).divideAndRemainderBurnikelZiegler(new MutableBigInteger(val), q);
2055 BigInteger qBigInt = q.isZero() ? ZERO : q.toBigInteger(signum*val.signum);
2056 BigInteger rBigInt = r.isZero() ? ZERO : r.toBigInteger(signum);
2057 return new BigInteger[] {qBigInt, rBigInt};
2058 }
2059
2060 /**
2061 * Returns a BigInteger whose value is <tt>(this<sup>exponent</sup>)</tt>.
2062 * Note that {@code exponent} is an integer rather than a BigInteger.
2063 *
2064 * @param exponent exponent to which this BigInteger is to be raised.
2065 * @return <tt>this<sup>exponent</sup></tt>
2066 * @throws ArithmeticException {@code exponent} is negative. (This would
2067 * cause the operation to yield a non-integer value.)
2068 */
2069 public BigInteger pow(int exponent) {
2070 if (exponent < 0) {
2071 throw new ArithmeticException("Negative exponent");
2072 }
2073 if (signum == 0) {
2074 return (exponent == 0 ? ONE : this);
2075 }
2076
2077 BigInteger partToSquare = this.abs();
2078
2079 // Factor out powers of two from the base, as the exponentiation of
2080 // these can be done by left shifts only.
2081 // The remaining part can then be exponentiated faster. The
2082 // powers of two will be multiplied back at the end.
2083 int powersOfTwo = partToSquare.getLowestSetBit();
2084
2085 int remainingBits;
2086
2087 // Factor the powers of two out quickly by shifting right, if needed.
2088 if (powersOfTwo > 0) {
2089 partToSquare = partToSquare.shiftRight(powersOfTwo);
2090 remainingBits = partToSquare.bitLength();
2091 if (remainingBits == 1) { // Nothing left but +/- 1?
2092 if (signum < 0 && (exponent&1) == 1) {
2093 return NEGATIVE_ONE.shiftLeft(powersOfTwo*exponent);
2094 } else {
2095 return ONE.shiftLeft(powersOfTwo*exponent);
2096 }
2097 }
2098 } else {
2099 remainingBits = partToSquare.bitLength();
2100 if (remainingBits == 1) { // Nothing left but +/- 1?
2101 if (signum < 0 && (exponent&1) == 1) {
2102 return NEGATIVE_ONE;
2103 } else {
2104 return ONE;
2105 }
2106 }
2107 }
2108
2109 // This is a quick way to approximate the size of the result,
2110 // similar to doing log2[n] * exponent. This will give an upper bound
2111 // of how big the result can be, and which algorithm to use.
2112 int scaleFactor = remainingBits * exponent;
2113
2114 // Use slightly different algorithms for small and large operands.
2115 // See if the result will safely fit into a long. (Largest 2^63-1)
2116 if (partToSquare.mag.length == 1 && scaleFactor <= 62) {
2117 // Small number algorithm. Everything fits into a long.
2118 int newSign = (signum <0 && (exponent&1) == 1 ? -1 : 1);
2119 long result = 1;
2120 long baseToPow2 = partToSquare.mag[0] & LONG_MASK;
2121
2122 int workingExponent = exponent;
2123
2124 // Perform exponentiation using repeated squaring trick
2125 while (workingExponent != 0) {
2126 if ((workingExponent & 1) == 1) {
2127 result = result * baseToPow2;
2128 }
2129
2130 if ((workingExponent >>>= 1) != 0) {
2131 baseToPow2 = baseToPow2 * baseToPow2;
2132 }
2133 }
2134
2135 // Multiply back the powers of two (quickly, by shifting left)
2136 if (powersOfTwo > 0) {
2137 int bitsToShift = powersOfTwo*exponent;
2138 if (bitsToShift + scaleFactor <= 62) { // Fits in long?
2139 return valueOf((result << bitsToShift) * newSign);
2140 } else {
2141 return valueOf(result*newSign).shiftLeft(bitsToShift);
2142 }
2143 }
2144 else {
2145 return valueOf(result*newSign);
2146 }
2147 } else {
2148 // Large number algorithm. This is basically identical to
2149 // the algorithm above, but calls multiply() and square()
2150 // which may use more efficient algorithms for large numbers.
2151 BigInteger answer = ONE;
2152
2153 int workingExponent = exponent;
2154 // Perform exponentiation using repeated squaring trick
2155 while (workingExponent != 0) {
2156 if ((workingExponent & 1) == 1) {
2157 answer = answer.multiply(partToSquare);
2158 }
2159
2160 if ((workingExponent >>>= 1) != 0) {
2161 partToSquare = partToSquare.square();
2162 }
2163 }
2164 // Multiply back the (exponentiated) powers of two (quickly,
2165 // by shifting left)
2166 if (powersOfTwo > 0) {
2167 answer = answer.shiftLeft(powersOfTwo*exponent);
2168 }
2169
2170 if (signum < 0 && (exponent&1) == 1) {
2171 return answer.negate();
2172 } else {
2173 return answer;
2174 }
2175 }
2176 }
2177
2178 /**
2179 * Returns a BigInteger whose value is the greatest common divisor of
2180 * {@code abs(this)} and {@code abs(val)}. Returns 0 if
2181 * {@code this == 0 && val == 0}.
2182 *
2183 * @param val value with which the GCD is to be computed.
2184 * @return {@code GCD(abs(this), abs(val))}
2185 */
2186 public BigInteger gcd(BigInteger val) {
2187 if (val.signum == 0)
2188 return this.abs();
2189 else if (this.signum == 0)
2190 return val.abs();
2191
2192 MutableBigInteger a = new MutableBigInteger(this);
2193 MutableBigInteger b = new MutableBigInteger(val);
2194
2195 MutableBigInteger result = a.hybridGCD(b);
2196
2197 return result.toBigInteger(1);
2198 }
2199
2200 /**
2201 * Package private method to return bit length for an integer.
2202 */
2203 static int bitLengthForInt(int n) {
2204 return 32 - Integer.numberOfLeadingZeros(n);
2205 }
2206
2207 /**
2208 * Left shift int array a up to len by n bits. Returns the array that
2209 * results from the shift since space may have to be reallocated.
2210 */
2211 private static int[] leftShift(int[] a, int len, int n) {
2212 int nInts = n >>> 5;
2213 int nBits = n&0x1F;
2214 int bitsInHighWord = bitLengthForInt(a[0]);
2215
2216 // If shift can be done without recopy, do so
2217 if (n <= (32-bitsInHighWord)) {
2218 primitiveLeftShift(a, len, nBits);
2219 return a;
2220 } else { // Array must be resized
2221 if (nBits <= (32-bitsInHighWord)) {
2222 int result[] = new int[nInts+len];
2223 System.arraycopy(a, 0, result, 0, len);
2224 primitiveLeftShift(result, result.length, nBits);
2225 return result;
2226 } else {
2227 int result[] = new int[nInts+len+1];
2228 System.arraycopy(a, 0, result, 0, len);
2229 primitiveRightShift(result, result.length, 32 - nBits);
2230 return result;
2231 }
2232 }
2233 }
2234
2235 // shifts a up to len right n bits assumes no leading zeros, 0<n<32
2236 static void primitiveRightShift(int[] a, int len, int n) {
2237 int n2 = 32 - n;
2238 for (int i=len-1, c=a[i]; i > 0; i--) {
2239 int b = c;
2240 c = a[i-1];
2241 a[i] = (c << n2) | (b >>> n);
2242 }
2243 a[0] >>>= n;
2244 }
2245
2246 // shifts a up to len left n bits assumes no leading zeros, 0<=n<32
2247 static void primitiveLeftShift(int[] a, int len, int n) {
2248 if (len == 0 || n == 0)
2249 return;
2250
2251 int n2 = 32 - n;
2252 for (int i=0, c=a[i], m=i+len-1; i < m; i++) {
2253 int b = c;
2254 c = a[i+1];
2255 a[i] = (b << n) | (c >>> n2);
2256 }
2257 a[len-1] <<= n;
2258 }
2259
2260 /**
2261 * Calculate bitlength of contents of the first len elements an int array,
2262 * assuming there are no leading zero ints.
2263 */
2264 private static int bitLength(int[] val, int len) {
2265 if (len == 0)
2266 return 0;
2267 return ((len - 1) << 5) + bitLengthForInt(val[0]);
2268 }
2269
2270 /**
2271 * Returns a BigInteger whose value is the absolute value of this
2272 * BigInteger.
2273 *
2274 * @return {@code abs(this)}
2275 */
2276 public BigInteger abs() {
2277 return (signum >= 0 ? this : this.negate());
2278 }
2279
2280 /**
2281 * Returns a BigInteger whose value is {@code (-this)}.
2282 *
2283 * @return {@code -this}
2284 */
2285 public BigInteger negate() {
2286 return new BigInteger(this.mag, -this.signum);
2287 }
2288
2289 /**
2290 * Returns the signum function of this BigInteger.
2291 *
2292 * @return -1, 0 or 1 as the value of this BigInteger is negative, zero or
2293 * positive.
2294 */
2295 public int signum() {
2296 return this.signum;
2297 }
2298
2299 // Modular Arithmetic Operations
2300
2301 /**
2302 * Returns a BigInteger whose value is {@code (this mod m}). This method
2303 * differs from {@code remainder} in that it always returns a
2304 * <i>non-negative</i> BigInteger.
2305 *
2306 * @param m the modulus.
2307 * @return {@code this mod m}
2308 * @throws ArithmeticException {@code m} ≤ 0
2309 * @see #remainder
2310 */
2311 public BigInteger mod(BigInteger m) {
2312 if (m.signum <= 0)
2313 throw new ArithmeticException("BigInteger: modulus not positive");
2314
2315 BigInteger result = this.remainder(m);
2316 return (result.signum >= 0 ? result : result.add(m));
2317 }
2318
2319 /**
2320 * Returns a BigInteger whose value is
2321 * <tt>(this<sup>exponent</sup> mod m)</tt>. (Unlike {@code pow}, this
2322 * method permits negative exponents.)
2323 *
2324 * @param exponent the exponent.
2325 * @param m the modulus.
2326 * @return <tt>this<sup>exponent</sup> mod m</tt>
2327 * @throws ArithmeticException {@code m} ≤ 0 or the exponent is
2328 * negative and this BigInteger is not <i>relatively
2329 * prime</i> to {@code m}.
2330 * @see #modInverse
2331 */
2332 public BigInteger modPow(BigInteger exponent, BigInteger m) {
2333 if (m.signum <= 0)
2334 throw new ArithmeticException("BigInteger: modulus not positive");
2335
2336 // Trivial cases
2337 if (exponent.signum == 0)
2338 return (m.equals(ONE) ? ZERO : ONE);
2339
2340 if (this.equals(ONE))
2341 return (m.equals(ONE) ? ZERO : ONE);
2342
2343 if (this.equals(ZERO) && exponent.signum >= 0)
2344 return ZERO;
2345
2346 if (this.equals(negConst[1]) && (!exponent.testBit(0)))
2347 return (m.equals(ONE) ? ZERO : ONE);
2348
2349 boolean invertResult;
2350 if ((invertResult = (exponent.signum < 0)))
2351 exponent = exponent.negate();
2352
2353 BigInteger base = (this.signum < 0 || this.compareTo(m) >= 0
2354 ? this.mod(m) : this);
2355 BigInteger result;
2356 if (m.testBit(0)) { // odd modulus
2357 result = base.oddModPow(exponent, m);
2358 } else {
2359 /*
2360 * Even modulus. Tear it into an "odd part" (m1) and power of two
2361 * (m2), exponentiate mod m1, manually exponentiate mod m2, and
2362 * use Chinese Remainder Theorem to combine results.
2363 */
2364
2365 // Tear m apart into odd part (m1) and power of 2 (m2)
2366 int p = m.getLowestSetBit(); // Max pow of 2 that divides m
2367
2368 BigInteger m1 = m.shiftRight(p); // m/2**p
2369 BigInteger m2 = ONE.shiftLeft(p); // 2**p
2370
2371 // Calculate new base from m1
2372 BigInteger base2 = (this.signum < 0 || this.compareTo(m1) >= 0
2373 ? this.mod(m1) : this);
2374
2375 // Caculate (base ** exponent) mod m1.
2376 BigInteger a1 = (m1.equals(ONE) ? ZERO :
2377 base2.oddModPow(exponent, m1));
2378
2379 // Calculate (this ** exponent) mod m2
2380 BigInteger a2 = base.modPow2(exponent, p);
2381
2382 // Combine results using Chinese Remainder Theorem
2383 BigInteger y1 = m2.modInverse(m1);
2384 BigInteger y2 = m1.modInverse(m2);
2385
2386 result = a1.multiply(m2).multiply(y1).add
2387 (a2.multiply(m1).multiply(y2)).mod(m);
2388 }
2389
2390 return (invertResult ? result.modInverse(m) : result);
2391 }
2392
2393 static int[] bnExpModThreshTable = {7, 25, 81, 241, 673, 1793,
2394 Integer.MAX_VALUE}; // Sentinel
2395
2396 /**
2397 * Returns a BigInteger whose value is x to the power of y mod z.
2398 * Assumes: z is odd && x < z.
2399 */
2400 private BigInteger oddModPow(BigInteger y, BigInteger z) {
2401 /*
2402 * The algorithm is adapted from Colin Plumb's C library.
2403 *
2404 * The window algorithm:
2405 * The idea is to keep a running product of b1 = n^(high-order bits of exp)
2406 * and then keep appending exponent bits to it. The following patterns
2407 * apply to a 3-bit window (k = 3):
2408 * To append 0: square
2409 * To append 1: square, multiply by n^1
2410 * To append 10: square, multiply by n^1, square
2411 * To append 11: square, square, multiply by n^3
2412 * To append 100: square, multiply by n^1, square, square
2413 * To append 101: square, square, square, multiply by n^5
2414 * To append 110: square, square, multiply by n^3, square
2415 * To append 111: square, square, square, multiply by n^7
2416 *
2417 * Since each pattern involves only one multiply, the longer the pattern
2418 * the better, except that a 0 (no multiplies) can be appended directly.
2419 * We precompute a table of odd powers of n, up to 2^k, and can then
2420 * multiply k bits of exponent at a time. Actually, assuming random
2421 * exponents, there is on average one zero bit between needs to
2422 * multiply (1/2 of the time there's none, 1/4 of the time there's 1,
2423 * 1/8 of the time, there's 2, 1/32 of the time, there's 3, etc.), so
2424 * you have to do one multiply per k+1 bits of exponent.
2425 *
2426 * The loop walks down the exponent, squaring the result buffer as
2427 * it goes. There is a wbits+1 bit lookahead buffer, buf, that is
2428 * filled with the upcoming exponent bits. (What is read after the
2429 * end of the exponent is unimportant, but it is filled with zero here.)
2430 * When the most-significant bit of this buffer becomes set, i.e.
2431 * (buf & tblmask) != 0, we have to decide what pattern to multiply
2432 * by, and when to do it. We decide, remember to do it in future
2433 * after a suitable number of squarings have passed (e.g. a pattern
2434 * of "100" in the buffer requires that we multiply by n^1 immediately;
2435 * a pattern of "110" calls for multiplying by n^3 after one more
2436 * squaring), clear the buffer, and continue.
2437 *
2438 * When we start, there is one more optimization: the result buffer
2439 * is implcitly one, so squaring it or multiplying by it can be
2440 * optimized away. Further, if we start with a pattern like "100"
2441 * in the lookahead window, rather than placing n into the buffer
2442 * and then starting to square it, we have already computed n^2
2443 * to compute the odd-powers table, so we can place that into
2444 * the buffer and save a squaring.
2445 *
2446 * This means that if you have a k-bit window, to compute n^z,
2447 * where z is the high k bits of the exponent, 1/2 of the time
2448 * it requires no squarings. 1/4 of the time, it requires 1
2449 * squaring, ... 1/2^(k-1) of the time, it reqires k-2 squarings.
2450 * And the remaining 1/2^(k-1) of the time, the top k bits are a
2451 * 1 followed by k-1 0 bits, so it again only requires k-2
2452 * squarings, not k-1. The average of these is 1. Add that
2453 * to the one squaring we have to do to compute the table,
2454 * and you'll see that a k-bit window saves k-2 squarings
2455 * as well as reducing the multiplies. (It actually doesn't
2456 * hurt in the case k = 1, either.)
2457 */
2458 // Special case for exponent of one
2459 if (y.equals(ONE))
2460 return this;
2461
2462 // Special case for base of zero
2463 if (signum == 0)
2464 return ZERO;
2465
2466 int[] base = mag.clone();
2467 int[] exp = y.mag;
2468 int[] mod = z.mag;
2469 int modLen = mod.length;
2470
2471 // Select an appropriate window size
2472 int wbits = 0;
2473 int ebits = bitLength(exp, exp.length);
2474 // if exponent is 65537 (0x10001), use minimum window size
2475 if ((ebits != 17) || (exp[0] != 65537)) {
2476 while (ebits > bnExpModThreshTable[wbits]) {
2477 wbits++;
2478 }
2479 }
2480
2481 // Calculate appropriate table size
2482 int tblmask = 1 << wbits;
2483
2484 // Allocate table for precomputed odd powers of base in Montgomery form
2485 int[][] table = new int[tblmask][];
2486 for (int i=0; i < tblmask; i++)
2487 table[i] = new int[modLen];
2488
2489 // Compute the modular inverse
2490 int inv = -MutableBigInteger.inverseMod32(mod[modLen-1]);
2491
2492 // Convert base to Montgomery form
2493 int[] a = leftShift(base, base.length, modLen << 5);
2494
2495 MutableBigInteger q = new MutableBigInteger(),
2496 a2 = new MutableBigInteger(a),
2497 b2 = new MutableBigInteger(mod);
2498
2499 MutableBigInteger r= a2.divide(b2, q);
2500 table[0] = r.toIntArray();
2501
2502 // Pad table[0] with leading zeros so its length is at least modLen
2503 if (table[0].length < modLen) {
2504 int offset = modLen - table[0].length;
2505 int[] t2 = new int[modLen];
2506 for (int i=0; i < table[0].length; i++)
2507 t2[i+offset] = table[0][i];
2508 table[0] = t2;
2509 }
2510
2511 // Set b to the square of the base
2512 int[] b = squareToLen(table[0], modLen, null);
2513 b = montReduce(b, mod, modLen, inv);
2514
2515 // Set t to high half of b
2516 int[] t = Arrays.copyOf(b, modLen);
2517
2518 // Fill in the table with odd powers of the base
2519 for (int i=1; i < tblmask; i++) {
2520 int[] prod = multiplyToLen(t, modLen, table[i-1], modLen, null);
2521 table[i] = montReduce(prod, mod, modLen, inv);
2522 }
2523
2524 // Pre load the window that slides over the exponent
2525 int bitpos = 1 << ((ebits-1) & (32-1));
2526
2527 int buf = 0;
2528 int elen = exp.length;
2529 int eIndex = 0;
2530 for (int i = 0; i <= wbits; i++) {
2531 buf = (buf << 1) | (((exp[eIndex] & bitpos) != 0)?1:0);
2532 bitpos >>>= 1;
2533 if (bitpos == 0) {
2534 eIndex++;
2535 bitpos = 1 << (32-1);
2536 elen--;
2537 }
2538 }
2539
2540 int multpos = ebits;
2541
2542 // The first iteration, which is hoisted out of the main loop
2543 ebits--;
2544 boolean isone = true;
2545
2546 multpos = ebits - wbits;
2547 while ((buf & 1) == 0) {
2548 buf >>>= 1;
2549 multpos++;
2550 }
2551
2552 int[] mult = table[buf >>> 1];
2553
2554 buf = 0;
2555 if (multpos == ebits)
2556 isone = false;
2557
2558 // The main loop
2559 while (true) {
2560 ebits--;
2561 // Advance the window
2562 buf <<= 1;
2563
2564 if (elen != 0) {
2565 buf |= ((exp[eIndex] & bitpos) != 0) ? 1 : 0;
2566 bitpos >>>= 1;
2567 if (bitpos == 0) {
2568 eIndex++;
2569 bitpos = 1 << (32-1);
2570 elen--;
2571 }
2572 }
2573
2574 // Examine the window for pending multiplies
2575 if ((buf & tblmask) != 0) {
2576 multpos = ebits - wbits;
2577 while ((buf & 1) == 0) {
2578 buf >>>= 1;
2579 multpos++;
2580 }
2581 mult = table[buf >>> 1];
2582 buf = 0;
2583 }
2584
2585 // Perform multiply
2586 if (ebits == multpos) {
2587 if (isone) {
2588 b = mult.clone();
2589 isone = false;
2590 } else {
2591 t = b;
2592 a = multiplyToLen(t, modLen, mult, modLen, a);
2593 a = montReduce(a, mod, modLen, inv);
2594 t = a; a = b; b = t;
2595 }
2596 }
2597
2598 // Check if done
2599 if (ebits == 0)
2600 break;
2601
2602 // Square the input
2603 if (!isone) {
2604 t = b;
2605 a = squareToLen(t, modLen, a);
2606 a = montReduce(a, mod, modLen, inv);
2607 t = a; a = b; b = t;
2608 }
2609 }
2610
2611 // Convert result out of Montgomery form and return
2612 int[] t2 = new int[2*modLen];
2613 System.arraycopy(b, 0, t2, modLen, modLen);
2614
2615 b = montReduce(t2, mod, modLen, inv);
2616
2617 t2 = Arrays.copyOf(b, modLen);
2618
2619 return new BigInteger(1, t2);
2620 }
2621
2622 /**
2623 * Montgomery reduce n, modulo mod. This reduces modulo mod and divides
2624 * by 2^(32*mlen). Adapted from Colin Plumb's C library.
2625 */
2626 private static int[] montReduce(int[] n, int[] mod, int mlen, int inv) {
2627 int c=0;
2628 int len = mlen;
2629 int offset=0;
2630
2631 do {
2632 int nEnd = n[n.length-1-offset];
2633 int carry = mulAdd(n, mod, offset, mlen, inv * nEnd);
2634 c += addOne(n, offset, mlen, carry);
2635 offset++;
2636 } while (--len > 0);
2637
2638 while (c > 0)
2639 c += subN(n, mod, mlen);
2640
2641 while (intArrayCmpToLen(n, mod, mlen) >= 0)
2642 subN(n, mod, mlen);
2643
2644 return n;
2645 }
2646
2647
2648 /*
2649 * Returns -1, 0 or +1 as big-endian unsigned int array arg1 is less than,
2650 * equal to, or greater than arg2 up to length len.
2651 */
2652 private static int intArrayCmpToLen(int[] arg1, int[] arg2, int len) {
2653 for (int i=0; i < len; i++) {
2654 long b1 = arg1[i] & LONG_MASK;
2655 long b2 = arg2[i] & LONG_MASK;
2656 if (b1 < b2)
2657 return -1;
2658 if (b1 > b2)
2659 return 1;
2660 }
2661 return 0;
2662 }
2663
2664 /**
2665 * Subtracts two numbers of same length, returning borrow.
2666 */
2667 private static int subN(int[] a, int[] b, int len) {
2668 long sum = 0;
2669
2670 while (--len >= 0) {
2671 sum = (a[len] & LONG_MASK) -
2672 (b[len] & LONG_MASK) + (sum >> 32);
2673 a[len] = (int)sum;
2674 }
2675
2676 return (int)(sum >> 32);
2677 }
2678
2679 /**
2680 * Multiply an array by one word k and add to result, return the carry
2681 */
2682 static int mulAdd(int[] out, int[] in, int offset, int len, int k) {
2683 long kLong = k & LONG_MASK;
2684 long carry = 0;
2685
2686 offset = out.length-offset - 1;
2687 for (int j=len-1; j >= 0; j--) {
2688 long product = (in[j] & LONG_MASK) * kLong +
2689 (out[offset] & LONG_MASK) + carry;
2690 out[offset--] = (int)product;
2691 carry = product >>> 32;
2692 }
2693 return (int)carry;
2694 }
2695
2696 /**
2697 * Add one word to the number a mlen words into a. Return the resulting
2698 * carry.
2699 */
2700 static int addOne(int[] a, int offset, int mlen, int carry) {
2701 offset = a.length-1-mlen-offset;
2702 long t = (a[offset] & LONG_MASK) + (carry & LONG_MASK);
2703
2704 a[offset] = (int)t;
2705 if ((t >>> 32) == 0)
2706 return 0;
2707 while (--mlen >= 0) {
2708 if (--offset < 0) { // Carry out of number
2709 return 1;
2710 } else {
2711 a[offset]++;
2712 if (a[offset] != 0)
2713 return 0;
2714 }
2715 }
2716 return 1;
2717 }
2718
2719 /**
2720 * Returns a BigInteger whose value is (this ** exponent) mod (2**p)
2721 */
2722 private BigInteger modPow2(BigInteger exponent, int p) {
2723 /*
2724 * Perform exponentiation using repeated squaring trick, chopping off
2725 * high order bits as indicated by modulus.
2726 */
2727 BigInteger result = ONE;
2728 BigInteger baseToPow2 = this.mod2(p);
2729 int expOffset = 0;
2730
2731 int limit = exponent.bitLength();
2732
2733 if (this.testBit(0))
2734 limit = (p-1) < limit ? (p-1) : limit;
2735
2736 while (expOffset < limit) {
2737 if (exponent.testBit(expOffset))
2738 result = result.multiply(baseToPow2).mod2(p);
2739 expOffset++;
2740 if (expOffset < limit)
2741 baseToPow2 = baseToPow2.square().mod2(p);
2742 }
2743
2744 return result;
2745 }
2746
2747 /**
2748 * Returns a BigInteger whose value is this mod(2**p).
2749 * Assumes that this {@code BigInteger >= 0} and {@code p > 0}.
2750 */
2751 private BigInteger mod2(int p) {
2752 if (bitLength() <= p)
2753 return this;
2754
2755 // Copy remaining ints of mag
2756 int numInts = (p + 31) >>> 5;
2757 int[] mag = new int[numInts];
2758 System.arraycopy(this.mag, (this.mag.length - numInts), mag, 0, numInts);
2759
2760 // Mask out any excess bits
2761 int excessBits = (numInts << 5) - p;
2762 mag[0] &= (1L << (32-excessBits)) - 1;
2763
2764 return (mag[0] == 0 ? new BigInteger(1, mag) : new BigInteger(mag, 1));
2765 }
2766
2767 /**
2768 * Returns a BigInteger whose value is {@code (this}<sup>-1</sup> {@code mod m)}.
2769 *
2770 * @param m the modulus.
2771 * @return {@code this}<sup>-1</sup> {@code mod m}.
2772 * @throws ArithmeticException {@code m} ≤ 0, or this BigInteger
2773 * has no multiplicative inverse mod m (that is, this BigInteger
2774 * is not <i>relatively prime</i> to m).
2775 */
2776 public BigInteger modInverse(BigInteger m) {
2777 if (m.signum != 1)
2778 throw new ArithmeticException("BigInteger: modulus not positive");
2779
2780 if (m.equals(ONE))
2781 return ZERO;
2782
2783 // Calculate (this mod m)
2784 BigInteger modVal = this;
2785 if (signum < 0 || (this.compareMagnitude(m) >= 0))
2786 modVal = this.mod(m);
2787
2788 if (modVal.equals(ONE))
2789 return ONE;
2790
2791 MutableBigInteger a = new MutableBigInteger(modVal);
2792 MutableBigInteger b = new MutableBigInteger(m);
2793
2794 MutableBigInteger result = a.mutableModInverse(b);
2795 return result.toBigInteger(1);
2796 }
2797
2798 // Shift Operations
2799
2800 /**
2801 * Returns a BigInteger whose value is {@code (this << n)}.
2802 * The shift distance, {@code n}, may be negative, in which case
2803 * this method performs a right shift.
2804 * (Computes <tt>floor(this * 2<sup>n</sup>)</tt>.)
2805 *
2806 * @param n shift distance, in bits.
2807 * @return {@code this << n}
2808 * @throws ArithmeticException if the shift distance is {@code
2809 * Integer.MIN_VALUE}.
2810 * @see #shiftRight
2811 */
2812 public BigInteger shiftLeft(int n) {
2813 if (signum == 0)
2814 return ZERO;
2815 if (n == 0)
2816 return this;
2817 if (n < 0) {
2818 if (n == Integer.MIN_VALUE) {
2819 throw new ArithmeticException("Shift distance of Integer.MIN_VALUE not supported.");
2820 } else {
2821 return shiftRight(-n);
2822 }
2823 }
2824 int[] newMag = shiftLeft(mag, n);
2825
2826 return new BigInteger(newMag, signum);
2827 }
2828
2829 private static int[] shiftLeft(int[] mag, int n) {
2830 int nInts = n >>> 5;
2831 int nBits = n & 0x1f;
2832 int magLen = mag.length;
2833 int newMag[] = null;
2834
2835 if (nBits == 0) {
2836 newMag = new int[magLen + nInts];
2837 System.arraycopy(mag, 0, newMag, 0, magLen);
2838 } else {
2839 int i = 0;
2840 int nBits2 = 32 - nBits;
2841 int highBits = mag[0] >>> nBits2;
2842 if (highBits != 0) {
2843 newMag = new int[magLen + nInts + 1];
2844 newMag[i++] = highBits;
2845 } else {
2846 newMag = new int[magLen + nInts];
2847 }
2848 int j=0;
2849 while (j < magLen-1)
2850 newMag[i++] = mag[j++] << nBits | mag[j] >>> nBits2;
2851 newMag[i] = mag[j] << nBits;
2852 }
2853 return newMag;
2854 }
2855
2856 /**
2857 * Returns a BigInteger whose value is {@code (this >> n)}. Sign
2858 * extension is performed. The shift distance, {@code n}, may be
2859 * negative, in which case this method performs a left shift.
2860 * (Computes <tt>floor(this / 2<sup>n</sup>)</tt>.)
2861 *
2862 * @param n shift distance, in bits.
2863 * @return {@code this >> n}
2864 * @throws ArithmeticException if the shift distance is {@code
2865 * Integer.MIN_VALUE}.
2866 * @see #shiftLeft
2867 */
2868 public BigInteger shiftRight(int n) {
2869 if (n == 0)
2870 return this;
2871 if (n < 0) {
2872 if (n == Integer.MIN_VALUE) {
2873 throw new ArithmeticException("Shift distance of Integer.MIN_VALUE not supported.");
2874 } else {
2875 return shiftLeft(-n);
2876 }
2877 }
2878
2879 int nInts = n >>> 5;
2880 int nBits = n & 0x1f;
2881 int magLen = mag.length;
2882 int newMag[] = null;
2883
2884 // Special case: entire contents shifted off the end
2885 if (nInts >= magLen)
2886 return (signum >= 0 ? ZERO : negConst[1]);
2887
2888 if (nBits == 0) {
2889 int newMagLen = magLen - nInts;
2890 newMag = Arrays.copyOf(mag, newMagLen);
2891 } else {
2892 int i = 0;
2893 int highBits = mag[0] >>> nBits;
2894 if (highBits != 0) {
2895 newMag = new int[magLen - nInts];
2896 newMag[i++] = highBits;
2897 } else {
2898 newMag = new int[magLen - nInts -1];
2899 }
2900
2901 int nBits2 = 32 - nBits;
2902 int j=0;
2903 while (j < magLen - nInts - 1)
2904 newMag[i++] = (mag[j++] << nBits2) | (mag[j] >>> nBits);
2905 }
2906
2907 if (signum < 0) {
2908 // Find out whether any one-bits were shifted off the end.
2909 boolean onesLost = false;
2910 for (int i=magLen-1, j=magLen-nInts; i >= j && !onesLost; i--)
2911 onesLost = (mag[i] != 0);
2912 if (!onesLost && nBits != 0)
2913 onesLost = (mag[magLen - nInts - 1] << (32 - nBits) != 0);
2914
2915 if (onesLost)
2916 newMag = javaIncrement(newMag);
2917 }
2918
2919 return new BigInteger(newMag, signum);
2920 }
2921
2922 int[] javaIncrement(int[] val) {
2923 int lastSum = 0;
2924 for (int i=val.length-1; i >= 0 && lastSum == 0; i--)
2925 lastSum = (val[i] += 1);
2926 if (lastSum == 0) {
2927 val = new int[val.length+1];
2928 val[0] = 1;
2929 }
2930 return val;
2931 }
2932
2933 // Bitwise Operations
2934
2935 /**
2936 * Returns a BigInteger whose value is {@code (this & val)}. (This
2937 * method returns a negative BigInteger if and only if this and val are
2938 * both negative.)
2939 *
2940 * @param val value to be AND'ed with this BigInteger.
2941 * @return {@code this & val}
2942 */
2943 public BigInteger and(BigInteger val) {
2944 int[] result = new int[Math.max(intLength(), val.intLength())];
2945 for (int i=0; i < result.length; i++)
2946 result[i] = (getInt(result.length-i-1)
2947 & val.getInt(result.length-i-1));
2948
2949 return valueOf(result);
2950 }
2951
2952 /**
2953 * Returns a BigInteger whose value is {@code (this | val)}. (This method
2954 * returns a negative BigInteger if and only if either this or val is
2955 * negative.)
2956 *
2957 * @param val value to be OR'ed with this BigInteger.
2958 * @return {@code this | val}
2959 */
2960 public BigInteger or(BigInteger val) {
2961 int[] result = new int[Math.max(intLength(), val.intLength())];
2962 for (int i=0; i < result.length; i++)
2963 result[i] = (getInt(result.length-i-1)
2964 | val.getInt(result.length-i-1));
2965
2966 return valueOf(result);
2967 }
2968
2969 /**
2970 * Returns a BigInteger whose value is {@code (this ^ val)}. (This method
2971 * returns a negative BigInteger if and only if exactly one of this and
2972 * val are negative.)
2973 *
2974 * @param val value to be XOR'ed with this BigInteger.
2975 * @return {@code this ^ val}
2976 */
2977 public BigInteger xor(BigInteger val) {
2978 int[] result = new int[Math.max(intLength(), val.intLength())];
2979 for (int i=0; i < result.length; i++)
2980 result[i] = (getInt(result.length-i-1)
2981 ^ val.getInt(result.length-i-1));
2982
2983 return valueOf(result);
2984 }
2985
2986 /**
2987 * Returns a BigInteger whose value is {@code (~this)}. (This method
2988 * returns a negative value if and only if this BigInteger is
2989 * non-negative.)
2990 *
2991 * @return {@code ~this}
2992 */
2993 public BigInteger not() {
2994 int[] result = new int[intLength()];
2995 for (int i=0; i < result.length; i++)
2996 result[i] = ~getInt(result.length-i-1);
2997
2998 return valueOf(result);
2999 }
3000
3001 /**
3002 * Returns a BigInteger whose value is {@code (this & ~val)}. This
3003 * method, which is equivalent to {@code and(val.not())}, is provided as
3004 * a convenience for masking operations. (This method returns a negative
3005 * BigInteger if and only if {@code this} is negative and {@code val} is
3006 * positive.)
3007 *
3008 * @param val value to be complemented and AND'ed with this BigInteger.
3009 * @return {@code this & ~val}
3010 */
3011 public BigInteger andNot(BigInteger val) {
3012 int[] result = new int[Math.max(intLength(), val.intLength())];
3013 for (int i=0; i < result.length; i++)
3014 result[i] = (getInt(result.length-i-1)
3015 & ~val.getInt(result.length-i-1));
3016
3017 return valueOf(result);
3018 }
3019
3020
3021 // Single Bit Operations
3022
3023 /**
3024 * Returns {@code true} if and only if the designated bit is set.
3025 * (Computes {@code ((this & (1<<n)) != 0)}.)
3026 *
3027 * @param n index of bit to test.
3028 * @return {@code true} if and only if the designated bit is set.
3029 * @throws ArithmeticException {@code n} is negative.
3030 */
3031 public boolean testBit(int n) {
3032 if (n < 0)
3033 throw new ArithmeticException("Negative bit address");
3034
3035 return (getInt(n >>> 5) & (1 << (n & 31))) != 0;
3036 }
3037
3038 /**
3039 * Returns a BigInteger whose value is equivalent to this BigInteger
3040 * with the designated bit set. (Computes {@code (this | (1<<n))}.)
3041 *
3042 * @param n index of bit to set.
3043 * @return {@code this | (1<<n)}
3044 * @throws ArithmeticException {@code n} is negative.
3045 */
3046 public BigInteger setBit(int n) {
3047 if (n < 0)
3048 throw new ArithmeticException("Negative bit address");
3049
3050 int intNum = n >>> 5;
3051 int[] result = new int[Math.max(intLength(), intNum+2)];
3052
3053 for (int i=0; i < result.length; i++)
3054 result[result.length-i-1] = getInt(i);
3055
3056 result[result.length-intNum-1] |= (1 << (n & 31));
3057
3058 return valueOf(result);
3059 }
3060
3061 /**
3062 * Returns a BigInteger whose value is equivalent to this BigInteger
3063 * with the designated bit cleared.
3064 * (Computes {@code (this & ~(1<<n))}.)
3065 *
3066 * @param n index of bit to clear.
3067 * @return {@code this & ~(1<<n)}
3068 * @throws ArithmeticException {@code n} is negative.
3069 */
3070 public BigInteger clearBit(int n) {
3071 if (n < 0)
3072 throw new ArithmeticException("Negative bit address");
3073
3074 int intNum = n >>> 5;
3075 int[] result = new int[Math.max(intLength(), ((n + 1) >>> 5) + 1)];
3076
3077 for (int i=0; i < result.length; i++)
3078 result[result.length-i-1] = getInt(i);
3079
3080 result[result.length-intNum-1] &= ~(1 << (n & 31));
3081
3082 return valueOf(result);
3083 }
3084
3085 /**
3086 * Returns a BigInteger whose value is equivalent to this BigInteger
3087 * with the designated bit flipped.
3088 * (Computes {@code (this ^ (1<<n))}.)
3089 *
3090 * @param n index of bit to flip.
3091 * @return {@code this ^ (1<<n)}
3092 * @throws ArithmeticException {@code n} is negative.
3093 */
3094 public BigInteger flipBit(int n) {
3095 if (n < 0)
3096 throw new ArithmeticException("Negative bit address");
3097
3098 int intNum = n >>> 5;
3099 int[] result = new int[Math.max(intLength(), intNum+2)];
3100
3101 for (int i=0; i < result.length; i++)
3102 result[result.length-i-1] = getInt(i);
3103
3104 result[result.length-intNum-1] ^= (1 << (n & 31));
3105
3106 return valueOf(result);
3107 }
3108
3109 /**
3110 * Returns the index of the rightmost (lowest-order) one bit in this
3111 * BigInteger (the number of zero bits to the right of the rightmost
3112 * one bit). Returns -1 if this BigInteger contains no one bits.
3113 * (Computes {@code (this == 0? -1 : log2(this & -this))}.)
3114 *
3115 * @return index of the rightmost one bit in this BigInteger.
3116 */
3117 public int getLowestSetBit() {
3118 @SuppressWarnings("deprecation") int lsb = lowestSetBit - 2;
3119 if (lsb == -2) { // lowestSetBit not initialized yet
3120 lsb = 0;
3121 if (signum == 0) {
3122 lsb -= 1;
3123 } else {
3124 // Search for lowest order nonzero int
3125 int i,b;
3126 for (i=0; (b = getInt(i)) == 0; i++)
3127 ;
3128 lsb += (i << 5) + Integer.numberOfTrailingZeros(b);
3129 }
3130 lowestSetBit = lsb + 2;
3131 }
3132 return lsb;
3133 }
3134
3135
3136 // Miscellaneous Bit Operations
3137
3138 /**
3139 * Returns the number of bits in the minimal two's-complement
3140 * representation of this BigInteger, <i>excluding</i> a sign bit.
3141 * For positive BigIntegers, this is equivalent to the number of bits in
3142 * the ordinary binary representation. (Computes
3143 * {@code (ceil(log2(this < 0 ? -this : this+1)))}.)
3144 *
3145 * @return number of bits in the minimal two's-complement
3146 * representation of this BigInteger, <i>excluding</i> a sign bit.
3147 */
3148 public int bitLength() {
3149 @SuppressWarnings("deprecation") int n = bitLength - 1;
3150 if (n == -1) { // bitLength not initialized yet
3151 int[] m = mag;
3152 int len = m.length;
3153 if (len == 0) {
3154 n = 0; // offset by one to initialize
3155 } else {
3156 // Calculate the bit length of the magnitude
3157 int magBitLength = ((len - 1) << 5) + bitLengthForInt(mag[0]);
3158 if (signum < 0) {
3159 // Check if magnitude is a power of two
3160 boolean pow2 = (Integer.bitCount(mag[0]) == 1);
3161 for (int i=1; i< len && pow2; i++)
3162 pow2 = (mag[i] == 0);
3163
3164 n = (pow2 ? magBitLength -1 : magBitLength);
3165 } else {
3166 n = magBitLength;
3167 }
3168 }
3169 bitLength = n + 1;
3170 }
3171 return n;
3172 }
3173
3174 /**
3175 * Returns the number of bits in the two's complement representation
3176 * of this BigInteger that differ from its sign bit. This method is
3177 * useful when implementing bit-vector style sets atop BigIntegers.
3178 *
3179 * @return number of bits in the two's complement representation
3180 * of this BigInteger that differ from its sign bit.
3181 */
3182 public int bitCount() {
3183 @SuppressWarnings("deprecation") int bc = bitCount - 1;
3184 if (bc == -1) { // bitCount not initialized yet
3185 bc = 0; // offset by one to initialize
3186 // Count the bits in the magnitude
3187 for (int i=0; i < mag.length; i++)
3188 bc += Integer.bitCount(mag[i]);
3189 if (signum < 0) {
3190 // Count the trailing zeros in the magnitude
3191 int magTrailingZeroCount = 0, j;
3192 for (j=mag.length-1; mag[j] == 0; j--)
3193 magTrailingZeroCount += 32;
3194 magTrailingZeroCount += Integer.numberOfTrailingZeros(mag[j]);
3195 bc += magTrailingZeroCount - 1;
3196 }
3197 bitCount = bc + 1;
3198 }
3199 return bc;
3200 }
3201
3202 // Primality Testing
3203
3204 /**
3205 * Returns {@code true} if this BigInteger is probably prime,
3206 * {@code false} if it's definitely composite. If
3207 * {@code certainty} is ≤ 0, {@code true} is
3208 * returned.
3209 *
3210 * @param certainty a measure of the uncertainty that the caller is
3211 * willing to tolerate: if the call returns {@code true}
3212 * the probability that this BigInteger is prime exceeds
3213 * (1 - 1/2<sup>{@code certainty}</sup>). The execution time of
3214 * this method is proportional to the value of this parameter.
3215 * @return {@code true} if this BigInteger is probably prime,
3216 * {@code false} if it's definitely composite.
3217 */
3218 public boolean isProbablePrime(int certainty) {
3219 if (certainty <= 0)
3220 return true;
3221 BigInteger w = this.abs();
3222 if (w.equals(TWO))
3223 return true;
3224 if (!w.testBit(0) || w.equals(ONE))
3225 return false;
3226
3227 return w.primeToCertainty(certainty, null);
3228 }
3229
3230 // Comparison Operations
3231
3232 /**
3233 * Compares this BigInteger with the specified BigInteger. This
3234 * method is provided in preference to individual methods for each
3235 * of the six boolean comparison operators ({@literal <}, ==,
3236 * {@literal >}, {@literal >=}, !=, {@literal <=}). The suggested
3237 * idiom for performing these comparisons is: {@code
3238 * (x.compareTo(y)} <<i>op</i>> {@code 0)}, where
3239 * <<i>op</i>> is one of the six comparison operators.
3240 *
3241 * @param val BigInteger to which this BigInteger is to be compared.
3242 * @return -1, 0 or 1 as this BigInteger is numerically less than, equal
3243 * to, or greater than {@code val}.
3244 */
3245 public int compareTo(BigInteger val) {
3246 if (signum == val.signum) {
3247 switch (signum) {
3248 case 1:
3249 return compareMagnitude(val);
3250 case -1:
3251 return val.compareMagnitude(this);
3252 default:
3253 return 0;
3254 }
3255 }
3256 return signum > val.signum ? 1 : -1;
3257 }
3258
3259 /**
3260 * Compares the magnitude array of this BigInteger with the specified
3261 * BigInteger's. This is the version of compareTo ignoring sign.
3262 *
3263 * @param val BigInteger whose magnitude array to be compared.
3264 * @return -1, 0 or 1 as this magnitude array is less than, equal to or
3265 * greater than the magnitude aray for the specified BigInteger's.
3266 */
3267 final int compareMagnitude(BigInteger val) {
3268 int[] m1 = mag;
3269 int len1 = m1.length;
3270 int[] m2 = val.mag;
3271 int len2 = m2.length;
3272 if (len1 < len2)
3273 return -1;
3274 if (len1 > len2)
3275 return 1;
3276 for (int i = 0; i < len1; i++) {
3277 int a = m1[i];
3278 int b = m2[i];
3279 if (a != b)
3280 return ((a & LONG_MASK) < (b & LONG_MASK)) ? -1 : 1;
3281 }
3282 return 0;
3283 }
3284
3285 /**
3286 * Version of compareMagnitude that compares magnitude with long value.
3287 * val can't be Long.MIN_VALUE.
3288 */
3289 final int compareMagnitude(long val) {
3290 assert val != Long.MIN_VALUE;
3291 int[] m1 = mag;
3292 int len = m1.length;
3293 if (len > 2) {
3294 return 1;
3295 }
3296 if (val < 0) {
3297 val = -val;
3298 }
3299 int highWord = (int)(val >>> 32);
3300 if (highWord == 0) {
3301 if (len < 1)
3302 return -1;
3303 if (len > 1)
3304 return 1;
3305 int a = m1[0];
3306 int b = (int)val;
3307 if (a != b) {
3308 return ((a & LONG_MASK) < (b & LONG_MASK))? -1 : 1;
3309 }
3310 return 0;
3311 } else {
3312 if (len < 2)
3313 return -1;
3314 int a = m1[0];
3315 int b = highWord;
3316 if (a != b) {
3317 return ((a & LONG_MASK) < (b & LONG_MASK))? -1 : 1;
3318 }
3319 a = m1[1];
3320 b = (int)val;
3321 if (a != b) {
3322 return ((a & LONG_MASK) < (b & LONG_MASK))? -1 : 1;
3323 }
3324 return 0;
3325 }
3326 }
3327
3328 /**
3329 * Compares this BigInteger with the specified Object for equality.
3330 *
3331 * @param x Object to which this BigInteger is to be compared.
3332 * @return {@code true} if and only if the specified Object is a
3333 * BigInteger whose value is numerically equal to this BigInteger.
3334 */
3335 public boolean equals(Object x) {
3336 // This test is just an optimization, which may or may not help
3337 if (x == this)
3338 return true;
3339
3340 if (!(x instanceof BigInteger))
3341 return false;
3342
3343 BigInteger xInt = (BigInteger) x;
3344 if (xInt.signum != signum)
3345 return false;
3346
3347 int[] m = mag;
3348 int len = m.length;
3349 int[] xm = xInt.mag;
3350 if (len != xm.length)
3351 return false;
3352
3353 for (int i = 0; i < len; i++)
3354 if (xm[i] != m[i])
3355 return false;
3356
3357 return true;
3358 }
3359
3360 /**
3361 * Returns the minimum of this BigInteger and {@code val}.
3362 *
3363 * @param val value with which the minimum is to be computed.
3364 * @return the BigInteger whose value is the lesser of this BigInteger and
3365 * {@code val}. If they are equal, either may be returned.
3366 */
3367 public BigInteger min(BigInteger val) {
3368 return (compareTo(val) < 0 ? this : val);
3369 }
3370
3371 /**
3372 * Returns the maximum of this BigInteger and {@code val}.
3373 *
3374 * @param val value with which the maximum is to be computed.
3375 * @return the BigInteger whose value is the greater of this and
3376 * {@code val}. If they are equal, either may be returned.
3377 */
3378 public BigInteger max(BigInteger val) {
3379 return (compareTo(val) > 0 ? this : val);
3380 }
3381
3382
3383 // Hash Function
3384
3385 /**
3386 * Returns the hash code for this BigInteger.
3387 *
3388 * @return hash code for this BigInteger.
3389 */
3390 public int hashCode() {
3391 int hashCode = 0;
3392
3393 for (int i=0; i < mag.length; i++)
3394 hashCode = (int)(31*hashCode + (mag[i] & LONG_MASK));
3395
3396 return hashCode * signum;
3397 }
3398
3399 /**
3400 * Returns the String representation of this BigInteger in the
3401 * given radix. If the radix is outside the range from {@link
3402 * Character#MIN_RADIX} to {@link Character#MAX_RADIX} inclusive,
3403 * it will default to 10 (as is the case for
3404 * {@code Integer.toString}). The digit-to-character mapping
3405 * provided by {@code Character.forDigit} is used, and a minus
3406 * sign is prepended if appropriate. (This representation is
3407 * compatible with the {@link #BigInteger(String, int) (String,
3408 * int)} constructor.)
3409 *
3410 * @param radix radix of the String representation.
3411 * @return String representation of this BigInteger in the given radix.
3412 * @see Integer#toString
3413 * @see Character#forDigit
3414 * @see #BigInteger(java.lang.String, int)
3415 */
3416 public String toString(int radix) {
3417 if (signum == 0)
3418 return "0";
3419 if (radix < Character.MIN_RADIX || radix > Character.MAX_RADIX)
3420 radix = 10;
3421
3422 // If it's small enough, use smallToString.
3423 if (mag.length <= SCHOENHAGE_BASE_CONVERSION_THRESHOLD)
3424 return smallToString(radix);
3425
3426 // Otherwise use recursive toString, which requires positive arguments.
3427 // The results will be concatenated into this StringBuilder
3428 StringBuilder sb = new StringBuilder();
3429 if (signum < 0) {
3430 toString(this.negate(), sb, radix, 0);
3431 sb.insert(0, '-');
3432 }
3433 else
3434 toString(this, sb, radix, 0);
3435
3436 return sb.toString();
3437 }
3438
3439 /** This method is used to perform toString when arguments are small. */
3440 private String smallToString(int radix) {
3441 if (signum == 0) {
3442 return "0";
3443 }
3444
3445 // Compute upper bound on number of digit groups and allocate space
3446 int maxNumDigitGroups = (4*mag.length + 6)/7;
3447 String digitGroup[] = new String[maxNumDigitGroups];
3448
3449 // Translate number to string, a digit group at a time
3450 BigInteger tmp = this.abs();
3451 int numGroups = 0;
3452 while (tmp.signum != 0) {
3453 BigInteger d = longRadix[radix];
3454
3455 MutableBigInteger q = new MutableBigInteger(),
3456 a = new MutableBigInteger(tmp.mag),
3457 b = new MutableBigInteger(d.mag);
3458 MutableBigInteger r = a.divide(b, q);
3459 BigInteger q2 = q.toBigInteger(tmp.signum * d.signum);
3460 BigInteger r2 = r.toBigInteger(tmp.signum * d.signum);
3461
3462 digitGroup[numGroups++] = Long.toString(r2.longValue(), radix);
3463 tmp = q2;
3464 }
3465
3466 // Put sign (if any) and first digit group into result buffer
3467 StringBuilder buf = new StringBuilder(numGroups*digitsPerLong[radix]+1);
3468 if (signum < 0) {
3469 buf.append('-');
3470 }
3471 buf.append(digitGroup[numGroups-1]);
3472
3473 // Append remaining digit groups padded with leading zeros
3474 for (int i=numGroups-2; i >= 0; i--) {
3475 // Prepend (any) leading zeros for this digit group
3476 int numLeadingZeros = digitsPerLong[radix]-digitGroup[i].length();
3477 if (numLeadingZeros != 0) {
3478 buf.append(zeros[numLeadingZeros]);
3479 }
3480 buf.append(digitGroup[i]);
3481 }
3482 return buf.toString();
3483 }
3484
3485 /**
3486 * Converts the specified BigInteger to a string and appends to
3487 * {@code sb}. This implements the recursive Schoenhage algorithm
3488 * for base conversions.
3489 * <p/>
3490 * See Knuth, Donald, _The Art of Computer Programming_, Vol. 2,
3491 * Answers to Exercises (4.4) Question 14.
3492 *
3493 * @param u The number to convert to a string.
3494 * @param sb The StringBuilder that will be appended to in place.
3495 * @param radix The base to convert to.
3496 * @param digits The minimum number of digits to pad to.
3497 */
3498 private static void toString(BigInteger u, StringBuilder sb, int radix,
3499 int digits) {
3500 /* If we're smaller than a certain threshold, use the smallToString
3501 method, padding with leading zeroes when necessary. */
3502 if (u.mag.length <= SCHOENHAGE_BASE_CONVERSION_THRESHOLD) {
3503 String s = u.smallToString(radix);
3504
3505 // Pad with internal zeros if necessary.
3506 // Don't pad if we're at the beginning of the string.
3507 if ((s.length() < digits) && (sb.length() > 0)) {
3508 for (int i=s.length(); i < digits; i++) { // May be a faster way to
3509 sb.append('0'); // do this?
3510 }
3511 }
3512
3513 sb.append(s);
3514 return;
3515 }
3516
3517 int b, n;
3518 b = u.bitLength();
3519
3520 // Calculate a value for n in the equation radix^(2^n) = u
3521 // and subtract 1 from that value. This is used to find the
3522 // cache index that contains the best value to divide u.
3523 n = (int) Math.round(Math.log(b * LOG_TWO / logCache[radix]) / LOG_TWO - 1.0);
3524 BigInteger v = getRadixConversionCache(radix, n);
3525 BigInteger[] results;
3526 results = u.divideAndRemainder(v);
3527
3528 int expectedDigits = 1 << n;
3529
3530 // Now recursively build the two halves of each number.
3531 toString(results[0], sb, radix, digits-expectedDigits);
3532 toString(results[1], sb, radix, expectedDigits);
3533 }
3534
3535 /**
3536 * Returns the value radix^(2^exponent) from the cache.
3537 * If this value doesn't already exist in the cache, it is added.
3538 * <p/>
3539 * This could be changed to a more complicated caching method using
3540 * {@code Future}.
3541 */
3542 private static BigInteger getRadixConversionCache(int radix, int exponent) {
3543 BigInteger[] cacheLine = powerCache[radix]; // volatile read
3544 if (exponent < cacheLine.length) {
3545 return cacheLine[exponent];
3546 }
3547
3548 int oldLength = cacheLine.length;
3549 cacheLine = Arrays.copyOf(cacheLine, exponent + 1);
3550 for (int i = oldLength; i <= exponent; i++) {
3551 cacheLine[i] = cacheLine[i - 1].pow(2);
3552 }
3553
3554 BigInteger[][] pc = powerCache; // volatile read again
3555 if (exponent >= pc[radix].length) {
3556 pc = pc.clone();
3557 pc[radix] = cacheLine;
3558 powerCache = pc; // volatile write, publish
3559 }
3560 return cacheLine[exponent];
3561 }
3562
3563 /* zero[i] is a string of i consecutive zeros. */
3564 private static String zeros[] = new String[64];
3565 static {
3566 zeros[63] =
3567 "000000000000000000000000000000000000000000000000000000000000000";
3568 for (int i=0; i < 63; i++)
3569 zeros[i] = zeros[63].substring(0, i);
3570 }
3571
3572 /**
3573 * Returns the decimal String representation of this BigInteger.
3574 * The digit-to-character mapping provided by
3575 * {@code Character.forDigit} is used, and a minus sign is
3576 * prepended if appropriate. (This representation is compatible
3577 * with the {@link #BigInteger(String) (String)} constructor, and
3578 * allows for String concatenation with Java's + operator.)
3579 *
3580 * @return decimal String representation of this BigInteger.
3581 * @see Character#forDigit
3582 * @see #BigInteger(java.lang.String)
3583 */
3584 public String toString() {
3585 return toString(10);
3586 }
3587
3588 /**
3589 * Returns a byte array containing the two's-complement
3590 * representation of this BigInteger. The byte array will be in
3591 * <i>big-endian</i> byte-order: the most significant byte is in
3592 * the zeroth element. The array will contain the minimum number
3593 * of bytes required to represent this BigInteger, including at
3594 * least one sign bit, which is {@code (ceil((this.bitLength() +
3595 * 1)/8))}. (This representation is compatible with the
3596 * {@link #BigInteger(byte[]) (byte[])} constructor.)
3597 *
3598 * @return a byte array containing the two's-complement representation of
3599 * this BigInteger.
3600 * @see #BigInteger(byte[])
3601 */
3602 public byte[] toByteArray() {
3603 int byteLen = bitLength()/8 + 1;
3604 byte[] byteArray = new byte[byteLen];
3605
3606 for (int i=byteLen-1, bytesCopied=4, nextInt=0, intIndex=0; i >= 0; i--) {
3607 if (bytesCopied == 4) {
3608 nextInt = getInt(intIndex++);
3609 bytesCopied = 1;
3610 } else {
3611 nextInt >>>= 8;
3612 bytesCopied++;
3613 }
3614 byteArray[i] = (byte)nextInt;
3615 }
3616 return byteArray;
3617 }
3618
3619 /**
3620 * Converts this BigInteger to an {@code int}. This
3621 * conversion is analogous to a
3622 * <i>narrowing primitive conversion</i> from {@code long} to
3623 * {@code int} as defined in section 5.1.3 of
3624 * <cite>The Java™ Language Specification</cite>:
3625 * if this BigInteger is too big to fit in an
3626 * {@code int}, only the low-order 32 bits are returned.
3627 * Note that this conversion can lose information about the
3628 * overall magnitude of the BigInteger value as well as return a
3629 * result with the opposite sign.
3630 *
3631 * @return this BigInteger converted to an {@code int}.
3632 * @see #intValueExact()
3633 */
3634 public int intValue() {
3635 int result = 0;
3636 result = getInt(0);
3637 return result;
3638 }
3639
3640 /**
3641 * Converts this BigInteger to a {@code long}. This
3642 * conversion is analogous to a
3643 * <i>narrowing primitive conversion</i> from {@code long} to
3644 * {@code int} as defined in section 5.1.3 of
3645 * <cite>The Java™ Language Specification</cite>:
3646 * if this BigInteger is too big to fit in a
3647 * {@code long}, only the low-order 64 bits are returned.
3648 * Note that this conversion can lose information about the
3649 * overall magnitude of the BigInteger value as well as return a
3650 * result with the opposite sign.
3651 *
3652 * @return this BigInteger converted to a {@code long}.
3653 * @see #longValueExact()
3654 */
3655 public long longValue() {
3656 long result = 0;
3657
3658 for (int i=1; i >= 0; i--)
3659 result = (result << 32) + (getInt(i) & LONG_MASK);
3660 return result;
3661 }
3662
3663 /**
3664 * Converts this BigInteger to a {@code float}. This
3665 * conversion is similar to the
3666 * <i>narrowing primitive conversion</i> from {@code double} to
3667 * {@code float} as defined in section 5.1.3 of
3668 * <cite>The Java™ Language Specification</cite>:
3669 * if this BigInteger has too great a magnitude
3670 * to represent as a {@code float}, it will be converted to
3671 * {@link Float#NEGATIVE_INFINITY} or {@link
3672 * Float#POSITIVE_INFINITY} as appropriate. Note that even when
3673 * the return value is finite, this conversion can lose
3674 * information about the precision of the BigInteger value.
3675 *
3676 * @return this BigInteger converted to a {@code float}.
3677 */
3678 public float floatValue() {
3679 if (signum == 0) {
3680 return 0.0f;
3681 }
3682
3683 int exponent = ((mag.length - 1) << 5) + bitLengthForInt(mag[0]) - 1;
3684
3685 // exponent == floor(log2(abs(this)))
3686 if (exponent < Long.SIZE - 1) {
3687 return longValue();
3688 } else if (exponent > Float.MAX_EXPONENT) {
3689 return signum > 0 ? Float.POSITIVE_INFINITY : Float.NEGATIVE_INFINITY;
3690 }
3691
3692 /*
3693 * We need the top SIGNIFICAND_WIDTH bits, including the "implicit"
3694 * one bit. To make rounding easier, we pick out the top
3695 * SIGNIFICAND_WIDTH + 1 bits, so we have one to help us round up or
3696 * down. twiceSignifFloor will contain the top SIGNIFICAND_WIDTH + 1
3697 * bits, and signifFloor the top SIGNIFICAND_WIDTH.
3698 *
3699 * It helps to consider the real number signif = abs(this) *
3700 * 2^(SIGNIFICAND_WIDTH - 1 - exponent).
3701 */
3702 int shift = exponent - FloatConsts.SIGNIFICAND_WIDTH;
3703
3704 int twiceSignifFloor;
3705 // twiceSignifFloor will be == abs().shiftRight(shift).intValue()
3706 // We do the shift into an int directly to improve performance.
3707
3708 int nBits = shift & 0x1f;
3709 int nBits2 = 32 - nBits;
3710
3711 if (nBits == 0) {
3712 twiceSignifFloor = mag[0];
3713 } else {
3714 twiceSignifFloor = mag[0] >>> nBits;
3715 if (twiceSignifFloor == 0) {
3716 twiceSignifFloor = (mag[0] << nBits2) | (mag[1] >>> nBits);
3717 }
3718 }
3719
3720 int signifFloor = twiceSignifFloor >> 1;
3721 signifFloor &= FloatConsts.SIGNIF_BIT_MASK; // remove the implied bit
3722
3723 /*
3724 * We round up if either the fractional part of signif is strictly
3725 * greater than 0.5 (which is true if the 0.5 bit is set and any lower
3726 * bit is set), or if the fractional part of signif is >= 0.5 and
3727 * signifFloor is odd (which is true if both the 0.5 bit and the 1 bit
3728 * are set). This is equivalent to the desired HALF_EVEN rounding.
3729 */
3730 boolean increment = (twiceSignifFloor & 1) != 0
3731 && ((signifFloor & 1) != 0 || abs().getLowestSetBit() < shift);
3732 int signifRounded = increment ? signifFloor + 1 : signifFloor;
3733 int bits = ((exponent + FloatConsts.EXP_BIAS))
3734 << (FloatConsts.SIGNIFICAND_WIDTH - 1);
3735 bits += signifRounded;
3736 /*
3737 * If signifRounded == 2^24, we'd need to set all of the significand
3738 * bits to zero and add 1 to the exponent. This is exactly the behavior
3739 * we get from just adding signifRounded to bits directly. If the
3740 * exponent is Float.MAX_EXPONENT, we round up (correctly) to
3741 * Float.POSITIVE_INFINITY.
3742 */
3743 bits |= signum & FloatConsts.SIGN_BIT_MASK;
3744 return Float.intBitsToFloat(bits);
3745 }
3746
3747 /**
3748 * Converts this BigInteger to a {@code double}. This
3749 * conversion is similar to the
3750 * <i>narrowing primitive conversion</i> from {@code double} to
3751 * {@code float} as defined in section 5.1.3 of
3752 * <cite>The Java™ Language Specification</cite>:
3753 * if this BigInteger has too great a magnitude
3754 * to represent as a {@code double}, it will be converted to
3755 * {@link Double#NEGATIVE_INFINITY} or {@link
3756 * Double#POSITIVE_INFINITY} as appropriate. Note that even when
3757 * the return value is finite, this conversion can lose
3758 * information about the precision of the BigInteger value.
3759 *
3760 * @return this BigInteger converted to a {@code double}.
3761 */
3762 public double doubleValue() {
3763 if (signum == 0) {
3764 return 0.0;
3765 }
3766
3767 int exponent = ((mag.length - 1) << 5) + bitLengthForInt(mag[0]) - 1;
3768
3769 // exponent == floor(log2(abs(this))Double)
3770 if (exponent < Long.SIZE - 1) {
3771 return longValue();
3772 } else if (exponent > Double.MAX_EXPONENT) {
3773 return signum > 0 ? Double.POSITIVE_INFINITY : Double.NEGATIVE_INFINITY;
3774 }
3775
3776 /*
3777 * We need the top SIGNIFICAND_WIDTH bits, including the "implicit"
3778 * one bit. To make rounding easier, we pick out the top
3779 * SIGNIFICAND_WIDTH + 1 bits, so we have one to help us round up or
3780 * down. twiceSignifFloor will contain the top SIGNIFICAND_WIDTH + 1
3781 * bits, and signifFloor the top SIGNIFICAND_WIDTH.
3782 *
3783 * It helps to consider the real number signif = abs(this) *
3784 * 2^(SIGNIFICAND_WIDTH - 1 - exponent).
3785 */
3786 int shift = exponent - DoubleConsts.SIGNIFICAND_WIDTH;
3787
3788 long twiceSignifFloor;
3789 // twiceSignifFloor will be == abs().shiftRight(shift).longValue()
3790 // We do the shift into a long directly to improve performance.
3791
3792 int nBits = shift & 0x1f;
3793 int nBits2 = 32 - nBits;
3794
3795 int highBits;
3796 int lowBits;
3797 if (nBits == 0) {
3798 highBits = mag[0];
3799 lowBits = mag[1];
3800 } else {
3801 highBits = mag[0] >>> nBits;
3802 lowBits = (mag[0] << nBits2) | (mag[1] >>> nBits);
3803 if (highBits == 0) {
3804 highBits = lowBits;
3805 lowBits = (mag[1] << nBits2) | (mag[2] >>> nBits);
3806 }
3807 }
3808
3809 twiceSignifFloor = ((highBits & LONG_MASK) << 32)
3810 | (lowBits & LONG_MASK);
3811
3812 long signifFloor = twiceSignifFloor >> 1;
3813 signifFloor &= DoubleConsts.SIGNIF_BIT_MASK; // remove the implied bit
3814
3815 /*
3816 * We round up if either the fractional part of signif is strictly
3817 * greater than 0.5 (which is true if the 0.5 bit is set and any lower
3818 * bit is set), or if the fractional part of signif is >= 0.5 and
3819 * signifFloor is odd (which is true if both the 0.5 bit and the 1 bit
3820 * are set). This is equivalent to the desired HALF_EVEN rounding.
3821 */
3822 boolean increment = (twiceSignifFloor & 1) != 0
3823 && ((signifFloor & 1) != 0 || abs().getLowestSetBit() < shift);
3824 long signifRounded = increment ? signifFloor + 1 : signifFloor;
3825 long bits = (long) ((exponent + DoubleConsts.EXP_BIAS))
3826 << (DoubleConsts.SIGNIFICAND_WIDTH - 1);
3827 bits += signifRounded;
3828 /*
3829 * If signifRounded == 2^53, we'd need to set all of the significand
3830 * bits to zero and add 1 to the exponent. This is exactly the behavior
3831 * we get from just adding signifRounded to bits directly. If the
3832 * exponent is Double.MAX_EXPONENT, we round up (correctly) to
3833 * Double.POSITIVE_INFINITY.
3834 */
3835 bits |= signum & DoubleConsts.SIGN_BIT_MASK;
3836 return Double.longBitsToDouble(bits);
3837 }
3838
3839 /**
3840 * Returns a copy of the input array stripped of any leading zero bytes.
3841 */
3842 private static int[] stripLeadingZeroInts(int val[]) {
3843 int vlen = val.length;
3844 int keep;
3845
3846 // Find first nonzero byte
3847 for (keep = 0; keep < vlen && val[keep] == 0; keep++)
3848 ;
3849 return java.util.Arrays.copyOfRange(val, keep, vlen);
3850 }
3851
3852 /**
3853 * Returns the input array stripped of any leading zero bytes.
3854 * Since the source is trusted the copying may be skipped.
3855 */
3856 private static int[] trustedStripLeadingZeroInts(int val[]) {
3857 int vlen = val.length;
3858 int keep;
3859
3860 // Find first nonzero byte
3861 for (keep = 0; keep < vlen && val[keep] == 0; keep++)
3862 ;
3863 return keep == 0 ? val : java.util.Arrays.copyOfRange(val, keep, vlen);
3864 }
3865
3866 /**
3867 * Returns a copy of the input array stripped of any leading zero bytes.
3868 */
3869 private static int[] stripLeadingZeroBytes(byte a[]) {
3870 int byteLength = a.length;
3871 int keep;
3872
3873 // Find first nonzero byte
3874 for (keep = 0; keep < byteLength && a[keep] == 0; keep++)
3875 ;
3876
3877 // Allocate new array and copy relevant part of input array
3878 int intLength = ((byteLength - keep) + 3) >>> 2;
3879 int[] result = new int[intLength];
3880 int b = byteLength - 1;
3881 for (int i = intLength-1; i >= 0; i--) {
3882 result[i] = a[b--] & 0xff;
3883 int bytesRemaining = b - keep + 1;
3884 int bytesToTransfer = Math.min(3, bytesRemaining);
3885 for (int j=8; j <= (bytesToTransfer << 3); j += 8)
3886 result[i] |= ((a[b--] & 0xff) << j);
3887 }
3888 return result;
3889 }
3890
3891 /**
3892 * Takes an array a representing a negative 2's-complement number and
3893 * returns the minimal (no leading zero bytes) unsigned whose value is -a.
3894 */
3895 private static int[] makePositive(byte a[]) {
3896 int keep, k;
3897 int byteLength = a.length;
3898
3899 // Find first non-sign (0xff) byte of input
3900 for (keep=0; keep < byteLength && a[keep] == -1; keep++)
3901 ;
3902
3903
3904 /* Allocate output array. If all non-sign bytes are 0x00, we must
3905 * allocate space for one extra output byte. */
3906 for (k=keep; k < byteLength && a[k] == 0; k++)
3907 ;
3908
3909 int extraByte = (k == byteLength) ? 1 : 0;
3910 int intLength = ((byteLength - keep + extraByte) + 3)/4;
3911 int result[] = new int[intLength];
3912
3913 /* Copy one's complement of input into output, leaving extra
3914 * byte (if it exists) == 0x00 */
3915 int b = byteLength - 1;
3916 for (int i = intLength-1; i >= 0; i--) {
3917 result[i] = a[b--] & 0xff;
3918 int numBytesToTransfer = Math.min(3, b-keep+1);
3919 if (numBytesToTransfer < 0)
3920 numBytesToTransfer = 0;
3921 for (int j=8; j <= 8*numBytesToTransfer; j += 8)
3922 result[i] |= ((a[b--] & 0xff) << j);
3923
3924 // Mask indicates which bits must be complemented
3925 int mask = -1 >>> (8*(3-numBytesToTransfer));
3926 result[i] = ~result[i] & mask;
3927 }
3928
3929 // Add one to one's complement to generate two's complement
3930 for (int i=result.length-1; i >= 0; i--) {
3931 result[i] = (int)((result[i] & LONG_MASK) + 1);
3932 if (result[i] != 0)
3933 break;
3934 }
3935
3936 return result;
3937 }
3938
3939 /**
3940 * Takes an array a representing a negative 2's-complement number and
3941 * returns the minimal (no leading zero ints) unsigned whose value is -a.
3942 */
3943 private static int[] makePositive(int a[]) {
3944 int keep, j;
3945
3946 // Find first non-sign (0xffffffff) int of input
3947 for (keep=0; keep < a.length && a[keep] == -1; keep++)
3948 ;
3949
3950 /* Allocate output array. If all non-sign ints are 0x00, we must
3951 * allocate space for one extra output int. */
3952 for (j=keep; j < a.length && a[j] == 0; j++)
3953 ;
3954 int extraInt = (j == a.length ? 1 : 0);
3955 int result[] = new int[a.length - keep + extraInt];
3956
3957 /* Copy one's complement of input into output, leaving extra
3958 * int (if it exists) == 0x00 */
3959 for (int i = keep; i < a.length; i++)
3960 result[i - keep + extraInt] = ~a[i];
3961
3962 // Add one to one's complement to generate two's complement
3963 for (int i=result.length-1; ++result[i] == 0; i--)
3964 ;
3965
3966 return result;
3967 }
3968
3969 /*
3970 * The following two arrays are used for fast String conversions. Both
3971 * are indexed by radix. The first is the number of digits of the given
3972 * radix that can fit in a Java long without "going negative", i.e., the
3973 * highest integer n such that radix**n < 2**63. The second is the
3974 * "long radix" that tears each number into "long digits", each of which
3975 * consists of the number of digits in the corresponding element in
3976 * digitsPerLong (longRadix[i] = i**digitPerLong[i]). Both arrays have
3977 * nonsense values in their 0 and 1 elements, as radixes 0 and 1 are not
3978 * used.
3979 */
3980 private static int digitsPerLong[] = {0, 0,
3981 62, 39, 31, 27, 24, 22, 20, 19, 18, 18, 17, 17, 16, 16, 15, 15, 15, 14,
3982 14, 14, 14, 13, 13, 13, 13, 13, 13, 12, 12, 12, 12, 12, 12, 12, 12};
3983
3984 private static BigInteger longRadix[] = {null, null,
3985 valueOf(0x4000000000000000L), valueOf(0x383d9170b85ff80bL),
3986 valueOf(0x4000000000000000L), valueOf(0x6765c793fa10079dL),
3987 valueOf(0x41c21cb8e1000000L), valueOf(0x3642798750226111L),
3988 valueOf(0x1000000000000000L), valueOf(0x12bf307ae81ffd59L),
3989 valueOf( 0xde0b6b3a7640000L), valueOf(0x4d28cb56c33fa539L),
3990 valueOf(0x1eca170c00000000L), valueOf(0x780c7372621bd74dL),
3991 valueOf(0x1e39a5057d810000L), valueOf(0x5b27ac993df97701L),
3992 valueOf(0x1000000000000000L), valueOf(0x27b95e997e21d9f1L),
3993 valueOf(0x5da0e1e53c5c8000L), valueOf( 0xb16a458ef403f19L),
3994 valueOf(0x16bcc41e90000000L), valueOf(0x2d04b7fdd9c0ef49L),
3995 valueOf(0x5658597bcaa24000L), valueOf( 0x6feb266931a75b7L),
3996 valueOf( 0xc29e98000000000L), valueOf(0x14adf4b7320334b9L),
3997 valueOf(0x226ed36478bfa000L), valueOf(0x383d9170b85ff80bL),
3998 valueOf(0x5a3c23e39c000000L), valueOf( 0x4e900abb53e6b71L),
3999 valueOf( 0x7600ec618141000L), valueOf( 0xaee5720ee830681L),
4000 valueOf(0x1000000000000000L), valueOf(0x172588ad4f5f0981L),
4001 valueOf(0x211e44f7d02c1000L), valueOf(0x2ee56725f06e5c71L),
4002 valueOf(0x41c21cb8e1000000L)};
4003
4004 /*
4005 * These two arrays are the integer analogue of above.
4006 */
4007 private static int digitsPerInt[] = {0, 0, 30, 19, 15, 13, 11,
4008 11, 10, 9, 9, 8, 8, 8, 8, 7, 7, 7, 7, 7, 7, 7, 6, 6, 6, 6,
4009 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 5};
4010
4011 private static int intRadix[] = {0, 0,
4012 0x40000000, 0x4546b3db, 0x40000000, 0x48c27395, 0x159fd800,
4013 0x75db9c97, 0x40000000, 0x17179149, 0x3b9aca00, 0xcc6db61,
4014 0x19a10000, 0x309f1021, 0x57f6c100, 0xa2f1b6f, 0x10000000,
4015 0x18754571, 0x247dbc80, 0x3547667b, 0x4c4b4000, 0x6b5a6e1d,
4016 0x6c20a40, 0x8d2d931, 0xb640000, 0xe8d4a51, 0x1269ae40,
4017 0x17179149, 0x1cb91000, 0x23744899, 0x2b73a840, 0x34e63b41,
4018 0x40000000, 0x4cfa3cc1, 0x5c13d840, 0x6d91b519, 0x39aa400
4019 };
4020
4021 /**
4022 * These routines provide access to the two's complement representation
4023 * of BigIntegers.
4024 */
4025
4026 /**
4027 * Returns the length of the two's complement representation in ints,
4028 * including space for at least one sign bit.
4029 */
4030 private int intLength() {
4031 return (bitLength() >>> 5) + 1;
4032 }
4033
4034 /* Returns sign bit */
4035 private int signBit() {
4036 return signum < 0 ? 1 : 0;
4037 }
4038
4039 /* Returns an int of sign bits */
4040 private int signInt() {
4041 return signum < 0 ? -1 : 0;
4042 }
4043
4044 /**
4045 * Returns the specified int of the little-endian two's complement
4046 * representation (int 0 is the least significant). The int number can
4047 * be arbitrarily high (values are logically preceded by infinitely many
4048 * sign ints).
4049 */
4050 private int getInt(int n) {
4051 if (n < 0)
4052 return 0;
4053 if (n >= mag.length)
4054 return signInt();
4055
4056 int magInt = mag[mag.length-n-1];
4057
4058 return (signum >= 0 ? magInt :
4059 (n <= firstNonzeroIntNum() ? -magInt : ~magInt));
4060 }
4061
4062 /**
4063 * Returns the index of the int that contains the first nonzero int in the
4064 * little-endian binary representation of the magnitude (int 0 is the
4065 * least significant). If the magnitude is zero, return value is undefined.
4066 */
4067 private int firstNonzeroIntNum() {
4068 int fn = firstNonzeroIntNum - 2;
4069 if (fn == -2) { // firstNonzeroIntNum not initialized yet
4070 fn = 0;
4071
4072 // Search for the first nonzero int
4073 int i;
4074 int mlen = mag.length;
4075 for (i = mlen - 1; i >= 0 && mag[i] == 0; i--)
4076 ;
4077 fn = mlen - i - 1;
4078 firstNonzeroIntNum = fn + 2; // offset by two to initialize
4079 }
4080 return fn;
4081 }
4082
4083 /** use serialVersionUID from JDK 1.1. for interoperability */
4084 private static final long serialVersionUID = -8287574255936472291L;
4085
4086 /**
4087 * Serializable fields for BigInteger.
4088 *
4089 * @serialField signum int
4090 * signum of this BigInteger.
4091 * @serialField magnitude int[]
4092 * magnitude array of this BigInteger.
4093 * @serialField bitCount int
4094 * number of bits in this BigInteger
4095 * @serialField bitLength int
4096 * the number of bits in the minimal two's-complement
4097 * representation of this BigInteger
4098 * @serialField lowestSetBit int
4099 * lowest set bit in the twos complement representation
4100 */
4101 private static final ObjectStreamField[] serialPersistentFields = {
4102 new ObjectStreamField("signum", Integer.TYPE),
4103 new ObjectStreamField("magnitude", byte[].class),
4104 new ObjectStreamField("bitCount", Integer.TYPE),
4105 new ObjectStreamField("bitLength", Integer.TYPE),
4106 new ObjectStreamField("firstNonzeroByteNum", Integer.TYPE),
4107 new ObjectStreamField("lowestSetBit", Integer.TYPE)
4108 };
4109
4110 /**
4111 * Reconstitute the {@code BigInteger} instance from a stream (that is,
4112 * deserialize it). The magnitude is read in as an array of bytes
4113 * for historical reasons, but it is converted to an array of ints
4114 * and the byte array is discarded.
4115 * Note:
4116 * The current convention is to initialize the cache fields, bitCount,
4117 * bitLength and lowestSetBit, to 0 rather than some other marker value.
4118 * Therefore, no explicit action to set these fields needs to be taken in
4119 * readObject because those fields already have a 0 value be default since
4120 * defaultReadObject is not being used.
4121 */
4122 private void readObject(java.io.ObjectInputStream s)
4123 throws java.io.IOException, ClassNotFoundException {
4124 /*
4125 * In order to maintain compatibility with previous serialized forms,
4126 * the magnitude of a BigInteger is serialized as an array of bytes.
4127 * The magnitude field is used as a temporary store for the byte array
4128 * that is deserialized. The cached computation fields should be
4129 * transient but are serialized for compatibility reasons.
4130 */
4131
4132 // prepare to read the alternate persistent fields
4133 ObjectInputStream.GetField fields = s.readFields();
4134
4135 // Read the alternate persistent fields that we care about
4136 int sign = fields.get("signum", -2);
4137 byte[] magnitude = (byte[])fields.get("magnitude", null);
4138
4139 // Validate signum
4140 if (sign < -1 || sign > 1) {
4141 String message = "BigInteger: Invalid signum value";
4142 if (fields.defaulted("signum"))
4143 message = "BigInteger: Signum not present in stream";
4144 throw new java.io.StreamCorruptedException(message);
4145 }
4146 if ((magnitude.length == 0) != (sign == 0)) {
4147 String message = "BigInteger: signum-magnitude mismatch";
4148 if (fields.defaulted("magnitude"))
4149 message = "BigInteger: Magnitude not present in stream";
4150 throw new java.io.StreamCorruptedException(message);
4151 }
4152
4153 // Commit final fields via Unsafe
4154 UnsafeHolder.putSign(this, sign);
4155
4156 // Calculate mag field from magnitude and discard magnitude
4157 UnsafeHolder.putMag(this, stripLeadingZeroBytes(magnitude));
4158 }
4159
4160 // Support for resetting final fields while deserializing
4161 private static class UnsafeHolder {
4162 private static final sun.misc.Unsafe unsafe;
4163 private static final long signumOffset;
4164 private static final long magOffset;
4165 static {
4166 try {
4167 unsafe = sun.misc.Unsafe.getUnsafe();
4168 signumOffset = unsafe.objectFieldOffset
4169 (BigInteger.class.getDeclaredField("signum"));
4170 magOffset = unsafe.objectFieldOffset
4171 (BigInteger.class.getDeclaredField("mag"));
4172 } catch (Exception ex) {
4173 throw new ExceptionInInitializerError(ex);
4174 }
4175 }
4176
4177 static void putSign(BigInteger bi, int sign) {
4178 unsafe.putIntVolatile(bi, signumOffset, sign);
4179 }
4180
4181 static void putMag(BigInteger bi, int[] magnitude) {
4182 unsafe.putObjectVolatile(bi, magOffset, magnitude);
4183 }
4184 }
4185
4186 /**
4187 * Save the {@code BigInteger} instance to a stream.
4188 * The magnitude of a BigInteger is serialized as a byte array for
4189 * historical reasons.
4190 *
4191 * @serialData two necessary fields are written as well as obsolete
4192 * fields for compatibility with older versions.
4193 */
4194 private void writeObject(ObjectOutputStream s) throws IOException {
4195 // set the values of the Serializable fields
4196 ObjectOutputStream.PutField fields = s.putFields();
4197 fields.put("signum", signum);
4198 fields.put("magnitude", magSerializedForm());
4199 // The values written for cached fields are compatible with older
4200 // versions, but are ignored in readObject so don't otherwise matter.
4201 fields.put("bitCount", -1);
4202 fields.put("bitLength", -1);
4203 fields.put("lowestSetBit", -2);
4204 fields.put("firstNonzeroByteNum", -2);
4205
4206 // save them
4207 s.writeFields();
4208 }
4209
4210 /**
4211 * Returns the mag array as an array of bytes.
4212 */
4213 private byte[] magSerializedForm() {
4214 int len = mag.length;
4215
4216 int bitLen = (len == 0 ? 0 : ((len - 1) << 5) + bitLengthForInt(mag[0]));
4217 int byteLen = (bitLen + 7) >>> 3;
4218 byte[] result = new byte[byteLen];
4219
4220 for (int i = byteLen - 1, bytesCopied = 4, intIndex = len - 1, nextInt = 0;
4221 i >= 0; i--) {
4222 if (bytesCopied == 4) {
4223 nextInt = mag[intIndex--];
4224 bytesCopied = 1;
4225 } else {
4226 nextInt >>>= 8;
4227 bytesCopied++;
4228 }
4229 result[i] = (byte)nextInt;
4230 }
4231 return result;
4232 }
4233
4234 /**
4235 * Converts this {@code BigInteger} to a {@code long}, checking
4236 * for lost information. If the value of this {@code BigInteger}
4237 * is out of the range of the {@code long} type, then an
4238 * {@code ArithmeticException} is thrown.
4239 *
4240 * @return this {@code BigInteger} converted to a {@code long}.
4241 * @throws ArithmeticException if the value of {@code this} will
4242 * not exactly fit in a {@code long}.
4243 * @see BigInteger#longValue
4244 * @since 1.8
4245 */
4246 public long longValueExact() {
4247 if (mag.length <= 2 && bitLength() <= 63)
4248 return longValue();
4249 else
4250 throw new ArithmeticException("BigInteger out of long range");
4251 }
4252
4253 /**
4254 * Converts this {@code BigInteger} to an {@code int}, checking
4255 * for lost information. If the value of this {@code BigInteger}
4256 * is out of the range of the {@code int} type, then an
4257 * {@code ArithmeticException} is thrown.
4258 *
4259 * @return this {@code BigInteger} converted to an {@code int}.
4260 * @throws ArithmeticException if the value of {@code this} will
4261 * not exactly fit in a {@code int}.
4262 * @see BigInteger#intValue
4263 * @since 1.8
4264 */
4265 public int intValueExact() {
4266 if (mag.length <= 1 && bitLength() <= 31)
4267 return intValue();
4268 else
4269 throw new ArithmeticException("BigInteger out of int range");
4270 }
4271
4272 /**
4273 * Converts this {@code BigInteger} to a {@code short}, checking
4274 * for lost information. If the value of this {@code BigInteger}
4275 * is out of the range of the {@code short} type, then an
4276 * {@code ArithmeticException} is thrown.
4277 *
4278 * @return this {@code BigInteger} converted to a {@code short}.
4279 * @throws ArithmeticException if the value of {@code this} will
4280 * not exactly fit in a {@code short}.
4281 * @see BigInteger#shortValue
4282 * @since 1.8
4283 */
4284 public short shortValueExact() {
4285 if (mag.length <= 1 && bitLength() <= 31) {
4286 int value = intValue();
4287 if (value >= Short.MIN_VALUE && value <= Short.MAX_VALUE)
4288 return shortValue();
4289 }
4290 throw new ArithmeticException("BigInteger out of short range");
4291 }
4292
4293 /**
4294 * Converts this {@code BigInteger} to a {@code byte}, checking
4295 * for lost information. If the value of this {@code BigInteger}
4296 * is out of the range of the {@code byte} type, then an
4297 * {@code ArithmeticException} is thrown.
4298 *
4299 * @return this {@code BigInteger} converted to a {@code byte}.
4300 * @throws ArithmeticException if the value of {@code this} will
4301 * not exactly fit in a {@code byte}.
4302 * @see BigInteger#byteValue
4303 * @since 1.8
4304 */
4305 public byte byteValueExact() {
4306 if (mag.length <= 1 && bitLength() <= 31) {
4307 int value = intValue();
4308 if (value >= Byte.MIN_VALUE && value <= Byte.MAX_VALUE)
4309 return byteValue();
4310 }
4311 throw new ArithmeticException("BigInteger out of byte range");
4312 }
4313 }