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