1 /*
2 * Copyright (c) 1996, 2013, Oracle and/or its affiliates. All rights reserved.
3 * DO NOT ALTER OR REMOVE COPYRIGHT NOTICES OR THIS FILE HEADER.
4 *
5 * This code is free software; you can redistribute it and/or modify it
6 * under the terms of the GNU General Public License version 2 only, as
7 * published by the Free Software Foundation. Oracle designates this
8 * particular file as subject to the "Classpath" exception as provided
9 * by Oracle in the LICENSE file that accompanied this code.
10 *
11 * This code is distributed in the hope that it will be useful, but WITHOUT
12 * ANY WARRANTY; without even the implied warranty of MERCHANTABILITY or
13 * FITNESS FOR A PARTICULAR PURPOSE. See the GNU General Public License
14 * version 2 for more details (a copy is included in the LICENSE file that
15 * accompanied this code).
16 *
17 * You should have received a copy of the GNU General Public License version
18 * 2 along with this work; if not, write to the Free Software Foundation,
19 * Inc., 51 Franklin St, Fifth Floor, Boston, MA 02110-1301 USA.
20 *
21 * Please contact Oracle, 500 Oracle Parkway, Redwood Shores, CA 94065 USA
22 * or visit www.oracle.com if you need additional information or have any
23 * questions.
24 */
25
26 /*
27 * Portions Copyright (c) 1995 Colin Plumb. All rights reserved.
28 */
29
30 package java.math;
31
32 import java.io.IOException;
33 import java.io.ObjectInputStream;
34 import java.io.ObjectOutputStream;
35 import java.io.ObjectStreamField;
36 import java.util.Arrays;
37 import java.util.Random;
38 import sun.misc.DoubleConsts;
39 import sun.misc.FloatConsts;
40
41 /**
42 * Immutable arbitrary-precision integers. All operations behave as if
43 * BigIntegers were represented in two's-complement notation (like Java's
44 * primitive integer types). BigInteger provides analogues to all of Java's
45 * primitive integer operators, and all relevant methods from java.lang.Math.
46 * Additionally, BigInteger provides operations for modular arithmetic, GCD
47 * calculation, primality testing, prime generation, bit manipulation,
48 * and a few other miscellaneous operations.
49 *
50 * <p>Semantics of arithmetic operations exactly mimic those of Java's integer
51 * arithmetic operators, as defined in <i>The Java Language Specification</i>.
52 * For example, division by zero throws an {@code ArithmeticException}, and
53 * division of a negative by a positive yields a negative (or zero) remainder.
54 * All of the details in the Spec concerning overflow are ignored, as
55 * BigIntegers are made as large as necessary to accommodate the results of an
56 * operation.
57 *
58 * <p>Semantics of shift operations extend those of Java's shift operators
59 * to allow for negative shift distances. A right-shift with a negative
60 * shift distance results in a left shift, and vice-versa. The unsigned
61 * right shift operator ({@code >>>}) is omitted, as this operation makes
62 * little sense in combination with the "infinite word size" abstraction
63 * provided by this class.
64 *
65 * <p>Semantics of bitwise logical operations exactly mimic those of Java's
66 * bitwise integer operators. The binary operators ({@code and},
67 * {@code or}, {@code xor}) implicitly perform sign extension on the shorter
68 * of the two operands prior to performing the operation.
69 *
70 * <p>Comparison operations perform signed integer comparisons, analogous to
71 * those performed by Java's relational and equality operators.
72 *
73 * <p>Modular arithmetic operations are provided to compute residues, perform
74 * exponentiation, and compute multiplicative inverses. These methods always
75 * return a non-negative result, between {@code 0} and {@code (modulus - 1)},
76 * inclusive.
77 *
78 * <p>Bit operations operate on a single bit of the two's-complement
79 * representation of their operand. If necessary, the operand is sign-
80 * extended so that it contains the designated bit. None of the single-bit
81 * operations can produce a BigInteger with a different sign from the
82 * BigInteger being operated on, as they affect only a single bit, and the
83 * "infinite word size" abstraction provided by this class ensures that there
84 * are infinitely many "virtual sign bits" preceding each BigInteger.
85 *
86 * <p>For the sake of brevity and clarity, pseudo-code is used throughout the
87 * descriptions of BigInteger methods. The pseudo-code expression
88 * {@code (i + j)} is shorthand for "a BigInteger whose value is
89 * that of the BigInteger {@code i} plus that of the BigInteger {@code j}."
90 * The pseudo-code expression {@code (i == j)} is shorthand for
91 * "{@code true} if and only if the BigInteger {@code i} represents the same
92 * value as the BigInteger {@code j}." Other pseudo-code expressions are
93 * interpreted similarly.
94 *
95 * <p>All methods and constructors in this class throw
96 * {@code NullPointerException} when passed
97 * a null object reference for any input parameter.
98 *
99 * @see BigDecimal
100 * @author Josh Bloch
101 * @author Michael McCloskey
102 * @author Alan Eliasen
103 * @since JDK1.1
104 */
105
106 public class BigInteger extends Number implements Comparable<BigInteger> {
107 /**
108 * The signum of this BigInteger: -1 for negative, 0 for zero, or
109 * 1 for positive. Note that the BigInteger zero <i>must</i> have
110 * a signum of 0. This is necessary to ensures that there is exactly one
111 * representation for each BigInteger value.
112 *
113 * @serial
114 */
115 final int signum;
116
117 /**
118 * The magnitude of this BigInteger, in <i>big-endian</i> order: the
119 * zeroth element of this array is the most-significant int of the
120 * magnitude. The magnitude must be "minimal" in that the most-significant
121 * int ({@code mag[0]}) must be non-zero. This is necessary to
122 * ensure that there is exactly one representation for each BigInteger
123 * value. Note that this implies that the BigInteger zero has a
124 * zero-length mag array.
125 */
126 final int[] mag;
127
128 // These "redundant fields" are initialized with recognizable nonsense
129 // values, and cached the first time they are needed (or never, if they
130 // aren't needed).
131
132 /**
133 * One plus the bitCount of this BigInteger. Zeros means unitialized.
134 *
135 * @serial
136 * @see #bitCount
137 * @deprecated Deprecated since logical value is offset from stored
138 * value and correction factor is applied in accessor method.
139 */
140 @Deprecated
141 private int bitCount;
142
143 /**
144 * One plus the bitLength of this BigInteger. Zeros means unitialized.
145 * (either value is acceptable).
146 *
147 * @serial
148 * @see #bitLength()
149 * @deprecated Deprecated since logical value is offset from stored
150 * value and correction factor is applied in accessor method.
151 */
152 @Deprecated
153 private int bitLength;
154
155 /**
156 * Two plus the lowest set bit of this BigInteger, as returned by
157 * getLowestSetBit().
158 *
159 * @serial
160 * @see #getLowestSetBit
161 * @deprecated Deprecated since logical value is offset from stored
162 * value and correction factor is applied in accessor method.
163 */
164 @Deprecated
165 private int lowestSetBit;
166
167 /**
168 * Two plus the index of the lowest-order int in the magnitude of this
169 * BigInteger that contains a nonzero int, or -2 (either value is acceptable).
170 * The least significant int has int-number 0, the next int in order of
171 * increasing significance has int-number 1, and so forth.
172 * @deprecated Deprecated since logical value is offset from stored
173 * value and correction factor is applied in accessor method.
174 */
175 @Deprecated
176 private int firstNonzeroIntNum;
177
178 /**
179 * This mask is used to obtain the value of an int as if it were unsigned.
180 */
181 final static long LONG_MASK = 0xffffffffL;
182
183 /**
184 * The threshold value for using Karatsuba multiplication. If the number
185 * of ints in both mag arrays are greater than this number, then
186 * Karatsuba multiplication will be used. This value is found
187 * experimentally to work well.
188 */
189 private static final int KARATSUBA_THRESHOLD = 50;
190
191 /**
192 * The threshold value for using 3-way Toom-Cook multiplication.
193 * If the number of ints in each mag array is greater than the
194 * Karatsuba threshold, and the number of ints in at least one of
195 * the mag arrays is greater than this threshold, then Toom-Cook
196 * multiplication will be used.
197 */
198 private static final int TOOM_COOK_THRESHOLD = 75;
199
200 /**
201 * The threshold value for using Karatsuba squaring. If the number
202 * of ints in the number are larger than this value,
203 * Karatsuba squaring will be used. This value is found
204 * experimentally to work well.
205 */
206 private static final int KARATSUBA_SQUARE_THRESHOLD = 90;
207
208 /**
209 * The threshold value for using Toom-Cook squaring. If the number
210 * of ints in the number are larger than this value,
211 * Toom-Cook squaring will be used. This value is found
212 * experimentally to work well.
213 */
214 private static final int TOOM_COOK_SQUARE_THRESHOLD = 140;
215
216 //Constructors
217
218 /**
219 * Translates a byte array containing the two's-complement binary
220 * representation of a BigInteger into a BigInteger. The input array is
221 * assumed to be in <i>big-endian</i> byte-order: the most significant
222 * byte is in the zeroth element.
223 *
224 * @param val big-endian two's-complement binary representation of
225 * BigInteger.
226 * @throws NumberFormatException {@code val} is zero bytes long.
227 */
228 public BigInteger(byte[] val) {
229 if (val.length == 0)
230 throw new NumberFormatException("Zero length BigInteger");
231
232 if (val[0] < 0) {
233 mag = makePositive(val);
234 signum = -1;
235 } else {
236 mag = stripLeadingZeroBytes(val);
237 signum = (mag.length == 0 ? 0 : 1);
238 }
239 }
240
241 /**
242 * This private constructor translates an int array containing the
243 * two's-complement binary representation of a BigInteger into a
244 * BigInteger. The input array is assumed to be in <i>big-endian</i>
245 * int-order: the most significant int is in the zeroth element.
246 */
247 private BigInteger(int[] val) {
248 if (val.length == 0)
249 throw new NumberFormatException("Zero length BigInteger");
250
251 if (val[0] < 0) {
252 mag = makePositive(val);
253 signum = -1;
254 } else {
255 mag = trustedStripLeadingZeroInts(val);
256 signum = (mag.length == 0 ? 0 : 1);
257 }
258 }
259
260 /**
261 * Translates the sign-magnitude representation of a BigInteger into a
262 * BigInteger. The sign is represented as an integer signum value: -1 for
263 * negative, 0 for zero, or 1 for positive. The magnitude is a byte array
264 * in <i>big-endian</i> byte-order: the most significant byte is in the
265 * zeroth element. A zero-length magnitude array is permissible, and will
266 * result in a BigInteger value of 0, whether signum is -1, 0 or 1.
267 *
268 * @param signum signum of the number (-1 for negative, 0 for zero, 1
269 * for positive).
270 * @param magnitude big-endian binary representation of the magnitude of
271 * the number.
272 * @throws NumberFormatException {@code signum} is not one of the three
273 * legal values (-1, 0, and 1), or {@code signum} is 0 and
274 * {@code magnitude} contains one or more non-zero bytes.
275 */
276 public BigInteger(int signum, byte[] magnitude) {
277 this.mag = stripLeadingZeroBytes(magnitude);
278
279 if (signum < -1 || signum > 1)
280 throw(new NumberFormatException("Invalid signum value"));
281
282 if (this.mag.length==0) {
283 this.signum = 0;
284 } else {
285 if (signum == 0)
286 throw(new NumberFormatException("signum-magnitude mismatch"));
287 this.signum = signum;
288 }
289 }
290
291 /**
292 * A constructor for internal use that translates the sign-magnitude
293 * representation of a BigInteger into a BigInteger. It checks the
294 * arguments and copies the magnitude so this constructor would be
295 * safe for external use.
296 */
297 private BigInteger(int signum, int[] magnitude) {
298 this.mag = stripLeadingZeroInts(magnitude);
299
300 if (signum < -1 || signum > 1)
301 throw(new NumberFormatException("Invalid signum value"));
302
303 if (this.mag.length==0) {
304 this.signum = 0;
305 } else {
306 if (signum == 0)
307 throw(new NumberFormatException("signum-magnitude mismatch"));
308 this.signum = signum;
309 }
310 }
311
312 /**
313 * Translates the String representation of a BigInteger in the
314 * specified radix into a BigInteger. The String representation
315 * consists of an optional minus or plus sign followed by a
316 * sequence of one or more digits in the specified radix. The
317 * character-to-digit mapping is provided by {@code
318 * Character.digit}. The String may not contain any extraneous
319 * characters (whitespace, for example).
320 *
321 * @param val String representation of BigInteger.
322 * @param radix radix to be used in interpreting {@code val}.
323 * @throws NumberFormatException {@code val} is not a valid representation
324 * of a BigInteger in the specified radix, or {@code radix} is
325 * outside the range from {@link Character#MIN_RADIX} to
326 * {@link Character#MAX_RADIX}, inclusive.
327 * @see Character#digit
328 */
329 public BigInteger(String val, int radix) {
330 int cursor = 0, numDigits;
331 final int len = val.length();
332
333 if (radix < Character.MIN_RADIX || radix > Character.MAX_RADIX)
334 throw new NumberFormatException("Radix out of range");
335 if (len == 0)
336 throw new NumberFormatException("Zero length BigInteger");
337
338 // Check for at most one leading sign
339 int sign = 1;
340 int index1 = val.lastIndexOf('-');
341 int index2 = val.lastIndexOf('+');
342 if ((index1 + index2) <= -1) {
343 // No leading sign character or at most one leading sign character
344 if (index1 == 0 || index2 == 0) {
345 cursor = 1;
346 if (len == 1)
347 throw new NumberFormatException("Zero length BigInteger");
348 }
349 if (index1 == 0)
350 sign = -1;
351 } else
352 throw new NumberFormatException("Illegal embedded sign character");
353
354 // Skip leading zeros and compute number of digits in magnitude
355 while (cursor < len &&
356 Character.digit(val.charAt(cursor), radix) == 0)
357 cursor++;
358 if (cursor == len) {
359 signum = 0;
360 mag = ZERO.mag;
361 return;
362 }
363
364 numDigits = len - cursor;
365 signum = sign;
366
367 // Pre-allocate array of expected size. May be too large but can
368 // never be too small. Typically exact.
369 int numBits = (int)(((numDigits * bitsPerDigit[radix]) >>> 10) + 1);
370 int numWords = (numBits + 31) >>> 5;
371 int[] magnitude = new int[numWords];
372
373 // Process first (potentially short) digit group
374 int firstGroupLen = numDigits % digitsPerInt[radix];
375 if (firstGroupLen == 0)
376 firstGroupLen = digitsPerInt[radix];
377 String group = val.substring(cursor, cursor += firstGroupLen);
378 magnitude[numWords - 1] = Integer.parseInt(group, radix);
379 if (magnitude[numWords - 1] < 0)
380 throw new NumberFormatException("Illegal digit");
381
382 // Process remaining digit groups
383 int superRadix = intRadix[radix];
384 int groupVal = 0;
385 while (cursor < len) {
386 group = val.substring(cursor, cursor += digitsPerInt[radix]);
387 groupVal = Integer.parseInt(group, radix);
388 if (groupVal < 0)
389 throw new NumberFormatException("Illegal digit");
390 destructiveMulAdd(magnitude, superRadix, groupVal);
391 }
392 // Required for cases where the array was overallocated.
393 mag = trustedStripLeadingZeroInts(magnitude);
394 }
395
396 /*
397 * Constructs a new BigInteger using a char array with radix=10.
398 * Sign is precalculated outside and not allowed in the val.
399 */
400 BigInteger(char[] val, int sign, int len) {
401 int cursor = 0, numDigits;
402
403 // Skip leading zeros and compute number of digits in magnitude
404 while (cursor < len && Character.digit(val[cursor], 10) == 0) {
405 cursor++;
406 }
407 if (cursor == len) {
408 signum = 0;
409 mag = ZERO.mag;
410 return;
411 }
412
413 numDigits = len - cursor;
414 signum = sign;
415 // Pre-allocate array of expected size
416 int numWords;
417 if (len < 10) {
418 numWords = 1;
419 } else {
420 int numBits = (int)(((numDigits * bitsPerDigit[10]) >>> 10) + 1);
421 numWords = (numBits + 31) >>> 5;
422 }
423 int[] magnitude = new int[numWords];
424
425 // Process first (potentially short) digit group
426 int firstGroupLen = numDigits % digitsPerInt[10];
427 if (firstGroupLen == 0)
428 firstGroupLen = digitsPerInt[10];
429 magnitude[numWords - 1] = parseInt(val, cursor, cursor += firstGroupLen);
430
431 // Process remaining digit groups
432 while (cursor < len) {
433 int groupVal = parseInt(val, cursor, cursor += digitsPerInt[10]);
434 destructiveMulAdd(magnitude, intRadix[10], groupVal);
435 }
436 mag = trustedStripLeadingZeroInts(magnitude);
437 }
438
439 // Create an integer with the digits between the two indexes
440 // Assumes start < end. The result may be negative, but it
441 // is to be treated as an unsigned value.
442 private int parseInt(char[] source, int start, int end) {
443 int result = Character.digit(source[start++], 10);
444 if (result == -1)
445 throw new NumberFormatException(new String(source));
446
447 for (int index = start; index<end; index++) {
448 int nextVal = Character.digit(source[index], 10);
449 if (nextVal == -1)
450 throw new NumberFormatException(new String(source));
451 result = 10*result + nextVal;
452 }
453
454 return result;
455 }
456
457 // bitsPerDigit in the given radix times 1024
458 // Rounded up to avoid underallocation.
459 private static long bitsPerDigit[] = { 0, 0,
460 1024, 1624, 2048, 2378, 2648, 2875, 3072, 3247, 3402, 3543, 3672,
461 3790, 3899, 4001, 4096, 4186, 4271, 4350, 4426, 4498, 4567, 4633,
462 4696, 4756, 4814, 4870, 4923, 4975, 5025, 5074, 5120, 5166, 5210,
463 5253, 5295};
464
465 // Multiply x array times word y in place, and add word z
466 private static void destructiveMulAdd(int[] x, int y, int z) {
467 // Perform the multiplication word by word
468 long ylong = y & LONG_MASK;
469 long zlong = z & LONG_MASK;
470 int len = x.length;
471
472 long product = 0;
473 long carry = 0;
474 for (int i = len-1; i >= 0; i--) {
475 product = ylong * (x[i] & LONG_MASK) + carry;
476 x[i] = (int)product;
477 carry = product >>> 32;
478 }
479
480 // Perform the addition
481 long sum = (x[len-1] & LONG_MASK) + zlong;
482 x[len-1] = (int)sum;
483 carry = sum >>> 32;
484 for (int i = len-2; i >= 0; i--) {
485 sum = (x[i] & LONG_MASK) + carry;
486 x[i] = (int)sum;
487 carry = sum >>> 32;
488 }
489 }
490
491 /**
492 * Translates the decimal String representation of a BigInteger into a
493 * BigInteger. The String representation consists of an optional minus
494 * sign followed by a sequence of one or more decimal digits. The
495 * character-to-digit mapping is provided by {@code Character.digit}.
496 * The String may not contain any extraneous characters (whitespace, for
497 * example).
498 *
499 * @param val decimal String representation of BigInteger.
500 * @throws NumberFormatException {@code val} is not a valid representation
501 * of a BigInteger.
502 * @see Character#digit
503 */
504 public BigInteger(String val) {
505 this(val, 10);
506 }
507
508 /**
509 * Constructs a randomly generated BigInteger, uniformly distributed over
510 * the range 0 to (2<sup>{@code numBits}</sup> - 1), inclusive.
511 * The uniformity of the distribution assumes that a fair source of random
512 * bits is provided in {@code rnd}. Note that this constructor always
513 * constructs a non-negative BigInteger.
514 *
515 * @param numBits maximum bitLength of the new BigInteger.
516 * @param rnd source of randomness to be used in computing the new
517 * BigInteger.
518 * @throws IllegalArgumentException {@code numBits} is negative.
519 * @see #bitLength()
520 */
521 public BigInteger(int numBits, Random rnd) {
522 this(1, randomBits(numBits, rnd));
523 }
524
525 private static byte[] randomBits(int numBits, Random rnd) {
526 if (numBits < 0)
527 throw new IllegalArgumentException("numBits must be non-negative");
528 int numBytes = (int)(((long)numBits+7)/8); // avoid overflow
529 byte[] randomBits = new byte[numBytes];
530
531 // Generate random bytes and mask out any excess bits
532 if (numBytes > 0) {
533 rnd.nextBytes(randomBits);
534 int excessBits = 8*numBytes - numBits;
535 randomBits[0] &= (1 << (8-excessBits)) - 1;
536 }
537 return randomBits;
538 }
539
540 /**
541 * Constructs a randomly generated positive BigInteger that is probably
542 * prime, with the specified bitLength.
543 *
544 * <p>It is recommended that the {@link #probablePrime probablePrime}
545 * method be used in preference to this constructor unless there
546 * is a compelling need to specify a certainty.
547 *
548 * @param bitLength bitLength of the returned BigInteger.
549 * @param certainty a measure of the uncertainty that the caller is
550 * willing to tolerate. The probability that the new BigInteger
551 * represents a prime number will exceed
552 * (1 - 1/2<sup>{@code certainty}</sup>). The execution time of
553 * this constructor is proportional to the value of this parameter.
554 * @param rnd source of random bits used to select candidates to be
555 * tested for primality.
556 * @throws ArithmeticException {@code bitLength < 2}.
557 * @see #bitLength()
558 */
559 public BigInteger(int bitLength, int certainty, Random rnd) {
560 BigInteger prime;
561
562 if (bitLength < 2)
563 throw new ArithmeticException("bitLength < 2");
564 prime = (bitLength < SMALL_PRIME_THRESHOLD
565 ? smallPrime(bitLength, certainty, rnd)
566 : largePrime(bitLength, certainty, rnd));
567 signum = 1;
568 mag = prime.mag;
569 }
570
571 // Minimum size in bits that the requested prime number has
572 // before we use the large prime number generating algorithms.
573 // The cutoff of 95 was chosen empirically for best performance.
574 private static final int SMALL_PRIME_THRESHOLD = 95;
575
576 // Certainty required to meet the spec of probablePrime
577 private static final int DEFAULT_PRIME_CERTAINTY = 100;
578
579 /**
580 * Returns a positive BigInteger that is probably prime, with the
581 * specified bitLength. The probability that a BigInteger returned
582 * by this method is composite does not exceed 2<sup>-100</sup>.
583 *
584 * @param bitLength bitLength of the returned BigInteger.
585 * @param rnd source of random bits used to select candidates to be
586 * tested for primality.
587 * @return a BigInteger of {@code bitLength} bits that is probably prime
588 * @throws ArithmeticException {@code bitLength < 2}.
589 * @see #bitLength()
590 * @since 1.4
591 */
592 public static BigInteger probablePrime(int bitLength, Random rnd) {
593 if (bitLength < 2)
594 throw new ArithmeticException("bitLength < 2");
595
596 return (bitLength < SMALL_PRIME_THRESHOLD ?
597 smallPrime(bitLength, DEFAULT_PRIME_CERTAINTY, rnd) :
598 largePrime(bitLength, DEFAULT_PRIME_CERTAINTY, rnd));
599 }
600
601 /**
602 * Find a random number of the specified bitLength that is probably prime.
603 * This method is used for smaller primes, its performance degrades on
604 * larger bitlengths.
605 *
606 * This method assumes bitLength > 1.
607 */
608 private static BigInteger smallPrime(int bitLength, int certainty, Random rnd) {
609 int magLen = (bitLength + 31) >>> 5;
610 int temp[] = new int[magLen];
611 int highBit = 1 << ((bitLength+31) & 0x1f); // High bit of high int
612 int highMask = (highBit << 1) - 1; // Bits to keep in high int
613
614 while(true) {
615 // Construct a candidate
616 for (int i=0; i<magLen; i++)
617 temp[i] = rnd.nextInt();
618 temp[0] = (temp[0] & highMask) | highBit; // Ensure exact length
619 if (bitLength > 2)
620 temp[magLen-1] |= 1; // Make odd if bitlen > 2
621
622 BigInteger p = new BigInteger(temp, 1);
623
624 // Do cheap "pre-test" if applicable
625 if (bitLength > 6) {
626 long r = p.remainder(SMALL_PRIME_PRODUCT).longValue();
627 if ((r%3==0) || (r%5==0) || (r%7==0) || (r%11==0) ||
628 (r%13==0) || (r%17==0) || (r%19==0) || (r%23==0) ||
629 (r%29==0) || (r%31==0) || (r%37==0) || (r%41==0))
630 continue; // Candidate is composite; try another
631 }
632
633 // All candidates of bitLength 2 and 3 are prime by this point
634 if (bitLength < 4)
635 return p;
636
637 // Do expensive test if we survive pre-test (or it's inapplicable)
638 if (p.primeToCertainty(certainty, rnd))
639 return p;
640 }
641 }
642
643 private static final BigInteger SMALL_PRIME_PRODUCT
644 = valueOf(3L*5*7*11*13*17*19*23*29*31*37*41);
645
646 /**
647 * Find a random number of the specified bitLength that is probably prime.
648 * This method is more appropriate for larger bitlengths since it uses
649 * a sieve to eliminate most composites before using a more expensive
650 * test.
651 */
652 private static BigInteger largePrime(int bitLength, int certainty, Random rnd) {
653 BigInteger p;
654 p = new BigInteger(bitLength, rnd).setBit(bitLength-1);
655 p.mag[p.mag.length-1] &= 0xfffffffe;
656
657 // Use a sieve length likely to contain the next prime number
658 int searchLen = (bitLength / 20) * 64;
659 BitSieve searchSieve = new BitSieve(p, searchLen);
660 BigInteger candidate = searchSieve.retrieve(p, certainty, rnd);
661
662 while ((candidate == null) || (candidate.bitLength() != bitLength)) {
663 p = p.add(BigInteger.valueOf(2*searchLen));
664 if (p.bitLength() != bitLength)
665 p = new BigInteger(bitLength, rnd).setBit(bitLength-1);
666 p.mag[p.mag.length-1] &= 0xfffffffe;
667 searchSieve = new BitSieve(p, searchLen);
668 candidate = searchSieve.retrieve(p, certainty, rnd);
669 }
670 return candidate;
671 }
672
673 /**
674 * Returns the first integer greater than this {@code BigInteger} that
675 * is probably prime. The probability that the number returned by this
676 * method is composite does not exceed 2<sup>-100</sup>. This method will
677 * never skip over a prime when searching: if it returns {@code p}, there
678 * is no prime {@code q} such that {@code this < q < p}.
679 *
680 * @return the first integer greater than this {@code BigInteger} that
681 * is probably prime.
682 * @throws ArithmeticException {@code this < 0}.
683 * @since 1.5
684 */
685 public BigInteger nextProbablePrime() {
686 if (this.signum < 0)
687 throw new ArithmeticException("start < 0: " + this);
688
689 // Handle trivial cases
690 if ((this.signum == 0) || this.equals(ONE))
691 return TWO;
692
693 BigInteger result = this.add(ONE);
694
695 // Fastpath for small numbers
696 if (result.bitLength() < SMALL_PRIME_THRESHOLD) {
697
698 // Ensure an odd number
699 if (!result.testBit(0))
700 result = result.add(ONE);
701
702 while(true) {
703 // Do cheap "pre-test" if applicable
704 if (result.bitLength() > 6) {
705 long r = result.remainder(SMALL_PRIME_PRODUCT).longValue();
706 if ((r%3==0) || (r%5==0) || (r%7==0) || (r%11==0) ||
707 (r%13==0) || (r%17==0) || (r%19==0) || (r%23==0) ||
708 (r%29==0) || (r%31==0) || (r%37==0) || (r%41==0)) {
709 result = result.add(TWO);
710 continue; // Candidate is composite; try another
711 }
712 }
713
714 // All candidates of bitLength 2 and 3 are prime by this point
715 if (result.bitLength() < 4)
716 return result;
717
718 // The expensive test
719 if (result.primeToCertainty(DEFAULT_PRIME_CERTAINTY, null))
720 return result;
721
722 result = result.add(TWO);
723 }
724 }
725
726 // Start at previous even number
727 if (result.testBit(0))
728 result = result.subtract(ONE);
729
730 // Looking for the next large prime
731 int searchLen = (result.bitLength() / 20) * 64;
732
733 while(true) {
734 BitSieve searchSieve = new BitSieve(result, searchLen);
735 BigInteger candidate = searchSieve.retrieve(result,
736 DEFAULT_PRIME_CERTAINTY, null);
737 if (candidate != null)
738 return candidate;
739 result = result.add(BigInteger.valueOf(2 * searchLen));
740 }
741 }
742
743 /**
744 * Returns {@code true} if this BigInteger is probably prime,
745 * {@code false} if it's definitely composite.
746 *
747 * This method assumes bitLength > 2.
748 *
749 * @param certainty a measure of the uncertainty that the caller is
750 * willing to tolerate: if the call returns {@code true}
751 * the probability that this BigInteger is prime exceeds
752 * {@code (1 - 1/2<sup>certainty</sup>)}. The execution time of
753 * this method is proportional to the value of this parameter.
754 * @return {@code true} if this BigInteger is probably prime,
755 * {@code false} if it's definitely composite.
756 */
757 boolean primeToCertainty(int certainty, Random random) {
758 int rounds = 0;
759 int n = (Math.min(certainty, Integer.MAX_VALUE-1)+1)/2;
760
761 // The relationship between the certainty and the number of rounds
762 // we perform is given in the draft standard ANSI X9.80, "PRIME
763 // NUMBER GENERATION, PRIMALITY TESTING, AND PRIMALITY CERTIFICATES".
764 int sizeInBits = this.bitLength();
765 if (sizeInBits < 100) {
766 rounds = 50;
767 rounds = n < rounds ? n : rounds;
768 return passesMillerRabin(rounds, random);
769 }
770
771 if (sizeInBits < 256) {
772 rounds = 27;
773 } else if (sizeInBits < 512) {
774 rounds = 15;
775 } else if (sizeInBits < 768) {
776 rounds = 8;
777 } else if (sizeInBits < 1024) {
778 rounds = 4;
779 } else {
780 rounds = 2;
781 }
782 rounds = n < rounds ? n : rounds;
783
784 return passesMillerRabin(rounds, random) && passesLucasLehmer();
785 }
786
787 /**
788 * Returns true iff this BigInteger is a Lucas-Lehmer probable prime.
789 *
790 * The following assumptions are made:
791 * This BigInteger is a positive, odd number.
792 */
793 private boolean passesLucasLehmer() {
794 BigInteger thisPlusOne = this.add(ONE);
795
796 // Step 1
797 int d = 5;
798 while (jacobiSymbol(d, this) != -1) {
799 // 5, -7, 9, -11, ...
800 d = (d<0) ? Math.abs(d)+2 : -(d+2);
801 }
802
803 // Step 2
804 BigInteger u = lucasLehmerSequence(d, thisPlusOne, this);
805
806 // Step 3
807 return u.mod(this).equals(ZERO);
808 }
809
810 /**
811 * Computes Jacobi(p,n).
812 * Assumes n positive, odd, n>=3.
813 */
814 private static int jacobiSymbol(int p, BigInteger n) {
815 if (p == 0)
816 return 0;
817
818 // Algorithm and comments adapted from Colin Plumb's C library.
819 int j = 1;
820 int u = n.mag[n.mag.length-1];
821
822 // Make p positive
823 if (p < 0) {
824 p = -p;
825 int n8 = u & 7;
826 if ((n8 == 3) || (n8 == 7))
827 j = -j; // 3 (011) or 7 (111) mod 8
828 }
829
830 // Get rid of factors of 2 in p
831 while ((p & 3) == 0)
832 p >>= 2;
833 if ((p & 1) == 0) {
834 p >>= 1;
835 if (((u ^ (u>>1)) & 2) != 0)
836 j = -j; // 3 (011) or 5 (101) mod 8
837 }
838 if (p == 1)
839 return j;
840 // Then, apply quadratic reciprocity
841 if ((p & u & 2) != 0) // p = u = 3 (mod 4)?
842 j = -j;
843 // And reduce u mod p
844 u = n.mod(BigInteger.valueOf(p)).intValue();
845
846 // Now compute Jacobi(u,p), u < p
847 while (u != 0) {
848 while ((u & 3) == 0)
849 u >>= 2;
850 if ((u & 1) == 0) {
851 u >>= 1;
852 if (((p ^ (p>>1)) & 2) != 0)
853 j = -j; // 3 (011) or 5 (101) mod 8
854 }
855 if (u == 1)
856 return j;
857 // Now both u and p are odd, so use quadratic reciprocity
858 assert (u < p);
859 int t = u; u = p; p = t;
860 if ((u & p & 2) != 0) // u = p = 3 (mod 4)?
861 j = -j;
862 // Now u >= p, so it can be reduced
863 u %= p;
864 }
865 return 0;
866 }
867
868 private static BigInteger lucasLehmerSequence(int z, BigInteger k, BigInteger n) {
869 BigInteger d = BigInteger.valueOf(z);
870 BigInteger u = ONE; BigInteger u2;
871 BigInteger v = ONE; BigInteger v2;
872
873 for (int i=k.bitLength()-2; i>=0; i--) {
874 u2 = u.multiply(v).mod(n);
875
876 v2 = v.square().add(d.multiply(u.square())).mod(n);
877 if (v2.testBit(0))
878 v2 = v2.subtract(n);
879
880 v2 = v2.shiftRight(1);
881
882 u = u2; v = v2;
883 if (k.testBit(i)) {
884 u2 = u.add(v).mod(n);
885 if (u2.testBit(0))
886 u2 = u2.subtract(n);
887
888 u2 = u2.shiftRight(1);
889 v2 = v.add(d.multiply(u)).mod(n);
890 if (v2.testBit(0))
891 v2 = v2.subtract(n);
892 v2 = v2.shiftRight(1);
893
894 u = u2; v = v2;
895 }
896 }
897 return u;
898 }
899
900 private static volatile Random staticRandom;
901
902 private static Random getSecureRandom() {
903 if (staticRandom == null) {
904 staticRandom = new java.security.SecureRandom();
905 }
906 return staticRandom;
907 }
908
909 /**
910 * Returns true iff this BigInteger passes the specified number of
911 * Miller-Rabin tests. This test is taken from the DSA spec (NIST FIPS
912 * 186-2).
913 *
914 * The following assumptions are made:
915 * This BigInteger is a positive, odd number greater than 2.
916 * iterations<=50.
917 */
918 private boolean passesMillerRabin(int iterations, Random rnd) {
919 // Find a and m such that m is odd and this == 1 + 2**a * m
920 BigInteger thisMinusOne = this.subtract(ONE);
921 BigInteger m = thisMinusOne;
922 int a = m.getLowestSetBit();
923 m = m.shiftRight(a);
924
925 // Do the tests
926 if (rnd == null) {
927 rnd = getSecureRandom();
928 }
929 for (int i=0; i<iterations; i++) {
930 // Generate a uniform random on (1, this)
931 BigInteger b;
932 do {
933 b = new BigInteger(this.bitLength(), rnd);
934 } while (b.compareTo(ONE) <= 0 || b.compareTo(this) >= 0);
935
936 int j = 0;
937 BigInteger z = b.modPow(m, this);
938 while(!((j==0 && z.equals(ONE)) || z.equals(thisMinusOne))) {
939 if (j>0 && z.equals(ONE) || ++j==a)
940 return false;
941 z = z.modPow(TWO, this);
942 }
943 }
944 return true;
945 }
946
947 /**
948 * This internal constructor differs from its public cousin
949 * with the arguments reversed in two ways: it assumes that its
950 * arguments are correct, and it doesn't copy the magnitude array.
951 */
952 BigInteger(int[] magnitude, int signum) {
953 this.signum = (magnitude.length==0 ? 0 : signum);
954 this.mag = magnitude;
955 }
956
957 /**
958 * This private constructor is for internal use and assumes that its
959 * arguments are correct.
960 */
961 private BigInteger(byte[] magnitude, int signum) {
962 this.signum = (magnitude.length==0 ? 0 : signum);
963 this.mag = stripLeadingZeroBytes(magnitude);
964 }
965
966 //Static Factory Methods
967
968 /**
969 * Returns a BigInteger whose value is equal to that of the
970 * specified {@code long}. This "static factory method" is
971 * provided in preference to a ({@code long}) constructor
972 * because it allows for reuse of frequently used BigIntegers.
973 *
974 * @param val value of the BigInteger to return.
975 * @return a BigInteger with the specified value.
976 */
977 public static BigInteger valueOf(long val) {
978 // If -MAX_CONSTANT < val < MAX_CONSTANT, return stashed constant
979 if (val == 0)
980 return ZERO;
981 if (val > 0 && val <= MAX_CONSTANT)
982 return posConst[(int) val];
983 else if (val < 0 && val >= -MAX_CONSTANT)
984 return negConst[(int) -val];
985
986 return new BigInteger(val);
987 }
988
989 /**
990 * Constructs a BigInteger with the specified value, which may not be zero.
991 */
992 private BigInteger(long val) {
993 if (val < 0) {
994 val = -val;
995 signum = -1;
996 } else {
997 signum = 1;
998 }
999
1000 int highWord = (int)(val >>> 32);
1001 if (highWord==0) {
1002 mag = new int[1];
1003 mag[0] = (int)val;
1004 } else {
1005 mag = new int[2];
1006 mag[0] = highWord;
1007 mag[1] = (int)val;
1008 }
1009 }
1010
1011 /**
1012 * Returns a BigInteger with the given two's complement representation.
1013 * Assumes that the input array will not be modified (the returned
1014 * BigInteger will reference the input array if feasible).
1015 */
1016 private static BigInteger valueOf(int val[]) {
1017 return (val[0]>0 ? new BigInteger(val, 1) : new BigInteger(val));
1018 }
1019
1020 // Constants
1021
1022 /**
1023 * Initialize static constant array when class is loaded.
1024 */
1025 private final static int MAX_CONSTANT = 16;
1026 private static BigInteger posConst[] = new BigInteger[MAX_CONSTANT+1];
1027 private static BigInteger negConst[] = new BigInteger[MAX_CONSTANT+1];
1028
1029 static {
1030 for (int i = 1; i <= MAX_CONSTANT; i++) {
1031 int[] magnitude = new int[1];
1032 magnitude[0] = i;
1033 posConst[i] = new BigInteger(magnitude, 1);
1034 negConst[i] = new BigInteger(magnitude, -1);
1035 }
1036 }
1037
1038 /**
1039 * The BigInteger constant zero.
1040 *
1041 * @since 1.2
1042 */
1043 public static final BigInteger ZERO = new BigInteger(new int[0], 0);
1044
1045 /**
1046 * The BigInteger constant one.
1047 *
1048 * @since 1.2
1049 */
1050 public static final BigInteger ONE = valueOf(1);
1051
1052 /**
1053 * The BigInteger constant two. (Not exported.)
1054 */
1055 private static final BigInteger TWO = valueOf(2);
1056
1057 /**
1058 * The BigInteger constant -1. (Not exported.)
1059 */
1060 private static final BigInteger NEGATIVE_ONE = valueOf(-1);
1061
1062 /**
1063 * The BigInteger constant ten.
1064 *
1065 * @since 1.5
1066 */
1067 public static final BigInteger TEN = valueOf(10);
1068
1069 // Arithmetic Operations
1070
1071 /**
1072 * Returns a BigInteger whose value is {@code (this + val)}.
1073 *
1074 * @param val value to be added to this BigInteger.
1075 * @return {@code this + val}
1076 */
1077 public BigInteger add(BigInteger val) {
1078 if (val.signum == 0)
1079 return this;
1080 if (signum == 0)
1081 return val;
1082 if (val.signum == signum)
1083 return new BigInteger(add(mag, val.mag), signum);
1084
1085 int cmp = compareMagnitude(val);
1086 if (cmp == 0)
1087 return ZERO;
1088 int[] resultMag = (cmp > 0 ? subtract(mag, val.mag)
1089 : subtract(val.mag, mag));
1090 resultMag = trustedStripLeadingZeroInts(resultMag);
1091
1092 return new BigInteger(resultMag, cmp == signum ? 1 : -1);
1093 }
1094
1095 /**
1096 * Package private methods used by BigDecimal code to add a BigInteger
1097 * with a long. Assumes val is not equal to INFLATED.
1098 */
1099 BigInteger add(long val) {
1100 if (val == 0)
1101 return this;
1102 if (signum == 0)
1103 return valueOf(val);
1104 if (Long.signum(val) == signum)
1105 return new BigInteger(add(mag, Math.abs(val)), signum);
1106 int cmp = compareMagnitude(val);
1107 if (cmp == 0)
1108 return ZERO;
1109 int[] resultMag = (cmp > 0 ? subtract(mag, Math.abs(val)) : subtract(Math.abs(val), mag));
1110 resultMag = trustedStripLeadingZeroInts(resultMag);
1111 return new BigInteger(resultMag, cmp == signum ? 1 : -1);
1112 }
1113
1114 /**
1115 * Adds the contents of the int array x and long value val. This
1116 * method allocates a new int array to hold the answer and returns
1117 * a reference to that array. Assumes x.length > 0 and val is
1118 * non-negative
1119 */
1120 private static int[] add(int[] x, long val) {
1121 int[] y;
1122 long sum = 0;
1123 int xIndex = x.length;
1124 int[] result;
1125 int highWord = (int)(val >>> 32);
1126 if (highWord==0) {
1127 result = new int[xIndex];
1128 sum = (x[--xIndex] & LONG_MASK) + val;
1129 result[xIndex] = (int)sum;
1130 } else {
1131 if (xIndex == 1) {
1132 result = new int[2];
1133 sum = val + (x[0] & LONG_MASK);
1134 result[1] = (int)sum;
1135 result[0] = (int)(sum >>> 32);
1136 return result;
1137 } else {
1138 result = new int[xIndex];
1139 sum = (x[--xIndex] & LONG_MASK) + (val & LONG_MASK);
1140 result[xIndex] = (int)sum;
1141 sum = (x[--xIndex] & LONG_MASK) + (highWord & LONG_MASK) + (sum >>> 32);
1142 result[xIndex] = (int)sum;
1143 }
1144 }
1145 // Copy remainder of longer number while carry propagation is required
1146 boolean carry = (sum >>> 32 != 0);
1147 while (xIndex > 0 && carry)
1148 carry = ((result[--xIndex] = x[xIndex] + 1) == 0);
1149 // Copy remainder of longer number
1150 while (xIndex > 0)
1151 result[--xIndex] = x[xIndex];
1152 // Grow result if necessary
1153 if (carry) {
1154 int bigger[] = new int[result.length + 1];
1155 System.arraycopy(result, 0, bigger, 1, result.length);
1156 bigger[0] = 0x01;
1157 return bigger;
1158 }
1159 return result;
1160 }
1161
1162 /**
1163 * Adds the contents of the int arrays x and y. This method allocates
1164 * a new int array to hold the answer and returns a reference to that
1165 * array.
1166 */
1167 private static int[] add(int[] x, int[] y) {
1168 // If x is shorter, swap the two arrays
1169 if (x.length < y.length) {
1170 int[] tmp = x;
1171 x = y;
1172 y = tmp;
1173 }
1174
1175 int xIndex = x.length;
1176 int yIndex = y.length;
1177 int result[] = new int[xIndex];
1178 long sum = 0;
1179 if(yIndex==1) {
1180 sum = (x[--xIndex] & LONG_MASK) + (y[0] & LONG_MASK) ;
1181 result[xIndex] = (int)sum;
1182 } else {
1183 // Add common parts of both numbers
1184 while(yIndex > 0) {
1185 sum = (x[--xIndex] & LONG_MASK) +
1186 (y[--yIndex] & LONG_MASK) + (sum >>> 32);
1187 result[xIndex] = (int)sum;
1188 }
1189 }
1190 // Copy remainder of longer number while carry propagation is required
1191 boolean carry = (sum >>> 32 != 0);
1192 while (xIndex > 0 && carry)
1193 carry = ((result[--xIndex] = x[xIndex] + 1) == 0);
1194
1195 // Copy remainder of longer number
1196 while (xIndex > 0)
1197 result[--xIndex] = x[xIndex];
1198
1199 // Grow result if necessary
1200 if (carry) {
1201 int bigger[] = new int[result.length + 1];
1202 System.arraycopy(result, 0, bigger, 1, result.length);
1203 bigger[0] = 0x01;
1204 return bigger;
1205 }
1206 return result;
1207 }
1208
1209 private static int[] subtract(long val, int[] little) {
1210 int highWord = (int)(val >>> 32);
1211 if (highWord==0) {
1212 int result[] = new int[1];
1213 result[0] = (int)(val - (little[0] & LONG_MASK));
1214 return result;
1215 } else {
1216 int result[] = new int[2];
1217 if(little.length==1) {
1218 long difference = ((int)val & LONG_MASK) - (little[0] & LONG_MASK);
1219 result[1] = (int)difference;
1220 // Subtract remainder of longer number while borrow propagates
1221 boolean borrow = (difference >> 32 != 0);
1222 if(borrow) {
1223 result[0] = highWord - 1;
1224 } else { // Copy remainder of longer number
1225 result[0] = highWord;
1226 }
1227 return result;
1228 } else { // little.length==2
1229 long difference = ((int)val & LONG_MASK) - (little[1] & LONG_MASK);
1230 result[1] = (int)difference;
1231 difference = (highWord & LONG_MASK) - (little[0] & LONG_MASK) + (difference >> 32);
1232 result[0] = (int)difference;
1233 return result;
1234 }
1235 }
1236 }
1237
1238 /**
1239 * Subtracts the contents of the second argument (val) from the
1240 * first (big). The first int array (big) must represent a larger number
1241 * than the second. This method allocates the space necessary to hold the
1242 * answer.
1243 * assumes val >= 0
1244 */
1245 private static int[] subtract(int[] big, long val) {
1246 int highWord = (int)(val >>> 32);
1247 int bigIndex = big.length;
1248 int result[] = new int[bigIndex];
1249 long difference = 0;
1250
1251 if (highWord==0) {
1252 difference = (big[--bigIndex] & LONG_MASK) - val;
1253 result[bigIndex] = (int)difference;
1254 } else {
1255 difference = (big[--bigIndex] & LONG_MASK) - (val & LONG_MASK);
1256 result[bigIndex] = (int)difference;
1257 difference = (big[--bigIndex] & LONG_MASK) - (highWord & LONG_MASK) + (difference >> 32);
1258 result[bigIndex] = (int)difference;
1259 }
1260
1261
1262 // Subtract remainder of longer number while borrow propagates
1263 boolean borrow = (difference >> 32 != 0);
1264 while (bigIndex > 0 && borrow)
1265 borrow = ((result[--bigIndex] = big[bigIndex] - 1) == -1);
1266
1267 // Copy remainder of longer number
1268 while (bigIndex > 0)
1269 result[--bigIndex] = big[bigIndex];
1270
1271 return result;
1272 }
1273
1274 /**
1275 * Returns a BigInteger whose value is {@code (this - val)}.
1276 *
1277 * @param val value to be subtracted from this BigInteger.
1278 * @return {@code this - val}
1279 */
1280 public BigInteger subtract(BigInteger val) {
1281 if (val.signum == 0)
1282 return this;
1283 if (signum == 0)
1284 return val.negate();
1285 if (val.signum != signum)
1286 return new BigInteger(add(mag, val.mag), signum);
1287
1288 int cmp = compareMagnitude(val);
1289 if (cmp == 0)
1290 return ZERO;
1291 int[] resultMag = (cmp > 0 ? subtract(mag, val.mag)
1292 : subtract(val.mag, mag));
1293 resultMag = trustedStripLeadingZeroInts(resultMag);
1294 return new BigInteger(resultMag, cmp == signum ? 1 : -1);
1295 }
1296
1297 /**
1298 * Subtracts the contents of the second int arrays (little) from the
1299 * first (big). The first int array (big) must represent a larger number
1300 * than the second. This method allocates the space necessary to hold the
1301 * answer.
1302 */
1303 private static int[] subtract(int[] big, int[] little) {
1304 int bigIndex = big.length;
1305 int result[] = new int[bigIndex];
1306 int littleIndex = little.length;
1307 long difference = 0;
1308
1309 // Subtract common parts of both numbers
1310 while(littleIndex > 0) {
1311 difference = (big[--bigIndex] & LONG_MASK) -
1312 (little[--littleIndex] & LONG_MASK) +
1313 (difference >> 32);
1314 result[bigIndex] = (int)difference;
1315 }
1316
1317 // Subtract remainder of longer number while borrow propagates
1318 boolean borrow = (difference >> 32 != 0);
1319 while (bigIndex > 0 && borrow)
1320 borrow = ((result[--bigIndex] = big[bigIndex] - 1) == -1);
1321
1322 // Copy remainder of longer number
1323 while (bigIndex > 0)
1324 result[--bigIndex] = big[bigIndex];
1325
1326 return result;
1327 }
1328
1329 /**
1330 * Returns a BigInteger whose value is {@code (this * val)}.
1331 *
1332 * @param val value to be multiplied by this BigInteger.
1333 * @return {@code this * val}
1334 */
1335 public BigInteger multiply(BigInteger val) {
1336 if (val.signum == 0 || signum == 0)
1337 return ZERO;
1338
1339 int xlen = mag.length;
1340 int ylen = val.mag.length;
1341
1342 if ((xlen < KARATSUBA_THRESHOLD) || (ylen < KARATSUBA_THRESHOLD))
1343 {
1344 int resultSign = signum == val.signum ? 1 : -1;
1345 if (val.mag.length == 1) {
1346 return multiplyByInt(mag,val.mag[0], resultSign);
1347 }
1348 if(mag.length == 1) {
1349 return multiplyByInt(val.mag,mag[0], resultSign);
1350 }
1351 int[] result = multiplyToLen(mag, xlen,
1352 val.mag, ylen, null);
1353 result = trustedStripLeadingZeroInts(result);
1354 return new BigInteger(result, resultSign);
1355 }
1356 else
1357 if ((xlen < TOOM_COOK_THRESHOLD) && (ylen < TOOM_COOK_THRESHOLD))
1358 return multiplyKaratsuba(this, val);
1359 else
1360 return multiplyToomCook3(this, val);
1361 }
1362
1363 private static BigInteger multiplyByInt(int[] x, int y, int sign) {
1364 if(Integer.bitCount(y)==1) {
1365 return new BigInteger(shiftLeft(x,Integer.numberOfTrailingZeros(y)), sign);
1366 }
1367 int xlen = x.length;
1368 int[] rmag = new int[xlen + 1];
1369 long carry = 0;
1370 long yl = y & LONG_MASK;
1371 int rstart = rmag.length - 1;
1372 for (int i = xlen - 1; i >= 0; i--) {
1373 long product = (x[i] & LONG_MASK) * yl + carry;
1374 rmag[rstart--] = (int)product;
1375 carry = product >>> 32;
1376 }
1377 if (carry == 0L) {
1378 rmag = java.util.Arrays.copyOfRange(rmag, 1, rmag.length);
1379 } else {
1380 rmag[rstart] = (int)carry;
1381 }
1382 return new BigInteger(rmag, sign);
1383 }
1384
1385 /**
1386 * Package private methods used by BigDecimal code to multiply a BigInteger
1387 * with a long. Assumes v is not equal to INFLATED.
1388 */
1389 BigInteger multiply(long v) {
1390 if (v == 0 || signum == 0)
1391 return ZERO;
1392 if (v == BigDecimal.INFLATED)
1393 return multiply(BigInteger.valueOf(v));
1394 int rsign = (v > 0 ? signum : -signum);
1395 if (v < 0)
1396 v = -v;
1397 long dh = v >>> 32; // higher order bits
1398 long dl = v & LONG_MASK; // lower order bits
1399
1400 int xlen = mag.length;
1401 int[] value = mag;
1402 int[] rmag = (dh == 0L) ? (new int[xlen + 1]) : (new int[xlen + 2]);
1403 long carry = 0;
1404 int rstart = rmag.length - 1;
1405 for (int i = xlen - 1; i >= 0; i--) {
1406 long product = (value[i] & LONG_MASK) * dl + carry;
1407 rmag[rstart--] = (int)product;
1408 carry = product >>> 32;
1409 }
1410 rmag[rstart] = (int)carry;
1411 if (dh != 0L) {
1412 carry = 0;
1413 rstart = rmag.length - 2;
1414 for (int i = xlen - 1; i >= 0; i--) {
1415 long product = (value[i] & LONG_MASK) * dh +
1416 (rmag[rstart] & LONG_MASK) + carry;
1417 rmag[rstart--] = (int)product;
1418 carry = product >>> 32;
1419 }
1420 rmag[0] = (int)carry;
1421 }
1422 if (carry == 0L)
1423 rmag = java.util.Arrays.copyOfRange(rmag, 1, rmag.length);
1424 return new BigInteger(rmag, rsign);
1425 }
1426
1427 /**
1428 * Multiplies int arrays x and y to the specified lengths and places
1429 * the result into z. There will be no leading zeros in the resultant array.
1430 */
1431 private int[] multiplyToLen(int[] x, int xlen, int[] y, int ylen, int[] z) {
1432 int xstart = xlen - 1;
1433 int ystart = ylen - 1;
1434
1435 if (z == null || z.length < (xlen+ ylen))
1436 z = new int[xlen+ylen];
1437
1438 long carry = 0;
1439 for (int j=ystart, k=ystart+1+xstart; j>=0; j--, k--) {
1440 long product = (y[j] & LONG_MASK) *
1441 (x[xstart] & LONG_MASK) + carry;
1442 z[k] = (int)product;
1443 carry = product >>> 32;
1444 }
1445 z[xstart] = (int)carry;
1446
1447 for (int i = xstart-1; i >= 0; i--) {
1448 carry = 0;
1449 for (int j=ystart, k=ystart+1+i; j>=0; j--, k--) {
1450 long product = (y[j] & LONG_MASK) *
1451 (x[i] & LONG_MASK) +
1452 (z[k] & LONG_MASK) + carry;
1453 z[k] = (int)product;
1454 carry = product >>> 32;
1455 }
1456 z[i] = (int)carry;
1457 }
1458 return z;
1459 }
1460
1461 /**
1462 * Multiplies two BigIntegers using the Karatsuba multiplication
1463 * algorithm. This is a recursive divide-and-conquer algorithm which is
1464 * more efficient for large numbers than what is commonly called the
1465 * "grade-school" algorithm used in multiplyToLen. If the numbers to be
1466 * multiplied have length n, the "grade-school" algorithm has an
1467 * asymptotic complexity of O(n^2). In contrast, the Karatsuba algorithm
1468 * has complexity of O(n^(log2(3))), or O(n^1.585). It achieves this
1469 * increased performance by doing 3 multiplies instead of 4 when
1470 * evaluating the product. As it has some overhead, should be used when
1471 * both numbers are larger than a certain threshold (found
1472 * experimentally).
1473 *
1474 * See: http://en.wikipedia.org/wiki/Karatsuba_algorithm
1475 */
1476 private static BigInteger multiplyKaratsuba(BigInteger x, BigInteger y)
1477 {
1478 int xlen = x.mag.length;
1479 int ylen = y.mag.length;
1480
1481 // The number of ints in each half of the number.
1482 int half = (Math.max(xlen, ylen)+1) / 2;
1483
1484 // xl and yl are the lower halves of x and y respectively,
1485 // xh and yh are the upper halves.
1486 BigInteger xl = x.getLower(half);
1487 BigInteger xh = x.getUpper(half);
1488 BigInteger yl = y.getLower(half);
1489 BigInteger yh = y.getUpper(half);
1490
1491 BigInteger p1 = xh.multiply(yh); // p1 = xh*yh
1492 BigInteger p2 = xl.multiply(yl); // p2 = xl*yl
1493
1494 // p3=(xh+xl)*(yh+yl)
1495 BigInteger p3 = xh.add(xl).multiply(yh.add(yl));
1496
1497 // result = p1 * 2^(32*2*half) + (p3 - p1 - p2) * 2^(32*half) + p2
1498 BigInteger result = p1.shiftLeft(32*half).add(p3.subtract(p1).subtract(p2)).shiftLeft(32*half).add(p2);
1499
1500 if (x.signum != y.signum)
1501 return result.negate();
1502 else
1503 return result;
1504 }
1505
1506 /**
1507 * Multiplies two BigIntegers using a 3-way Toom-Cook multiplication
1508 * algorithm. This is a recursive divide-and-conquer algorithm which is
1509 * more efficient for large numbers than what is commonly called the
1510 * "grade-school" algorithm used in multiplyToLen. If the numbers to be
1511 * multiplied have length n, the "grade-school" algorithm has an
1512 * asymptotic complexity of O(n^2). In contrast, 3-way Toom-Cook has a
1513 * complexity of about O(n^1.465). It achieves this increased asymptotic
1514 * performance by breaking each number into three parts and by doing 5
1515 * multiplies instead of 9 when evaluating the product. Due to overhead
1516 * (additions, shifts, and one division) in the Toom-Cook algorithm, it
1517 * should only be used when both numbers are larger than a certain
1518 * threshold (found experimentally). This threshold is generally larger
1519 * than that for Karatsuba multiplication, so this algorithm is generally
1520 * only used when numbers become significantly larger.
1521 *
1522 * The algorithm used is the "optimal" 3-way Toom-Cook algorithm outlined
1523 * by Marco Bodrato.
1524 *
1525 * See: http://bodrato.it/toom-cook/
1526 * http://bodrato.it/papers/#WAIFI2007
1527 *
1528 * "Towards Optimal Toom-Cook Multiplication for Univariate and
1529 * Multivariate Polynomials in Characteristic 2 and 0." by Marco BODRATO;
1530 * In C.Carlet and B.Sunar, Eds., "WAIFI'07 proceedings", p. 116-133,
1531 * LNCS #4547. Springer, Madrid, Spain, June 21-22, 2007.
1532 *
1533 */
1534 private static BigInteger multiplyToomCook3(BigInteger a, BigInteger b)
1535 {
1536 int alen = a.mag.length;
1537 int blen = b.mag.length;
1538
1539 int largest = Math.max(alen, blen);
1540
1541 // k is the size (in ints) of the lower-order slices.
1542 int k = (largest+2)/3; // Equal to ceil(largest/3)
1543
1544 // r is the size (in ints) of the highest-order slice.
1545 int r = largest - 2*k;
1546
1547 // Obtain slices of the numbers. a2 and b2 are the most significant
1548 // bits of the numbers a and b, and a0 and b0 the least significant.
1549 BigInteger a0, a1, a2, b0, b1, b2;
1550 a2 = a.getToomSlice(k, r, 0, largest);
1551 a1 = a.getToomSlice(k, r, 1, largest);
1552 a0 = a.getToomSlice(k, r, 2, largest);
1553 b2 = b.getToomSlice(k, r, 0, largest);
1554 b1 = b.getToomSlice(k, r, 1, largest);
1555 b0 = b.getToomSlice(k, r, 2, largest);
1556
1557 BigInteger v0, v1, v2, vm1, vinf, t1, t2, tm1, da1, db1;
1558
1559 v0 = a0.multiply(b0);
1560 da1 = a2.add(a0);
1561 db1 = b2.add(b0);
1562 vm1 = da1.subtract(a1).multiply(db1.subtract(b1));
1563 da1 = da1.add(a1);
1564 db1 = db1.add(b1);
1565 v1 = da1.multiply(db1);
1566 v2 = da1.add(a2).shiftLeft(1).subtract(a0).multiply(
1567 db1.add(b2).shiftLeft(1).subtract(b0));
1568 vinf = a2.multiply(b2);
1569
1570 /* The algorithm requires two divisions by 2 and one by 3.
1571 All divisions are known to be exact, that is, they do not produce
1572 remainders, and all results are positive. The divisions by 2 are
1573 implemented as right shifts which are relatively efficient, leaving
1574 only an exact division by 3, which is done by a specialized
1575 linear-time algorithm. */
1576 t2 = v2.subtract(vm1).exactDivideBy3();
1577 tm1 = v1.subtract(vm1).shiftRight(1);
1578 t1 = v1.subtract(v0);
1579 t2 = t2.subtract(t1).shiftRight(1);
1580 t1 = t1.subtract(tm1).subtract(vinf);
1581 t2 = t2.subtract(vinf.shiftLeft(1));
1582 tm1 = tm1.subtract(t2);
1583
1584 // Number of bits to shift left.
1585 int ss = k*32;
1586
1587 BigInteger result = vinf.shiftLeft(ss).add(t2).shiftLeft(ss).add(t1).shiftLeft(ss).add(tm1).shiftLeft(ss).add(v0);
1588
1589 if (a.signum != b.signum)
1590 return result.negate();
1591 else
1592 return result;
1593 }
1594
1595
1596 /**
1597 * Returns a slice of a BigInteger for use in Toom-Cook multiplication.
1598 *
1599 * @param lowerSize The size of the lower-order bit slices.
1600 * @param upperSize The size of the higher-order bit slices.
1601 * @param slice The index of which slice is requested, which must be a
1602 * number from 0 to size-1. Slice 0 is the highest-order bits, and slice
1603 * size-1 are the lowest-order bits. Slice 0 may be of different size than
1604 * the other slices.
1605 * @param fullsize The size of the larger integer array, used to align
1606 * slices to the appropriate position when multiplying different-sized
1607 * numbers.
1608 */
1609 private BigInteger getToomSlice(int lowerSize, int upperSize, int slice,
1610 int fullsize)
1611 {
1612 int start, end, sliceSize, len, offset;
1613
1614 len = mag.length;
1615 offset = fullsize - len;
1616
1617 if (slice == 0)
1618 {
1619 start = 0 - offset;
1620 end = upperSize - 1 - offset;
1621 }
1622 else
1623 {
1624 start = upperSize + (slice-1)*lowerSize - offset;
1625 end = start + lowerSize - 1;
1626 }
1627
1628 if (start < 0)
1629 start = 0;
1630 if (end < 0)
1631 return ZERO;
1632
1633 sliceSize = (end-start) + 1;
1634
1635 if (sliceSize <= 0)
1636 return ZERO;
1637
1638 // While performing Toom-Cook, all slices are positive and
1639 // the sign is adjusted when the final number is composed.
1640 if (start==0 && sliceSize >= len)
1641 return this.abs();
1642
1643 int intSlice[] = new int[sliceSize];
1644 System.arraycopy(mag, start, intSlice, 0, sliceSize);
1645
1646 return new BigInteger(trustedStripLeadingZeroInts(intSlice), 1);
1647 }
1648
1649 /**
1650 * Does an exact division (that is, the remainder is known to be zero)
1651 * of the specified number by 3. This is used in Toom-Cook
1652 * multiplication. This is an efficient algorithm that runs in linear
1653 * time. If the argument is not exactly divisible by 3, results are
1654 * undefined. Note that this is expected to be called with positive
1655 * arguments only.
1656 */
1657 private BigInteger exactDivideBy3()
1658 {
1659 int len = mag.length;
1660 int[] result = new int[len];
1661 long x, w, q, borrow;
1662 borrow = 0L;
1663 for (int i=len-1; i>=0; i--)
1664 {
1665 x = (mag[i] & LONG_MASK);
1666 w = x - borrow;
1667 if (borrow > x) // Did we make the number go negative?
1668 borrow = 1L;
1669 else
1670 borrow = 0L;
1671
1672 // 0xAAAAAAAB is the modular inverse of 3 (mod 2^32). Thus,
1673 // the effect of this is to divide by 3 (mod 2^32).
1674 // This is much faster than division on most architectures.
1675 q = (w * 0xAAAAAAABL) & LONG_MASK;
1676 result[i] = (int) q;
1677
1678 // Now check the borrow. The second check can of course be
1679 // eliminated if the first fails.
1680 if (q >= 0x55555556L)
1681 {
1682 borrow++;
1683 if (q >= 0xAAAAAAABL)
1684 borrow++;
1685 }
1686 }
1687 result = trustedStripLeadingZeroInts(result);
1688 return new BigInteger(result, signum);
1689 }
1690
1691 /**
1692 * Returns a new BigInteger representing n lower ints of the number.
1693 * This is used by Karatsuba multiplication and Karatsuba squaring.
1694 */
1695 private BigInteger getLower(int n) {
1696 int len = mag.length;
1697
1698 if (len <= n)
1699 return this;
1700
1701 int lowerInts[] = new int[n];
1702 System.arraycopy(mag, len-n, lowerInts, 0, n);
1703
1704 return new BigInteger(trustedStripLeadingZeroInts(lowerInts), 1);
1705 }
1706
1707 /**
1708 * Returns a new BigInteger representing mag.length-n upper
1709 * ints of the number. This is used by Karatsuba multiplication and
1710 * Karatsuba squaring.
1711 */
1712 private BigInteger getUpper(int n) {
1713 int len = mag.length;
1714
1715 if (len <= n)
1716 return ZERO;
1717
1718 int upperLen = len - n;
1719 int upperInts[] = new int[upperLen];
1720 System.arraycopy(mag, 0, upperInts, 0, upperLen);
1721
1722 return new BigInteger(trustedStripLeadingZeroInts(upperInts), 1);
1723 }
1724
1725 // Squaring
1726
1727 /**
1728 * Returns a BigInteger whose value is {@code (this<sup>2</sup>)}.
1729 *
1730 * @return {@code this<sup>2</sup>}
1731 */
1732 private BigInteger square() {
1733 if (signum == 0)
1734 return ZERO;
1735 int len = mag.length;
1736
1737 if (len < KARATSUBA_SQUARE_THRESHOLD)
1738 {
1739 int[] z = squareToLen(mag, len, null);
1740 return new BigInteger(trustedStripLeadingZeroInts(z), 1);
1741 }
1742 else
1743 if (len < TOOM_COOK_SQUARE_THRESHOLD)
1744 return squareKaratsuba();
1745 else
1746 return squareToomCook3();
1747 }
1748
1749 /**
1750 * Squares the contents of the int array x. The result is placed into the
1751 * int array z. The contents of x are not changed.
1752 */
1753 private static final int[] squareToLen(int[] x, int len, int[] z) {
1754 /*
1755 * The algorithm used here is adapted from Colin Plumb's C library.
1756 * Technique: Consider the partial products in the multiplication
1757 * of "abcde" by itself:
1758 *
1759 * a b c d e
1760 * * a b c d e
1761 * ==================
1762 * ae be ce de ee
1763 * ad bd cd dd de
1764 * ac bc cc cd ce
1765 * ab bb bc bd be
1766 * aa ab ac ad ae
1767 *
1768 * Note that everything above the main diagonal:
1769 * ae be ce de = (abcd) * e
1770 * ad bd cd = (abc) * d
1771 * ac bc = (ab) * c
1772 * ab = (a) * b
1773 *
1774 * is a copy of everything below the main diagonal:
1775 * de
1776 * cd ce
1777 * bc bd be
1778 * ab ac ad ae
1779 *
1780 * Thus, the sum is 2 * (off the diagonal) + diagonal.
1781 *
1782 * This is accumulated beginning with the diagonal (which
1783 * consist of the squares of the digits of the input), which is then
1784 * divided by two, the off-diagonal added, and multiplied by two
1785 * again. The low bit is simply a copy of the low bit of the
1786 * input, so it doesn't need special care.
1787 */
1788 int zlen = len << 1;
1789 if (z == null || z.length < zlen)
1790 z = new int[zlen];
1791
1792 // Store the squares, right shifted one bit (i.e., divided by 2)
1793 int lastProductLowWord = 0;
1794 for (int j=0, i=0; j<len; j++) {
1795 long piece = (x[j] & LONG_MASK);
1796 long product = piece * piece;
1797 z[i++] = (lastProductLowWord << 31) | (int)(product >>> 33);
1798 z[i++] = (int)(product >>> 1);
1799 lastProductLowWord = (int)product;
1800 }
1801
1802 // Add in off-diagonal sums
1803 for (int i=len, offset=1; i>0; i--, offset+=2) {
1804 int t = x[i-1];
1805 t = mulAdd(z, x, offset, i-1, t);
1806 addOne(z, offset-1, i, t);
1807 }
1808
1809 // Shift back up and set low bit
1810 primitiveLeftShift(z, zlen, 1);
1811 z[zlen-1] |= x[len-1] & 1;
1812
1813 return z;
1814 }
1815
1816 /**
1817 * Squares a BigInteger using the Karatsuba squaring algorithm. It should
1818 * be used when both numbers are larger than a certain threshold (found
1819 * experimentally). It is a recursive divide-and-conquer algorithm that
1820 * has better asymptotic performance than the algorithm used in
1821 * squareToLen.
1822 */
1823 private BigInteger squareKaratsuba()
1824 {
1825 int half = (mag.length+1) / 2;
1826
1827 BigInteger xl = getLower(half);
1828 BigInteger xh = getUpper(half);
1829
1830 BigInteger xhs = xh.square(); // xhs = xh^2
1831 BigInteger xls = xl.square(); // xls = xl^2
1832
1833 // xh^2 << 64 + (((xl+xh)^2 - (xh^2 + xl^2)) << 32) + xl^2
1834 return xhs.shiftLeft(half*32).add(xl.add(xh).square().subtract(xhs.add(xls))).shiftLeft(half*32).add(xls);
1835 }
1836
1837 /**
1838 * Squares a BigInteger using the 3-way Toom-Cook squaring algorithm. It
1839 * should be used when both numbers are larger than a certain threshold
1840 * (found experimentally). It is a recursive divide-and-conquer algorithm
1841 * that has better asymptotic performance than the algorithm used in
1842 * squareToLen or squareKaratsuba.
1843 */
1844 private BigInteger squareToomCook3()
1845 {
1846 int len = mag.length;
1847
1848 // k is the size (in ints) of the lower-order slices.
1849 int k = (len+2)/3; // Equal to ceil(largest/3)
1850
1851 // r is the size (in ints) of the highest-order slice.
1852 int r = len - 2*k;
1853
1854 // Obtain slices of the numbers. a2 is the most significant
1855 // bits of the number, and a0 the least significant.
1856 BigInteger a0, a1, a2;
1857 a2 = getToomSlice(k, r, 0, len);
1858 a1 = getToomSlice(k, r, 1, len);
1859 a0 = getToomSlice(k, r, 2, len);
1860 BigInteger v0, v1, v2, vm1, vinf, t1, t2, tm1, da1;
1861
1862 v0 = a0.square();
1863 da1 = a2.add(a0);
1864 vm1 = da1.subtract(a1).square();
1865 da1 = da1.add(a1);
1866 v1 = da1.square();
1867 vinf = a2.square();
1868 v2 = da1.add(a2).shiftLeft(1).subtract(a0).square();
1869
1870 /* The algorithm requires two divisions by 2 and one by 3.
1871 All divisions are known to be exact, that is, they do not produce
1872 remainders, and all results are positive. The divisions by 2 are
1873 implemented as right shifts which are relatively efficient, leaving
1874 only a division by 3.
1875 The division by 3 is done by an optimized algorithm for this case.
1876 */
1877 t2 = v2.subtract(vm1).exactDivideBy3();
1878 tm1 = v1.subtract(vm1).shiftRight(1);
1879 t1 = v1.subtract(v0);
1880 t2 = t2.subtract(t1).shiftRight(1);
1881 t1 = t1.subtract(tm1).subtract(vinf);
1882 t2 = t2.subtract(vinf.shiftLeft(1));
1883 tm1 = tm1.subtract(t2);
1884
1885 // Number of bits to shift left.
1886 int ss = k*32;
1887
1888 return vinf.shiftLeft(ss).add(t2).shiftLeft(ss).add(t1).shiftLeft(ss).add(tm1).shiftLeft(ss).add(v0);
1889 }
1890
1891 // Division
1892
1893 /**
1894 * Returns a BigInteger whose value is {@code (this / val)}.
1895 *
1896 * @param val value by which this BigInteger is to be divided.
1897 * @return {@code this / val}
1898 * @throws ArithmeticException if {@code val} is zero.
1899 */
1900 public BigInteger divide(BigInteger val) {
1901 MutableBigInteger q = new MutableBigInteger(),
1902 a = new MutableBigInteger(this.mag),
1903 b = new MutableBigInteger(val.mag);
1904
1905 a.divide(b, q, false);
1906 return q.toBigInteger(this.signum * val.signum);
1907 }
1908
1909 /**
1910 * Returns an array of two BigIntegers containing {@code (this / val)}
1911 * followed by {@code (this % val)}.
1912 *
1913 * @param val value by which this BigInteger is to be divided, and the
1914 * remainder computed.
1915 * @return an array of two BigIntegers: the quotient {@code (this / val)}
1916 * is the initial element, and the remainder {@code (this % val)}
1917 * is the final element.
1918 * @throws ArithmeticException if {@code val} is zero.
1919 */
1920 public BigInteger[] divideAndRemainder(BigInteger val) {
1921 BigInteger[] result = new BigInteger[2];
1922 MutableBigInteger q = new MutableBigInteger(),
1923 a = new MutableBigInteger(this.mag),
1924 b = new MutableBigInteger(val.mag);
1925 MutableBigInteger r = a.divide(b, q);
1926 result[0] = q.toBigInteger(this.signum == val.signum ? 1 : -1);
1927 result[1] = r.toBigInteger(this.signum);
1928 return result;
1929 }
1930
1931 /**
1932 * Returns a BigInteger whose value is {@code (this % val)}.
1933 *
1934 * @param val value by which this BigInteger is to be divided, and the
1935 * remainder computed.
1936 * @return {@code this % val}
1937 * @throws ArithmeticException if {@code val} is zero.
1938 */
1939 public BigInteger remainder(BigInteger val) {
1940 MutableBigInteger q = new MutableBigInteger(),
1941 a = new MutableBigInteger(this.mag),
1942 b = new MutableBigInteger(val.mag);
1943
1944 return a.divide(b, q).toBigInteger(this.signum);
1945 }
1946
1947 /**
1948 * Returns a BigInteger whose value is <tt>(this<sup>exponent</sup>)</tt>.
1949 * Note that {@code exponent} is an integer rather than a BigInteger.
1950 *
1951 * @param exponent exponent to which this BigInteger is to be raised.
1952 * @return <tt>this<sup>exponent</sup></tt>
1953 * @throws ArithmeticException {@code exponent} is negative. (This would
1954 * cause the operation to yield a non-integer value.)
1955 */
1956 public BigInteger pow(int exponent) {
1957 if (exponent < 0)
1958 throw new ArithmeticException("Negative exponent");
1959 if (signum==0)
1960 return (exponent==0 ? ONE : this);
1961
1962 BigInteger partToSquare = this.abs();
1963
1964 // Factor out powers of two from the base, as the exponentiation of
1965 // these can be done by left shifts only.
1966 // The remaining part can then be exponentiated faster. The
1967 // powers of two will be multiplied back at the end.
1968 int powersOfTwo = partToSquare.getLowestSetBit();
1969
1970 int remainingBits;
1971
1972 // Factor the powers of two out quickly by shifting right, if needed.
1973 if (powersOfTwo > 0)
1974 {
1975 partToSquare = partToSquare.shiftRight(powersOfTwo);
1976 remainingBits = partToSquare.bitLength();
1977 if (remainingBits == 1) // Nothing left but +/- 1?
1978 if (signum<0 && (exponent&1)==1)
1979 return NEGATIVE_ONE.shiftLeft(powersOfTwo*exponent);
1980 else
1981 return ONE.shiftLeft(powersOfTwo*exponent);
1982 }
1983 else
1984 {
1985 remainingBits = partToSquare.bitLength();
1986 if (remainingBits == 1) // Nothing left but +/- 1?
1987 if (signum<0 && (exponent&1)==1)
1988 return NEGATIVE_ONE;
1989 else
1990 return ONE;
1991 }
1992
1993 // This is a quick way to approximate the size of the result,
1994 // similar to doing log2[n] * exponent. This will give an upper bound
1995 // of how big the result can be, and which algorithm to use.
1996 int scaleFactor = remainingBits * exponent;
1997
1998 // Use slightly different algorithms for small and large operands.
1999 // See if the result will safely fit into a long. (Largest 2^63-1)
2000 if (partToSquare.mag.length==1 && scaleFactor <= 62)
2001 {
2002 // Small number algorithm. Everything fits into a long.
2003 int newSign = (signum<0 && (exponent&1)==1 ? -1 : 1);
2004 long result = 1;
2005 long baseToPow2 = partToSquare.mag[0] & LONG_MASK;
2006
2007 int workingExponent = exponent;
2008
2009 // Perform exponentiation using repeated squaring trick
2010 while (workingExponent != 0) {
2011 if ((workingExponent & 1)==1)
2012 result = result * baseToPow2;
2013
2014 if ((workingExponent >>>= 1) != 0)
2015 baseToPow2 = baseToPow2 * baseToPow2;
2016 }
2017
2018 // Multiply back the powers of two (quickly, by shifting left)
2019 if (powersOfTwo > 0)
2020 {
2021 int bitsToShift = powersOfTwo*exponent;
2022 if (bitsToShift + scaleFactor <= 62) // Fits in long?
2023 return valueOf((result << bitsToShift) * newSign);
2024 else
2025 return valueOf(result*newSign).shiftLeft(bitsToShift);
2026 }
2027 else
2028 return valueOf(result*newSign);
2029 }
2030 else
2031 {
2032 // Large number algorithm. This is basically identical to
2033 // the algorithm above, but calls multiply() and square()
2034 // which may use more efficient algorithms for large numbers.
2035 BigInteger answer = ONE;
2036
2037 int workingExponent = exponent;
2038 // Perform exponentiation using repeated squaring trick
2039 while (workingExponent != 0) {
2040 if ((workingExponent & 1)==1)
2041 answer = answer.multiply(partToSquare);
2042
2043 if ((workingExponent >>>= 1) != 0)
2044 partToSquare = partToSquare.square();
2045 }
2046 // Multiply back the (exponentiated) powers of two (quickly,
2047 // by shifting left)
2048 if (powersOfTwo > 0)
2049 answer = answer.shiftLeft(powersOfTwo*exponent);
2050
2051 if (signum<0 && (exponent&1)==1)
2052 return answer.negate();
2053 else
2054 return answer;
2055 }
2056 }
2057
2058 /**
2059 * Returns a BigInteger whose value is the greatest common divisor of
2060 * {@code abs(this)} and {@code abs(val)}. Returns 0 if
2061 * {@code this==0 && val==0}.
2062 *
2063 * @param val value with which the GCD is to be computed.
2064 * @return {@code GCD(abs(this), abs(val))}
2065 */
2066 public BigInteger gcd(BigInteger val) {
2067 if (val.signum == 0)
2068 return this.abs();
2069 else if (this.signum == 0)
2070 return val.abs();
2071
2072 MutableBigInteger a = new MutableBigInteger(this);
2073 MutableBigInteger b = new MutableBigInteger(val);
2074
2075 MutableBigInteger result = a.hybridGCD(b);
2076
2077 return result.toBigInteger(1);
2078 }
2079
2080 /**
2081 * Package private method to return bit length for an integer.
2082 */
2083 static int bitLengthForInt(int n) {
2084 return 32 - Integer.numberOfLeadingZeros(n);
2085 }
2086
2087 /**
2088 * Left shift int array a up to len by n bits. Returns the array that
2089 * results from the shift since space may have to be reallocated.
2090 */
2091 private static int[] leftShift(int[] a, int len, int n) {
2092 int nInts = n >>> 5;
2093 int nBits = n&0x1F;
2094 int bitsInHighWord = bitLengthForInt(a[0]);
2095
2096 // If shift can be done without recopy, do so
2097 if (n <= (32-bitsInHighWord)) {
2098 primitiveLeftShift(a, len, nBits);
2099 return a;
2100 } else { // Array must be resized
2101 if (nBits <= (32-bitsInHighWord)) {
2102 int result[] = new int[nInts+len];
2103 System.arraycopy(a, 0, result, 0, len);
2104 primitiveLeftShift(result, result.length, nBits);
2105 return result;
2106 } else {
2107 int result[] = new int[nInts+len+1];
2108 System.arraycopy(a, 0, result, 0, len);
2109 primitiveRightShift(result, result.length, 32 - nBits);
2110 return result;
2111 }
2112 }
2113 }
2114
2115 // shifts a up to len right n bits assumes no leading zeros, 0<n<32
2116 static void primitiveRightShift(int[] a, int len, int n) {
2117 int n2 = 32 - n;
2118 for (int i=len-1, c=a[i]; i>0; i--) {
2119 int b = c;
2120 c = a[i-1];
2121 a[i] = (c << n2) | (b >>> n);
2122 }
2123 a[0] >>>= n;
2124 }
2125
2126 // shifts a up to len left n bits assumes no leading zeros, 0<=n<32
2127 static void primitiveLeftShift(int[] a, int len, int n) {
2128 if (len == 0 || n == 0)
2129 return;
2130
2131 int n2 = 32 - n;
2132 for (int i=0, c=a[i], m=i+len-1; i<m; i++) {
2133 int b = c;
2134 c = a[i+1];
2135 a[i] = (b << n) | (c >>> n2);
2136 }
2137 a[len-1] <<= n;
2138 }
2139
2140 /**
2141 * Calculate bitlength of contents of the first len elements an int array,
2142 * assuming there are no leading zero ints.
2143 */
2144 private static int bitLength(int[] val, int len) {
2145 if (len == 0)
2146 return 0;
2147 return ((len - 1) << 5) + bitLengthForInt(val[0]);
2148 }
2149
2150 /**
2151 * Returns a BigInteger whose value is the absolute value of this
2152 * BigInteger.
2153 *
2154 * @return {@code abs(this)}
2155 */
2156 public BigInteger abs() {
2157 return (signum >= 0 ? this : this.negate());
2158 }
2159
2160 /**
2161 * Returns a BigInteger whose value is {@code (-this)}.
2162 *
2163 * @return {@code -this}
2164 */
2165 public BigInteger negate() {
2166 return new BigInteger(this.mag, -this.signum);
2167 }
2168
2169 /**
2170 * Returns the signum function of this BigInteger.
2171 *
2172 * @return -1, 0 or 1 as the value of this BigInteger is negative, zero or
2173 * positive.
2174 */
2175 public int signum() {
2176 return this.signum;
2177 }
2178
2179 // Modular Arithmetic Operations
2180
2181 /**
2182 * Returns a BigInteger whose value is {@code (this mod m}). This method
2183 * differs from {@code remainder} in that it always returns a
2184 * <i>non-negative</i> BigInteger.
2185 *
2186 * @param m the modulus.
2187 * @return {@code this mod m}
2188 * @throws ArithmeticException {@code m} ≤ 0
2189 * @see #remainder
2190 */
2191 public BigInteger mod(BigInteger m) {
2192 if (m.signum <= 0)
2193 throw new ArithmeticException("BigInteger: modulus not positive");
2194
2195 BigInteger result = this.remainder(m);
2196 return (result.signum >= 0 ? result : result.add(m));
2197 }
2198
2199 /**
2200 * Returns a BigInteger whose value is
2201 * <tt>(this<sup>exponent</sup> mod m)</tt>. (Unlike {@code pow}, this
2202 * method permits negative exponents.)
2203 *
2204 * @param exponent the exponent.
2205 * @param m the modulus.
2206 * @return <tt>this<sup>exponent</sup> mod m</tt>
2207 * @throws ArithmeticException {@code m} ≤ 0 or the exponent is
2208 * negative and this BigInteger is not <i>relatively
2209 * prime</i> to {@code m}.
2210 * @see #modInverse
2211 */
2212 public BigInteger modPow(BigInteger exponent, BigInteger m) {
2213 if (m.signum <= 0)
2214 throw new ArithmeticException("BigInteger: modulus not positive");
2215
2216 // Trivial cases
2217 if (exponent.signum == 0)
2218 return (m.equals(ONE) ? ZERO : ONE);
2219
2220 if (this.equals(ONE))
2221 return (m.equals(ONE) ? ZERO : ONE);
2222
2223 if (this.equals(ZERO) && exponent.signum >= 0)
2224 return ZERO;
2225
2226 if (this.equals(negConst[1]) && (!exponent.testBit(0)))
2227 return (m.equals(ONE) ? ZERO : ONE);
2228
2229 boolean invertResult;
2230 if ((invertResult = (exponent.signum < 0)))
2231 exponent = exponent.negate();
2232
2233 BigInteger base = (this.signum < 0 || this.compareTo(m) >= 0
2234 ? this.mod(m) : this);
2235 BigInteger result;
2236 if (m.testBit(0)) { // odd modulus
2237 result = base.oddModPow(exponent, m);
2238 } else {
2239 /*
2240 * Even modulus. Tear it into an "odd part" (m1) and power of two
2241 * (m2), exponentiate mod m1, manually exponentiate mod m2, and
2242 * use Chinese Remainder Theorem to combine results.
2243 */
2244
2245 // Tear m apart into odd part (m1) and power of 2 (m2)
2246 int p = m.getLowestSetBit(); // Max pow of 2 that divides m
2247
2248 BigInteger m1 = m.shiftRight(p); // m/2**p
2249 BigInteger m2 = ONE.shiftLeft(p); // 2**p
2250
2251 // Calculate new base from m1
2252 BigInteger base2 = (this.signum < 0 || this.compareTo(m1) >= 0
2253 ? this.mod(m1) : this);
2254
2255 // Caculate (base ** exponent) mod m1.
2256 BigInteger a1 = (m1.equals(ONE) ? ZERO :
2257 base2.oddModPow(exponent, m1));
2258
2259 // Calculate (this ** exponent) mod m2
2260 BigInteger a2 = base.modPow2(exponent, p);
2261
2262 // Combine results using Chinese Remainder Theorem
2263 BigInteger y1 = m2.modInverse(m1);
2264 BigInteger y2 = m1.modInverse(m2);
2265
2266 result = a1.multiply(m2).multiply(y1).add
2267 (a2.multiply(m1).multiply(y2)).mod(m);
2268 }
2269
2270 return (invertResult ? result.modInverse(m) : result);
2271 }
2272
2273 static int[] bnExpModThreshTable = {7, 25, 81, 241, 673, 1793,
2274 Integer.MAX_VALUE}; // Sentinel
2275
2276 /**
2277 * Returns a BigInteger whose value is x to the power of y mod z.
2278 * Assumes: z is odd && x < z.
2279 */
2280 private BigInteger oddModPow(BigInteger y, BigInteger z) {
2281 /*
2282 * The algorithm is adapted from Colin Plumb's C library.
2283 *
2284 * The window algorithm:
2285 * The idea is to keep a running product of b1 = n^(high-order bits of exp)
2286 * and then keep appending exponent bits to it. The following patterns
2287 * apply to a 3-bit window (k = 3):
2288 * To append 0: square
2289 * To append 1: square, multiply by n^1
2290 * To append 10: square, multiply by n^1, square
2291 * To append 11: square, square, multiply by n^3
2292 * To append 100: square, multiply by n^1, square, square
2293 * To append 101: square, square, square, multiply by n^5
2294 * To append 110: square, square, multiply by n^3, square
2295 * To append 111: square, square, square, multiply by n^7
2296 *
2297 * Since each pattern involves only one multiply, the longer the pattern
2298 * the better, except that a 0 (no multiplies) can be appended directly.
2299 * We precompute a table of odd powers of n, up to 2^k, and can then
2300 * multiply k bits of exponent at a time. Actually, assuming random
2301 * exponents, there is on average one zero bit between needs to
2302 * multiply (1/2 of the time there's none, 1/4 of the time there's 1,
2303 * 1/8 of the time, there's 2, 1/32 of the time, there's 3, etc.), so
2304 * you have to do one multiply per k+1 bits of exponent.
2305 *
2306 * The loop walks down the exponent, squaring the result buffer as
2307 * it goes. There is a wbits+1 bit lookahead buffer, buf, that is
2308 * filled with the upcoming exponent bits. (What is read after the
2309 * end of the exponent is unimportant, but it is filled with zero here.)
2310 * When the most-significant bit of this buffer becomes set, i.e.
2311 * (buf & tblmask) != 0, we have to decide what pattern to multiply
2312 * by, and when to do it. We decide, remember to do it in future
2313 * after a suitable number of squarings have passed (e.g. a pattern
2314 * of "100" in the buffer requires that we multiply by n^1 immediately;
2315 * a pattern of "110" calls for multiplying by n^3 after one more
2316 * squaring), clear the buffer, and continue.
2317 *
2318 * When we start, there is one more optimization: the result buffer
2319 * is implcitly one, so squaring it or multiplying by it can be
2320 * optimized away. Further, if we start with a pattern like "100"
2321 * in the lookahead window, rather than placing n into the buffer
2322 * and then starting to square it, we have already computed n^2
2323 * to compute the odd-powers table, so we can place that into
2324 * the buffer and save a squaring.
2325 *
2326 * This means that if you have a k-bit window, to compute n^z,
2327 * where z is the high k bits of the exponent, 1/2 of the time
2328 * it requires no squarings. 1/4 of the time, it requires 1
2329 * squaring, ... 1/2^(k-1) of the time, it reqires k-2 squarings.
2330 * And the remaining 1/2^(k-1) of the time, the top k bits are a
2331 * 1 followed by k-1 0 bits, so it again only requires k-2
2332 * squarings, not k-1. The average of these is 1. Add that
2333 * to the one squaring we have to do to compute the table,
2334 * and you'll see that a k-bit window saves k-2 squarings
2335 * as well as reducing the multiplies. (It actually doesn't
2336 * hurt in the case k = 1, either.)
2337 */
2338 // Special case for exponent of one
2339 if (y.equals(ONE))
2340 return this;
2341
2342 // Special case for base of zero
2343 if (signum==0)
2344 return ZERO;
2345
2346 int[] base = mag.clone();
2347 int[] exp = y.mag;
2348 int[] mod = z.mag;
2349 int modLen = mod.length;
2350
2351 // Select an appropriate window size
2352 int wbits = 0;
2353 int ebits = bitLength(exp, exp.length);
2354 // if exponent is 65537 (0x10001), use minimum window size
2355 if ((ebits != 17) || (exp[0] != 65537)) {
2356 while (ebits > bnExpModThreshTable[wbits]) {
2357 wbits++;
2358 }
2359 }
2360
2361 // Calculate appropriate table size
2362 int tblmask = 1 << wbits;
2363
2364 // Allocate table for precomputed odd powers of base in Montgomery form
2365 int[][] table = new int[tblmask][];
2366 for (int i=0; i<tblmask; i++)
2367 table[i] = new int[modLen];
2368
2369 // Compute the modular inverse
2370 int inv = -MutableBigInteger.inverseMod32(mod[modLen-1]);
2371
2372 // Convert base to Montgomery form
2373 int[] a = leftShift(base, base.length, modLen << 5);
2374
2375 MutableBigInteger q = new MutableBigInteger(),
2376 a2 = new MutableBigInteger(a),
2377 b2 = new MutableBigInteger(mod);
2378
2379 MutableBigInteger r= a2.divide(b2, q);
2380 table[0] = r.toIntArray();
2381
2382 // Pad table[0] with leading zeros so its length is at least modLen
2383 if (table[0].length < modLen) {
2384 int offset = modLen - table[0].length;
2385 int[] t2 = new int[modLen];
2386 for (int i=0; i<table[0].length; i++)
2387 t2[i+offset] = table[0][i];
2388 table[0] = t2;
2389 }
2390
2391 // Set b to the square of the base
2392 int[] b = squareToLen(table[0], modLen, null);
2393 b = montReduce(b, mod, modLen, inv);
2394
2395 // Set t to high half of b
2396 int[] t = Arrays.copyOf(b, modLen);
2397
2398 // Fill in the table with odd powers of the base
2399 for (int i=1; i<tblmask; i++) {
2400 int[] prod = multiplyToLen(t, modLen, table[i-1], modLen, null);
2401 table[i] = montReduce(prod, mod, modLen, inv);
2402 }
2403
2404 // Pre load the window that slides over the exponent
2405 int bitpos = 1 << ((ebits-1) & (32-1));
2406
2407 int buf = 0;
2408 int elen = exp.length;
2409 int eIndex = 0;
2410 for (int i = 0; i <= wbits; i++) {
2411 buf = (buf << 1) | (((exp[eIndex] & bitpos) != 0)?1:0);
2412 bitpos >>>= 1;
2413 if (bitpos == 0) {
2414 eIndex++;
2415 bitpos = 1 << (32-1);
2416 elen--;
2417 }
2418 }
2419
2420 int multpos = ebits;
2421
2422 // The first iteration, which is hoisted out of the main loop
2423 ebits--;
2424 boolean isone = true;
2425
2426 multpos = ebits - wbits;
2427 while ((buf & 1) == 0) {
2428 buf >>>= 1;
2429 multpos++;
2430 }
2431
2432 int[] mult = table[buf >>> 1];
2433
2434 buf = 0;
2435 if (multpos == ebits)
2436 isone = false;
2437
2438 // The main loop
2439 while(true) {
2440 ebits--;
2441 // Advance the window
2442 buf <<= 1;
2443
2444 if (elen != 0) {
2445 buf |= ((exp[eIndex] & bitpos) != 0) ? 1 : 0;
2446 bitpos >>>= 1;
2447 if (bitpos == 0) {
2448 eIndex++;
2449 bitpos = 1 << (32-1);
2450 elen--;
2451 }
2452 }
2453
2454 // Examine the window for pending multiplies
2455 if ((buf & tblmask) != 0) {
2456 multpos = ebits - wbits;
2457 while ((buf & 1) == 0) {
2458 buf >>>= 1;
2459 multpos++;
2460 }
2461 mult = table[buf >>> 1];
2462 buf = 0;
2463 }
2464
2465 // Perform multiply
2466 if (ebits == multpos) {
2467 if (isone) {
2468 b = mult.clone();
2469 isone = false;
2470 } else {
2471 t = b;
2472 a = multiplyToLen(t, modLen, mult, modLen, a);
2473 a = montReduce(a, mod, modLen, inv);
2474 t = a; a = b; b = t;
2475 }
2476 }
2477
2478 // Check if done
2479 if (ebits == 0)
2480 break;
2481
2482 // Square the input
2483 if (!isone) {
2484 t = b;
2485 a = squareToLen(t, modLen, a);
2486 a = montReduce(a, mod, modLen, inv);
2487 t = a; a = b; b = t;
2488 }
2489 }
2490
2491 // Convert result out of Montgomery form and return
2492 int[] t2 = new int[2*modLen];
2493 System.arraycopy(b, 0, t2, modLen, modLen);
2494
2495 b = montReduce(t2, mod, modLen, inv);
2496
2497 t2 = Arrays.copyOf(b, modLen);
2498
2499 return new BigInteger(1, t2);
2500 }
2501
2502 /**
2503 * Montgomery reduce n, modulo mod. This reduces modulo mod and divides
2504 * by 2^(32*mlen). Adapted from Colin Plumb's C library.
2505 */
2506 private static int[] montReduce(int[] n, int[] mod, int mlen, int inv) {
2507 int c=0;
2508 int len = mlen;
2509 int offset=0;
2510
2511 do {
2512 int nEnd = n[n.length-1-offset];
2513 int carry = mulAdd(n, mod, offset, mlen, inv * nEnd);
2514 c += addOne(n, offset, mlen, carry);
2515 offset++;
2516 } while(--len > 0);
2517
2518 while(c>0)
2519 c += subN(n, mod, mlen);
2520
2521 while (intArrayCmpToLen(n, mod, mlen) >= 0)
2522 subN(n, mod, mlen);
2523
2524 return n;
2525 }
2526
2527
2528 /*
2529 * Returns -1, 0 or +1 as big-endian unsigned int array arg1 is less than,
2530 * equal to, or greater than arg2 up to length len.
2531 */
2532 private static int intArrayCmpToLen(int[] arg1, int[] arg2, int len) {
2533 for (int i=0; i<len; i++) {
2534 long b1 = arg1[i] & LONG_MASK;
2535 long b2 = arg2[i] & LONG_MASK;
2536 if (b1 < b2)
2537 return -1;
2538 if (b1 > b2)
2539 return 1;
2540 }
2541 return 0;
2542 }
2543
2544 /**
2545 * Subtracts two numbers of same length, returning borrow.
2546 */
2547 private static int subN(int[] a, int[] b, int len) {
2548 long sum = 0;
2549
2550 while(--len >= 0) {
2551 sum = (a[len] & LONG_MASK) -
2552 (b[len] & LONG_MASK) + (sum >> 32);
2553 a[len] = (int)sum;
2554 }
2555
2556 return (int)(sum >> 32);
2557 }
2558
2559 /**
2560 * Multiply an array by one word k and add to result, return the carry
2561 */
2562 static int mulAdd(int[] out, int[] in, int offset, int len, int k) {
2563 long kLong = k & LONG_MASK;
2564 long carry = 0;
2565
2566 offset = out.length-offset - 1;
2567 for (int j=len-1; j >= 0; j--) {
2568 long product = (in[j] & LONG_MASK) * kLong +
2569 (out[offset] & LONG_MASK) + carry;
2570 out[offset--] = (int)product;
2571 carry = product >>> 32;
2572 }
2573 return (int)carry;
2574 }
2575
2576 /**
2577 * Add one word to the number a mlen words into a. Return the resulting
2578 * carry.
2579 */
2580 static int addOne(int[] a, int offset, int mlen, int carry) {
2581 offset = a.length-1-mlen-offset;
2582 long t = (a[offset] & LONG_MASK) + (carry & LONG_MASK);
2583
2584 a[offset] = (int)t;
2585 if ((t >>> 32) == 0)
2586 return 0;
2587 while (--mlen >= 0) {
2588 if (--offset < 0) { // Carry out of number
2589 return 1;
2590 } else {
2591 a[offset]++;
2592 if (a[offset] != 0)
2593 return 0;
2594 }
2595 }
2596 return 1;
2597 }
2598
2599 /**
2600 * Returns a BigInteger whose value is (this ** exponent) mod (2**p)
2601 */
2602 private BigInteger modPow2(BigInteger exponent, int p) {
2603 /*
2604 * Perform exponentiation using repeated squaring trick, chopping off
2605 * high order bits as indicated by modulus.
2606 */
2607 BigInteger result = ONE;
2608 BigInteger baseToPow2 = this.mod2(p);
2609 int expOffset = 0;
2610
2611 int limit = exponent.bitLength();
2612
2613 if (this.testBit(0))
2614 limit = (p-1) < limit ? (p-1) : limit;
2615
2616 while (expOffset < limit) {
2617 if (exponent.testBit(expOffset))
2618 result = result.multiply(baseToPow2).mod2(p);
2619 expOffset++;
2620 if (expOffset < limit)
2621 baseToPow2 = baseToPow2.square().mod2(p);
2622 }
2623
2624 return result;
2625 }
2626
2627 /**
2628 * Returns a BigInteger whose value is this mod(2**p).
2629 * Assumes that this {@code BigInteger >= 0} and {@code p > 0}.
2630 */
2631 private BigInteger mod2(int p) {
2632 if (bitLength() <= p)
2633 return this;
2634
2635 // Copy remaining ints of mag
2636 int numInts = (p + 31) >>> 5;
2637 int[] mag = new int[numInts];
2638 System.arraycopy(this.mag, (this.mag.length - numInts), mag, 0, numInts);
2639
2640 // Mask out any excess bits
2641 int excessBits = (numInts << 5) - p;
2642 mag[0] &= (1L << (32-excessBits)) - 1;
2643
2644 return (mag[0]==0 ? new BigInteger(1, mag) : new BigInteger(mag, 1));
2645 }
2646
2647 /**
2648 * Returns a BigInteger whose value is {@code (this}<sup>-1</sup> {@code mod m)}.
2649 *
2650 * @param m the modulus.
2651 * @return {@code this}<sup>-1</sup> {@code mod m}.
2652 * @throws ArithmeticException {@code m} ≤ 0, or this BigInteger
2653 * has no multiplicative inverse mod m (that is, this BigInteger
2654 * is not <i>relatively prime</i> to m).
2655 */
2656 public BigInteger modInverse(BigInteger m) {
2657 if (m.signum != 1)
2658 throw new ArithmeticException("BigInteger: modulus not positive");
2659
2660 if (m.equals(ONE))
2661 return ZERO;
2662
2663 // Calculate (this mod m)
2664 BigInteger modVal = this;
2665 if (signum < 0 || (this.compareMagnitude(m) >= 0))
2666 modVal = this.mod(m);
2667
2668 if (modVal.equals(ONE))
2669 return ONE;
2670
2671 MutableBigInteger a = new MutableBigInteger(modVal);
2672 MutableBigInteger b = new MutableBigInteger(m);
2673
2674 MutableBigInteger result = a.mutableModInverse(b);
2675 return result.toBigInteger(1);
2676 }
2677
2678 // Shift Operations
2679
2680 /**
2681 * Returns a BigInteger whose value is {@code (this << n)}.
2682 * The shift distance, {@code n}, may be negative, in which case
2683 * this method performs a right shift.
2684 * (Computes <tt>floor(this * 2<sup>n</sup>)</tt>.)
2685 *
2686 * @param n shift distance, in bits.
2687 * @return {@code this << n}
2688 * @throws ArithmeticException if the shift distance is {@code
2689 * Integer.MIN_VALUE}.
2690 * @see #shiftRight
2691 */
2692 public BigInteger shiftLeft(int n) {
2693 if (signum == 0)
2694 return ZERO;
2695 if (n==0)
2696 return this;
2697 if (n<0) {
2698 if (n == Integer.MIN_VALUE) {
2699 throw new ArithmeticException("Shift distance of Integer.MIN_VALUE not supported.");
2700 } else {
2701 return shiftRight(-n);
2702 }
2703 }
2704 int[] newMag = shiftLeft(mag, n);
2705
2706 return new BigInteger(newMag, signum);
2707 }
2708
2709 private static int[] shiftLeft(int[] mag, int n) {
2710 int nInts = n >>> 5;
2711 int nBits = n & 0x1f;
2712 int magLen = mag.length;
2713 int newMag[] = null;
2714
2715 if (nBits == 0) {
2716 newMag = new int[magLen + nInts];
2717 System.arraycopy(mag, 0, newMag, 0, magLen);
2718 } else {
2719 int i = 0;
2720 int nBits2 = 32 - nBits;
2721 int highBits = mag[0] >>> nBits2;
2722 if (highBits != 0) {
2723 newMag = new int[magLen + nInts + 1];
2724 newMag[i++] = highBits;
2725 } else {
2726 newMag = new int[magLen + nInts];
2727 }
2728 int j=0;
2729 while (j < magLen-1)
2730 newMag[i++] = mag[j++] << nBits | mag[j] >>> nBits2;
2731 newMag[i] = mag[j] << nBits;
2732 }
2733 return newMag;
2734 }
2735
2736 /**
2737 * Returns a BigInteger whose value is {@code (this >> n)}. Sign
2738 * extension is performed. The shift distance, {@code n}, may be
2739 * negative, in which case this method performs a left shift.
2740 * (Computes <tt>floor(this / 2<sup>n</sup>)</tt>.)
2741 *
2742 * @param n shift distance, in bits.
2743 * @return {@code this >> n}
2744 * @throws ArithmeticException if the shift distance is {@code
2745 * Integer.MIN_VALUE}.
2746 * @see #shiftLeft
2747 */
2748 public BigInteger shiftRight(int n) {
2749 if (n==0)
2750 return this;
2751 if (n<0) {
2752 if (n == Integer.MIN_VALUE) {
2753 throw new ArithmeticException("Shift distance of Integer.MIN_VALUE not supported.");
2754 } else {
2755 return shiftLeft(-n);
2756 }
2757 }
2758
2759 int nInts = n >>> 5;
2760 int nBits = n & 0x1f;
2761 int magLen = mag.length;
2762 int newMag[] = null;
2763
2764 // Special case: entire contents shifted off the end
2765 if (nInts >= magLen)
2766 return (signum >= 0 ? ZERO : negConst[1]);
2767
2768 if (nBits == 0) {
2769 int newMagLen = magLen - nInts;
2770 newMag = Arrays.copyOf(mag, newMagLen);
2771 } else {
2772 int i = 0;
2773 int highBits = mag[0] >>> nBits;
2774 if (highBits != 0) {
2775 newMag = new int[magLen - nInts];
2776 newMag[i++] = highBits;
2777 } else {
2778 newMag = new int[magLen - nInts -1];
2779 }
2780
2781 int nBits2 = 32 - nBits;
2782 int j=0;
2783 while (j < magLen - nInts - 1)
2784 newMag[i++] = (mag[j++] << nBits2) | (mag[j] >>> nBits);
2785 }
2786
2787 if (signum < 0) {
2788 // Find out whether any one-bits were shifted off the end.
2789 boolean onesLost = false;
2790 for (int i=magLen-1, j=magLen-nInts; i>=j && !onesLost; i--)
2791 onesLost = (mag[i] != 0);
2792 if (!onesLost && nBits != 0)
2793 onesLost = (mag[magLen - nInts - 1] << (32 - nBits) != 0);
2794
2795 if (onesLost)
2796 newMag = javaIncrement(newMag);
2797 }
2798
2799 return new BigInteger(newMag, signum);
2800 }
2801
2802 int[] javaIncrement(int[] val) {
2803 int lastSum = 0;
2804 for (int i=val.length-1; i >= 0 && lastSum == 0; i--)
2805 lastSum = (val[i] += 1);
2806 if (lastSum == 0) {
2807 val = new int[val.length+1];
2808 val[0] = 1;
2809 }
2810 return val;
2811 }
2812
2813 // Bitwise Operations
2814
2815 /**
2816 * Returns a BigInteger whose value is {@code (this & val)}. (This
2817 * method returns a negative BigInteger if and only if this and val are
2818 * both negative.)
2819 *
2820 * @param val value to be AND'ed with this BigInteger.
2821 * @return {@code this & val}
2822 */
2823 public BigInteger and(BigInteger val) {
2824 int[] result = new int[Math.max(intLength(), val.intLength())];
2825 for (int i=0; i<result.length; i++)
2826 result[i] = (getInt(result.length-i-1)
2827 & val.getInt(result.length-i-1));
2828
2829 return valueOf(result);
2830 }
2831
2832 /**
2833 * Returns a BigInteger whose value is {@code (this | val)}. (This method
2834 * returns a negative BigInteger if and only if either this or val is
2835 * negative.)
2836 *
2837 * @param val value to be OR'ed with this BigInteger.
2838 * @return {@code this | val}
2839 */
2840 public BigInteger or(BigInteger val) {
2841 int[] result = new int[Math.max(intLength(), val.intLength())];
2842 for (int i=0; i<result.length; i++)
2843 result[i] = (getInt(result.length-i-1)
2844 | val.getInt(result.length-i-1));
2845
2846 return valueOf(result);
2847 }
2848
2849 /**
2850 * Returns a BigInteger whose value is {@code (this ^ val)}. (This method
2851 * returns a negative BigInteger if and only if exactly one of this and
2852 * val are negative.)
2853 *
2854 * @param val value to be XOR'ed with this BigInteger.
2855 * @return {@code this ^ val}
2856 */
2857 public BigInteger xor(BigInteger val) {
2858 int[] result = new int[Math.max(intLength(), val.intLength())];
2859 for (int i=0; i<result.length; i++)
2860 result[i] = (getInt(result.length-i-1)
2861 ^ val.getInt(result.length-i-1));
2862
2863 return valueOf(result);
2864 }
2865
2866 /**
2867 * Returns a BigInteger whose value is {@code (~this)}. (This method
2868 * returns a negative value if and only if this BigInteger is
2869 * non-negative.)
2870 *
2871 * @return {@code ~this}
2872 */
2873 public BigInteger not() {
2874 int[] result = new int[intLength()];
2875 for (int i=0; i<result.length; i++)
2876 result[i] = ~getInt(result.length-i-1);
2877
2878 return valueOf(result);
2879 }
2880
2881 /**
2882 * Returns a BigInteger whose value is {@code (this & ~val)}. This
2883 * method, which is equivalent to {@code and(val.not())}, is provided as
2884 * a convenience for masking operations. (This method returns a negative
2885 * BigInteger if and only if {@code this} is negative and {@code val} is
2886 * positive.)
2887 *
2888 * @param val value to be complemented and AND'ed with this BigInteger.
2889 * @return {@code this & ~val}
2890 */
2891 public BigInteger andNot(BigInteger val) {
2892 int[] result = new int[Math.max(intLength(), val.intLength())];
2893 for (int i=0; i<result.length; i++)
2894 result[i] = (getInt(result.length-i-1)
2895 & ~val.getInt(result.length-i-1));
2896
2897 return valueOf(result);
2898 }
2899
2900
2901 // Single Bit Operations
2902
2903 /**
2904 * Returns {@code true} if and only if the designated bit is set.
2905 * (Computes {@code ((this & (1<<n)) != 0)}.)
2906 *
2907 * @param n index of bit to test.
2908 * @return {@code true} if and only if the designated bit is set.
2909 * @throws ArithmeticException {@code n} is negative.
2910 */
2911 public boolean testBit(int n) {
2912 if (n<0)
2913 throw new ArithmeticException("Negative bit address");
2914
2915 return (getInt(n >>> 5) & (1 << (n & 31))) != 0;
2916 }
2917
2918 /**
2919 * Returns a BigInteger whose value is equivalent to this BigInteger
2920 * with the designated bit set. (Computes {@code (this | (1<<n))}.)
2921 *
2922 * @param n index of bit to set.
2923 * @return {@code this | (1<<n)}
2924 * @throws ArithmeticException {@code n} is negative.
2925 */
2926 public BigInteger setBit(int n) {
2927 if (n<0)
2928 throw new ArithmeticException("Negative bit address");
2929
2930 int intNum = n >>> 5;
2931 int[] result = new int[Math.max(intLength(), intNum+2)];
2932
2933 for (int i=0; i<result.length; i++)
2934 result[result.length-i-1] = getInt(i);
2935
2936 result[result.length-intNum-1] |= (1 << (n & 31));
2937
2938 return valueOf(result);
2939 }
2940
2941 /**
2942 * Returns a BigInteger whose value is equivalent to this BigInteger
2943 * with the designated bit cleared.
2944 * (Computes {@code (this & ~(1<<n))}.)
2945 *
2946 * @param n index of bit to clear.
2947 * @return {@code this & ~(1<<n)}
2948 * @throws ArithmeticException {@code n} is negative.
2949 */
2950 public BigInteger clearBit(int n) {
2951 if (n<0)
2952 throw new ArithmeticException("Negative bit address");
2953
2954 int intNum = n >>> 5;
2955 int[] result = new int[Math.max(intLength(), ((n + 1) >>> 5) + 1)];
2956
2957 for (int i=0; i<result.length; i++)
2958 result[result.length-i-1] = getInt(i);
2959
2960 result[result.length-intNum-1] &= ~(1 << (n & 31));
2961
2962 return valueOf(result);
2963 }
2964
2965 /**
2966 * Returns a BigInteger whose value is equivalent to this BigInteger
2967 * with the designated bit flipped.
2968 * (Computes {@code (this ^ (1<<n))}.)
2969 *
2970 * @param n index of bit to flip.
2971 * @return {@code this ^ (1<<n)}
2972 * @throws ArithmeticException {@code n} is negative.
2973 */
2974 public BigInteger flipBit(int n) {
2975 if (n<0)
2976 throw new ArithmeticException("Negative bit address");
2977
2978 int intNum = n >>> 5;
2979 int[] result = new int[Math.max(intLength(), intNum+2)];
2980
2981 for (int i=0; i<result.length; i++)
2982 result[result.length-i-1] = getInt(i);
2983
2984 result[result.length-intNum-1] ^= (1 << (n & 31));
2985
2986 return valueOf(result);
2987 }
2988
2989 /**
2990 * Returns the index of the rightmost (lowest-order) one bit in this
2991 * BigInteger (the number of zero bits to the right of the rightmost
2992 * one bit). Returns -1 if this BigInteger contains no one bits.
2993 * (Computes {@code (this==0? -1 : log2(this & -this))}.)
2994 *
2995 * @return index of the rightmost one bit in this BigInteger.
2996 */
2997 public int getLowestSetBit() {
2998 @SuppressWarnings("deprecation") int lsb = lowestSetBit - 2;
2999 if (lsb == -2) { // lowestSetBit not initialized yet
3000 lsb = 0;
3001 if (signum == 0) {
3002 lsb -= 1;
3003 } else {
3004 // Search for lowest order nonzero int
3005 int i,b;
3006 for (i=0; (b = getInt(i))==0; i++)
3007 ;
3008 lsb += (i << 5) + Integer.numberOfTrailingZeros(b);
3009 }
3010 lowestSetBit = lsb + 2;
3011 }
3012 return lsb;
3013 }
3014
3015
3016 // Miscellaneous Bit Operations
3017
3018 /**
3019 * Returns the number of bits in the minimal two's-complement
3020 * representation of this BigInteger, <i>excluding</i> a sign bit.
3021 * For positive BigIntegers, this is equivalent to the number of bits in
3022 * the ordinary binary representation. (Computes
3023 * {@code (ceil(log2(this < 0 ? -this : this+1)))}.)
3024 *
3025 * @return number of bits in the minimal two's-complement
3026 * representation of this BigInteger, <i>excluding</i> a sign bit.
3027 */
3028 public int bitLength() {
3029 @SuppressWarnings("deprecation") int n = bitLength - 1;
3030 if (n == -1) { // bitLength not initialized yet
3031 int[] m = mag;
3032 int len = m.length;
3033 if (len == 0) {
3034 n = 0; // offset by one to initialize
3035 } else {
3036 // Calculate the bit length of the magnitude
3037 int magBitLength = ((len - 1) << 5) + bitLengthForInt(mag[0]);
3038 if (signum < 0) {
3039 // Check if magnitude is a power of two
3040 boolean pow2 = (Integer.bitCount(mag[0]) == 1);
3041 for (int i=1; i< len && pow2; i++)
3042 pow2 = (mag[i] == 0);
3043
3044 n = (pow2 ? magBitLength -1 : magBitLength);
3045 } else {
3046 n = magBitLength;
3047 }
3048 }
3049 bitLength = n + 1;
3050 }
3051 return n;
3052 }
3053
3054 /**
3055 * Returns the number of bits in the two's complement representation
3056 * of this BigInteger that differ from its sign bit. This method is
3057 * useful when implementing bit-vector style sets atop BigIntegers.
3058 *
3059 * @return number of bits in the two's complement representation
3060 * of this BigInteger that differ from its sign bit.
3061 */
3062 public int bitCount() {
3063 @SuppressWarnings("deprecation") int bc = bitCount - 1;
3064 if (bc == -1) { // bitCount not initialized yet
3065 bc = 0; // offset by one to initialize
3066 // Count the bits in the magnitude
3067 for (int i=0; i<mag.length; i++)
3068 bc += Integer.bitCount(mag[i]);
3069 if (signum < 0) {
3070 // Count the trailing zeros in the magnitude
3071 int magTrailingZeroCount = 0, j;
3072 for (j=mag.length-1; mag[j]==0; j--)
3073 magTrailingZeroCount += 32;
3074 magTrailingZeroCount += Integer.numberOfTrailingZeros(mag[j]);
3075 bc += magTrailingZeroCount - 1;
3076 }
3077 bitCount = bc + 1;
3078 }
3079 return bc;
3080 }
3081
3082 // Primality Testing
3083
3084 /**
3085 * Returns {@code true} if this BigInteger is probably prime,
3086 * {@code false} if it's definitely composite. If
3087 * {@code certainty} is ≤ 0, {@code true} is
3088 * returned.
3089 *
3090 * @param certainty a measure of the uncertainty that the caller is
3091 * willing to tolerate: if the call returns {@code true}
3092 * the probability that this BigInteger is prime exceeds
3093 * (1 - 1/2<sup>{@code certainty}</sup>). The execution time of
3094 * this method is proportional to the value of this parameter.
3095 * @return {@code true} if this BigInteger is probably prime,
3096 * {@code false} if it's definitely composite.
3097 */
3098 public boolean isProbablePrime(int certainty) {
3099 if (certainty <= 0)
3100 return true;
3101 BigInteger w = this.abs();
3102 if (w.equals(TWO))
3103 return true;
3104 if (!w.testBit(0) || w.equals(ONE))
3105 return false;
3106
3107 return w.primeToCertainty(certainty, null);
3108 }
3109
3110 // Comparison Operations
3111
3112 /**
3113 * Compares this BigInteger with the specified BigInteger. This
3114 * method is provided in preference to individual methods for each
3115 * of the six boolean comparison operators ({@literal <}, ==,
3116 * {@literal >}, {@literal >=}, !=, {@literal <=}). The suggested
3117 * idiom for performing these comparisons is: {@code
3118 * (x.compareTo(y)} <<i>op</i>> {@code 0)}, where
3119 * <<i>op</i>> is one of the six comparison operators.
3120 *
3121 * @param val BigInteger to which this BigInteger is to be compared.
3122 * @return -1, 0 or 1 as this BigInteger is numerically less than, equal
3123 * to, or greater than {@code val}.
3124 */
3125 public int compareTo(BigInteger val) {
3126 if (signum == val.signum) {
3127 switch (signum) {
3128 case 1:
3129 return compareMagnitude(val);
3130 case -1:
3131 return val.compareMagnitude(this);
3132 default:
3133 return 0;
3134 }
3135 }
3136 return signum > val.signum ? 1 : -1;
3137 }
3138
3139 /**
3140 * Compares the magnitude array of this BigInteger with the specified
3141 * BigInteger's. This is the version of compareTo ignoring sign.
3142 *
3143 * @param val BigInteger whose magnitude array to be compared.
3144 * @return -1, 0 or 1 as this magnitude array is less than, equal to or
3145 * greater than the magnitude aray for the specified BigInteger's.
3146 */
3147 final int compareMagnitude(BigInteger val) {
3148 int[] m1 = mag;
3149 int len1 = m1.length;
3150 int[] m2 = val.mag;
3151 int len2 = m2.length;
3152 if (len1 < len2)
3153 return -1;
3154 if (len1 > len2)
3155 return 1;
3156 for (int i = 0; i < len1; i++) {
3157 int a = m1[i];
3158 int b = m2[i];
3159 if (a != b)
3160 return ((a & LONG_MASK) < (b & LONG_MASK)) ? -1 : 1;
3161 }
3162 return 0;
3163 }
3164
3165 /**
3166 * Version of compareMagnitude that compares magnitude with long value.
3167 * val can't be Long.MIN_VALUE.
3168 */
3169 final int compareMagnitude(long val) {
3170 assert val != Long.MIN_VALUE;
3171 int[] m1 = mag;
3172 int len = m1.length;
3173 if(len > 2) {
3174 return 1;
3175 }
3176 if (val < 0) {
3177 val = -val;
3178 }
3179 int highWord = (int)(val >>> 32);
3180 if (highWord==0) {
3181 if (len < 1)
3182 return -1;
3183 if (len > 1)
3184 return 1;
3185 int a = m1[0];
3186 int b = (int)val;
3187 if (a != b) {
3188 return ((a & LONG_MASK) < (b & LONG_MASK))? -1 : 1;
3189 }
3190 return 0;
3191 } else {
3192 if (len < 2)
3193 return -1;
3194 int a = m1[0];
3195 int b = highWord;
3196 if (a != b) {
3197 return ((a & LONG_MASK) < (b & LONG_MASK))? -1 : 1;
3198 }
3199 a = m1[1];
3200 b = (int)val;
3201 if (a != b) {
3202 return ((a & LONG_MASK) < (b & LONG_MASK))? -1 : 1;
3203 }
3204 return 0;
3205 }
3206 }
3207
3208 /**
3209 * Compares this BigInteger with the specified Object for equality.
3210 *
3211 * @param x Object to which this BigInteger is to be compared.
3212 * @return {@code true} if and only if the specified Object is a
3213 * BigInteger whose value is numerically equal to this BigInteger.
3214 */
3215 public boolean equals(Object x) {
3216 // This test is just an optimization, which may or may not help
3217 if (x == this)
3218 return true;
3219
3220 if (!(x instanceof BigInteger))
3221 return false;
3222
3223 BigInteger xInt = (BigInteger) x;
3224 if (xInt.signum != signum)
3225 return false;
3226
3227 int[] m = mag;
3228 int len = m.length;
3229 int[] xm = xInt.mag;
3230 if (len != xm.length)
3231 return false;
3232
3233 for (int i = 0; i < len; i++)
3234 if (xm[i] != m[i])
3235 return false;
3236
3237 return true;
3238 }
3239
3240 /**
3241 * Returns the minimum of this BigInteger and {@code val}.
3242 *
3243 * @param val value with which the minimum is to be computed.
3244 * @return the BigInteger whose value is the lesser of this BigInteger and
3245 * {@code val}. If they are equal, either may be returned.
3246 */
3247 public BigInteger min(BigInteger val) {
3248 return (compareTo(val)<0 ? this : val);
3249 }
3250
3251 /**
3252 * Returns the maximum of this BigInteger and {@code val}.
3253 *
3254 * @param val value with which the maximum is to be computed.
3255 * @return the BigInteger whose value is the greater of this and
3256 * {@code val}. If they are equal, either may be returned.
3257 */
3258 public BigInteger max(BigInteger val) {
3259 return (compareTo(val)>0 ? this : val);
3260 }
3261
3262
3263 // Hash Function
3264
3265 /**
3266 * Returns the hash code for this BigInteger.
3267 *
3268 * @return hash code for this BigInteger.
3269 */
3270 public int hashCode() {
3271 int hashCode = 0;
3272
3273 for (int i=0; i<mag.length; i++)
3274 hashCode = (int)(31*hashCode + (mag[i] & LONG_MASK));
3275
3276 return hashCode * signum;
3277 }
3278
3279 /**
3280 * Returns the String representation of this BigInteger in the
3281 * given radix. If the radix is outside the range from {@link
3282 * Character#MIN_RADIX} to {@link Character#MAX_RADIX} inclusive,
3283 * it will default to 10 (as is the case for
3284 * {@code Integer.toString}). The digit-to-character mapping
3285 * provided by {@code Character.forDigit} is used, and a minus
3286 * sign is prepended if appropriate. (This representation is
3287 * compatible with the {@link #BigInteger(String, int) (String,
3288 * int)} constructor.)
3289 *
3290 * @param radix radix of the String representation.
3291 * @return String representation of this BigInteger in the given radix.
3292 * @see Integer#toString
3293 * @see Character#forDigit
3294 * @see #BigInteger(java.lang.String, int)
3295 */
3296 public String toString(int radix) {
3297 if (signum == 0)
3298 return "0";
3299 if (radix < Character.MIN_RADIX || radix > Character.MAX_RADIX)
3300 radix = 10;
3301
3302 // Compute upper bound on number of digit groups and allocate space
3303 int maxNumDigitGroups = (4*mag.length + 6)/7;
3304 String digitGroup[] = new String[maxNumDigitGroups];
3305
3306 // Translate number to string, a digit group at a time
3307 BigInteger tmp = this.abs();
3308 int numGroups = 0;
3309 while (tmp.signum != 0) {
3310 BigInteger d = longRadix[radix];
3311
3312 MutableBigInteger q = new MutableBigInteger(),
3313 a = new MutableBigInteger(tmp.mag),
3314 b = new MutableBigInteger(d.mag);
3315 MutableBigInteger r = a.divide(b, q);
3316 BigInteger q2 = q.toBigInteger(tmp.signum * d.signum);
3317 BigInteger r2 = r.toBigInteger(tmp.signum * d.signum);
3318
3319 digitGroup[numGroups++] = Long.toString(r2.longValue(), radix);
3320 tmp = q2;
3321 }
3322
3323 // Put sign (if any) and first digit group into result buffer
3324 StringBuilder buf = new StringBuilder(numGroups*digitsPerLong[radix]+1);
3325 if (signum<0)
3326 buf.append('-');
3327 buf.append(digitGroup[numGroups-1]);
3328
3329 // Append remaining digit groups padded with leading zeros
3330 for (int i=numGroups-2; i>=0; i--) {
3331 // Prepend (any) leading zeros for this digit group
3332 int numLeadingZeros = digitsPerLong[radix]-digitGroup[i].length();
3333 if (numLeadingZeros != 0)
3334 buf.append(zeros[numLeadingZeros]);
3335 buf.append(digitGroup[i]);
3336 }
3337 return buf.toString();
3338 }
3339
3340
3341 /* zero[i] is a string of i consecutive zeros. */
3342 private static String zeros[] = new String[64];
3343 static {
3344 zeros[63] =
3345 "000000000000000000000000000000000000000000000000000000000000000";
3346 for (int i=0; i<63; i++)
3347 zeros[i] = zeros[63].substring(0, i);
3348 }
3349
3350 /**
3351 * Returns the decimal String representation of this BigInteger.
3352 * The digit-to-character mapping provided by
3353 * {@code Character.forDigit} is used, and a minus sign is
3354 * prepended if appropriate. (This representation is compatible
3355 * with the {@link #BigInteger(String) (String)} constructor, and
3356 * allows for String concatenation with Java's + operator.)
3357 *
3358 * @return decimal String representation of this BigInteger.
3359 * @see Character#forDigit
3360 * @see #BigInteger(java.lang.String)
3361 */
3362 public String toString() {
3363 return toString(10);
3364 }
3365
3366 /**
3367 * Returns a byte array containing the two's-complement
3368 * representation of this BigInteger. The byte array will be in
3369 * <i>big-endian</i> byte-order: the most significant byte is in
3370 * the zeroth element. The array will contain the minimum number
3371 * of bytes required to represent this BigInteger, including at
3372 * least one sign bit, which is {@code (ceil((this.bitLength() +
3373 * 1)/8))}. (This representation is compatible with the
3374 * {@link #BigInteger(byte[]) (byte[])} constructor.)
3375 *
3376 * @return a byte array containing the two's-complement representation of
3377 * this BigInteger.
3378 * @see #BigInteger(byte[])
3379 */
3380 public byte[] toByteArray() {
3381 int byteLen = bitLength()/8 + 1;
3382 byte[] byteArray = new byte[byteLen];
3383
3384 for (int i=byteLen-1, bytesCopied=4, nextInt=0, intIndex=0; i>=0; i--) {
3385 if (bytesCopied == 4) {
3386 nextInt = getInt(intIndex++);
3387 bytesCopied = 1;
3388 } else {
3389 nextInt >>>= 8;
3390 bytesCopied++;
3391 }
3392 byteArray[i] = (byte)nextInt;
3393 }
3394 return byteArray;
3395 }
3396
3397 /**
3398 * Converts this BigInteger to an {@code int}. This
3399 * conversion is analogous to a
3400 * <i>narrowing primitive conversion</i> from {@code long} to
3401 * {@code int} as defined in section 5.1.3 of
3402 * <cite>The Java™ Language Specification</cite>:
3403 * if this BigInteger is too big to fit in an
3404 * {@code int}, only the low-order 32 bits are returned.
3405 * Note that this conversion can lose information about the
3406 * overall magnitude of the BigInteger value as well as return a
3407 * result with the opposite sign.
3408 *
3409 * @return this BigInteger converted to an {@code int}.
3410 * @see #intValueExact()
3411 */
3412 public int intValue() {
3413 int result = 0;
3414 result = getInt(0);
3415 return result;
3416 }
3417
3418 /**
3419 * Converts this BigInteger to a {@code long}. This
3420 * conversion is analogous to a
3421 * <i>narrowing primitive conversion</i> from {@code long} to
3422 * {@code int} as defined in section 5.1.3 of
3423 * <cite>The Java™ Language Specification</cite>:
3424 * if this BigInteger is too big to fit in a
3425 * {@code long}, only the low-order 64 bits are returned.
3426 * Note that this conversion can lose information about the
3427 * overall magnitude of the BigInteger value as well as return a
3428 * result with the opposite sign.
3429 *
3430 * @return this BigInteger converted to a {@code long}.
3431 * @see #longValueExact()
3432 */
3433 public long longValue() {
3434 long result = 0;
3435
3436 for (int i=1; i>=0; i--)
3437 result = (result << 32) + (getInt(i) & LONG_MASK);
3438 return result;
3439 }
3440
3441 /**
3442 * Converts this BigInteger to a {@code float}. This
3443 * conversion is similar to the
3444 * <i>narrowing primitive conversion</i> from {@code double} to
3445 * {@code float} as defined in section 5.1.3 of
3446 * <cite>The Java™ Language Specification</cite>:
3447 * if this BigInteger has too great a magnitude
3448 * to represent as a {@code float}, it will be converted to
3449 * {@link Float#NEGATIVE_INFINITY} or {@link
3450 * Float#POSITIVE_INFINITY} as appropriate. Note that even when
3451 * the return value is finite, this conversion can lose
3452 * information about the precision of the BigInteger value.
3453 *
3454 * @return this BigInteger converted to a {@code float}.
3455 */
3456 public float floatValue() {
3457 if (signum == 0) {
3458 return 0.0f;
3459 }
3460
3461 int exponent = ((mag.length - 1) << 5) + bitLengthForInt(mag[0]) - 1;
3462
3463 // exponent == floor(log2(abs(this)))
3464 if (exponent < Long.SIZE - 1) {
3465 return longValue();
3466 } else if (exponent > Float.MAX_EXPONENT) {
3467 return signum > 0 ? Float.POSITIVE_INFINITY : Float.NEGATIVE_INFINITY;
3468 }
3469
3470 /*
3471 * We need the top SIGNIFICAND_WIDTH bits, including the "implicit"
3472 * one bit. To make rounding easier, we pick out the top
3473 * SIGNIFICAND_WIDTH + 1 bits, so we have one to help us round up or
3474 * down. twiceSignifFloor will contain the top SIGNIFICAND_WIDTH + 1
3475 * bits, and signifFloor the top SIGNIFICAND_WIDTH.
3476 *
3477 * It helps to consider the real number signif = abs(this) *
3478 * 2^(SIGNIFICAND_WIDTH - 1 - exponent).
3479 */
3480 int shift = exponent - FloatConsts.SIGNIFICAND_WIDTH;
3481
3482 int twiceSignifFloor;
3483 // twiceSignifFloor will be == abs().shiftRight(shift).intValue()
3484 // We do the shift into an int directly to improve performance.
3485
3486 int nBits = shift & 0x1f;
3487 int nBits2 = 32 - nBits;
3488
3489 if (nBits == 0) {
3490 twiceSignifFloor = mag[0];
3491 } else {
3492 twiceSignifFloor = mag[0] >>> nBits;
3493 if (twiceSignifFloor == 0) {
3494 twiceSignifFloor = (mag[0] << nBits2) | (mag[1] >>> nBits);
3495 }
3496 }
3497
3498 int signifFloor = twiceSignifFloor >> 1;
3499 signifFloor &= FloatConsts.SIGNIF_BIT_MASK; // remove the implied bit
3500
3501 /*
3502 * We round up if either the fractional part of signif is strictly
3503 * greater than 0.5 (which is true if the 0.5 bit is set and any lower
3504 * bit is set), or if the fractional part of signif is >= 0.5 and
3505 * signifFloor is odd (which is true if both the 0.5 bit and the 1 bit
3506 * are set). This is equivalent to the desired HALF_EVEN rounding.
3507 */
3508 boolean increment = (twiceSignifFloor & 1) != 0
3509 && ((signifFloor & 1) != 0 || abs().getLowestSetBit() < shift);
3510 int signifRounded = increment ? signifFloor + 1 : signifFloor;
3511 int bits = ((exponent + FloatConsts.EXP_BIAS))
3512 << (FloatConsts.SIGNIFICAND_WIDTH - 1);
3513 bits += signifRounded;
3514 /*
3515 * If signifRounded == 2^24, we'd need to set all of the significand
3516 * bits to zero and add 1 to the exponent. This is exactly the behavior
3517 * we get from just adding signifRounded to bits directly. If the
3518 * exponent is Float.MAX_EXPONENT, we round up (correctly) to
3519 * Float.POSITIVE_INFINITY.
3520 */
3521 bits |= signum & FloatConsts.SIGN_BIT_MASK;
3522 return Float.intBitsToFloat(bits);
3523 }
3524
3525 /**
3526 * Converts this BigInteger to a {@code double}. This
3527 * conversion is similar to the
3528 * <i>narrowing primitive conversion</i> from {@code double} to
3529 * {@code float} as defined in section 5.1.3 of
3530 * <cite>The Java™ Language Specification</cite>:
3531 * if this BigInteger has too great a magnitude
3532 * to represent as a {@code double}, it will be converted to
3533 * {@link Double#NEGATIVE_INFINITY} or {@link
3534 * Double#POSITIVE_INFINITY} as appropriate. Note that even when
3535 * the return value is finite, this conversion can lose
3536 * information about the precision of the BigInteger value.
3537 *
3538 * @return this BigInteger converted to a {@code double}.
3539 */
3540 public double doubleValue() {
3541 if (signum == 0) {
3542 return 0.0;
3543 }
3544
3545 int exponent = ((mag.length - 1) << 5) + bitLengthForInt(mag[0]) - 1;
3546
3547 // exponent == floor(log2(abs(this))Double)
3548 if (exponent < Long.SIZE - 1) {
3549 return longValue();
3550 } else if (exponent > Double.MAX_EXPONENT) {
3551 return signum > 0 ? Double.POSITIVE_INFINITY : Double.NEGATIVE_INFINITY;
3552 }
3553
3554 /*
3555 * We need the top SIGNIFICAND_WIDTH bits, including the "implicit"
3556 * one bit. To make rounding easier, we pick out the top
3557 * SIGNIFICAND_WIDTH + 1 bits, so we have one to help us round up or
3558 * down. twiceSignifFloor will contain the top SIGNIFICAND_WIDTH + 1
3559 * bits, and signifFloor the top SIGNIFICAND_WIDTH.
3560 *
3561 * It helps to consider the real number signif = abs(this) *
3562 * 2^(SIGNIFICAND_WIDTH - 1 - exponent).
3563 */
3564 int shift = exponent - DoubleConsts.SIGNIFICAND_WIDTH;
3565
3566 long twiceSignifFloor;
3567 // twiceSignifFloor will be == abs().shiftRight(shift).longValue()
3568 // We do the shift into a long directly to improve performance.
3569
3570 int nBits = shift & 0x1f;
3571 int nBits2 = 32 - nBits;
3572
3573 int highBits;
3574 int lowBits;
3575 if (nBits == 0) {
3576 highBits = mag[0];
3577 lowBits = mag[1];
3578 } else {
3579 highBits = mag[0] >>> nBits;
3580 lowBits = (mag[0] << nBits2) | (mag[1] >>> nBits);
3581 if (highBits == 0) {
3582 highBits = lowBits;
3583 lowBits = (mag[1] << nBits2) | (mag[2] >>> nBits);
3584 }
3585 }
3586
3587 twiceSignifFloor = ((highBits & LONG_MASK) << 32)
3588 | (lowBits & LONG_MASK);
3589
3590 long signifFloor = twiceSignifFloor >> 1;
3591 signifFloor &= DoubleConsts.SIGNIF_BIT_MASK; // remove the implied bit
3592
3593 /*
3594 * We round up if either the fractional part of signif is strictly
3595 * greater than 0.5 (which is true if the 0.5 bit is set and any lower
3596 * bit is set), or if the fractional part of signif is >= 0.5 and
3597 * signifFloor is odd (which is true if both the 0.5 bit and the 1 bit
3598 * are set). This is equivalent to the desired HALF_EVEN rounding.
3599 */
3600 boolean increment = (twiceSignifFloor & 1) != 0
3601 && ((signifFloor & 1) != 0 || abs().getLowestSetBit() < shift);
3602 long signifRounded = increment ? signifFloor + 1 : signifFloor;
3603 long bits = (long) ((exponent + DoubleConsts.EXP_BIAS))
3604 << (DoubleConsts.SIGNIFICAND_WIDTH - 1);
3605 bits += signifRounded;
3606 /*
3607 * If signifRounded == 2^53, we'd need to set all of the significand
3608 * bits to zero and add 1 to the exponent. This is exactly the behavior
3609 * we get from just adding signifRounded to bits directly. If the
3610 * exponent is Double.MAX_EXPONENT, we round up (correctly) to
3611 * Double.POSITIVE_INFINITY.
3612 */
3613 bits |= signum & DoubleConsts.SIGN_BIT_MASK;
3614 return Double.longBitsToDouble(bits);
3615 }
3616
3617 /**
3618 * Returns a copy of the input array stripped of any leading zero bytes.
3619 */
3620 private static int[] stripLeadingZeroInts(int val[]) {
3621 int vlen = val.length;
3622 int keep;
3623
3624 // Find first nonzero byte
3625 for (keep = 0; keep < vlen && val[keep] == 0; keep++)
3626 ;
3627 return java.util.Arrays.copyOfRange(val, keep, vlen);
3628 }
3629
3630 /**
3631 * Returns the input array stripped of any leading zero bytes.
3632 * Since the source is trusted the copying may be skipped.
3633 */
3634 private static int[] trustedStripLeadingZeroInts(int val[]) {
3635 int vlen = val.length;
3636 int keep;
3637
3638 // Find first nonzero byte
3639 for (keep = 0; keep < vlen && val[keep] == 0; keep++)
3640 ;
3641 return keep == 0 ? val : java.util.Arrays.copyOfRange(val, keep, vlen);
3642 }
3643
3644 /**
3645 * Returns a copy of the input array stripped of any leading zero bytes.
3646 */
3647 private static int[] stripLeadingZeroBytes(byte a[]) {
3648 int byteLength = a.length;
3649 int keep;
3650
3651 // Find first nonzero byte
3652 for (keep = 0; keep < byteLength && a[keep]==0; keep++)
3653 ;
3654
3655 // Allocate new array and copy relevant part of input array
3656 int intLength = ((byteLength - keep) + 3) >>> 2;
3657 int[] result = new int[intLength];
3658 int b = byteLength - 1;
3659 for (int i = intLength-1; i >= 0; i--) {
3660 result[i] = a[b--] & 0xff;
3661 int bytesRemaining = b - keep + 1;
3662 int bytesToTransfer = Math.min(3, bytesRemaining);
3663 for (int j=8; j <= (bytesToTransfer << 3); j += 8)
3664 result[i] |= ((a[b--] & 0xff) << j);
3665 }
3666 return result;
3667 }
3668
3669 /**
3670 * Takes an array a representing a negative 2's-complement number and
3671 * returns the minimal (no leading zero bytes) unsigned whose value is -a.
3672 */
3673 private static int[] makePositive(byte a[]) {
3674 int keep, k;
3675 int byteLength = a.length;
3676
3677 // Find first non-sign (0xff) byte of input
3678 for (keep=0; keep<byteLength && a[keep]==-1; keep++)
3679 ;
3680
3681
3682 /* Allocate output array. If all non-sign bytes are 0x00, we must
3683 * allocate space for one extra output byte. */
3684 for (k=keep; k<byteLength && a[k]==0; k++)
3685 ;
3686
3687 int extraByte = (k==byteLength) ? 1 : 0;
3688 int intLength = ((byteLength - keep + extraByte) + 3)/4;
3689 int result[] = new int[intLength];
3690
3691 /* Copy one's complement of input into output, leaving extra
3692 * byte (if it exists) == 0x00 */
3693 int b = byteLength - 1;
3694 for (int i = intLength-1; i >= 0; i--) {
3695 result[i] = a[b--] & 0xff;
3696 int numBytesToTransfer = Math.min(3, b-keep+1);
3697 if (numBytesToTransfer < 0)
3698 numBytesToTransfer = 0;
3699 for (int j=8; j <= 8*numBytesToTransfer; j += 8)
3700 result[i] |= ((a[b--] & 0xff) << j);
3701
3702 // Mask indicates which bits must be complemented
3703 int mask = -1 >>> (8*(3-numBytesToTransfer));
3704 result[i] = ~result[i] & mask;
3705 }
3706
3707 // Add one to one's complement to generate two's complement
3708 for (int i=result.length-1; i>=0; i--) {
3709 result[i] = (int)((result[i] & LONG_MASK) + 1);
3710 if (result[i] != 0)
3711 break;
3712 }
3713
3714 return result;
3715 }
3716
3717 /**
3718 * Takes an array a representing a negative 2's-complement number and
3719 * returns the minimal (no leading zero ints) unsigned whose value is -a.
3720 */
3721 private static int[] makePositive(int a[]) {
3722 int keep, j;
3723
3724 // Find first non-sign (0xffffffff) int of input
3725 for (keep=0; keep<a.length && a[keep]==-1; keep++)
3726 ;
3727
3728 /* Allocate output array. If all non-sign ints are 0x00, we must
3729 * allocate space for one extra output int. */
3730 for (j=keep; j<a.length && a[j]==0; j++)
3731 ;
3732 int extraInt = (j==a.length ? 1 : 0);
3733 int result[] = new int[a.length - keep + extraInt];
3734
3735 /* Copy one's complement of input into output, leaving extra
3736 * int (if it exists) == 0x00 */
3737 for (int i = keep; i<a.length; i++)
3738 result[i - keep + extraInt] = ~a[i];
3739
3740 // Add one to one's complement to generate two's complement
3741 for (int i=result.length-1; ++result[i]==0; i--)
3742 ;
3743
3744 return result;
3745 }
3746
3747 /*
3748 * The following two arrays are used for fast String conversions. Both
3749 * are indexed by radix. The first is the number of digits of the given
3750 * radix that can fit in a Java long without "going negative", i.e., the
3751 * highest integer n such that radix**n < 2**63. The second is the
3752 * "long radix" that tears each number into "long digits", each of which
3753 * consists of the number of digits in the corresponding element in
3754 * digitsPerLong (longRadix[i] = i**digitPerLong[i]). Both arrays have
3755 * nonsense values in their 0 and 1 elements, as radixes 0 and 1 are not
3756 * used.
3757 */
3758 private static int digitsPerLong[] = {0, 0,
3759 62, 39, 31, 27, 24, 22, 20, 19, 18, 18, 17, 17, 16, 16, 15, 15, 15, 14,
3760 14, 14, 14, 13, 13, 13, 13, 13, 13, 12, 12, 12, 12, 12, 12, 12, 12};
3761
3762 private static BigInteger longRadix[] = {null, null,
3763 valueOf(0x4000000000000000L), valueOf(0x383d9170b85ff80bL),
3764 valueOf(0x4000000000000000L), valueOf(0x6765c793fa10079dL),
3765 valueOf(0x41c21cb8e1000000L), valueOf(0x3642798750226111L),
3766 valueOf(0x1000000000000000L), valueOf(0x12bf307ae81ffd59L),
3767 valueOf( 0xde0b6b3a7640000L), valueOf(0x4d28cb56c33fa539L),
3768 valueOf(0x1eca170c00000000L), valueOf(0x780c7372621bd74dL),
3769 valueOf(0x1e39a5057d810000L), valueOf(0x5b27ac993df97701L),
3770 valueOf(0x1000000000000000L), valueOf(0x27b95e997e21d9f1L),
3771 valueOf(0x5da0e1e53c5c8000L), valueOf( 0xb16a458ef403f19L),
3772 valueOf(0x16bcc41e90000000L), valueOf(0x2d04b7fdd9c0ef49L),
3773 valueOf(0x5658597bcaa24000L), valueOf( 0x6feb266931a75b7L),
3774 valueOf( 0xc29e98000000000L), valueOf(0x14adf4b7320334b9L),
3775 valueOf(0x226ed36478bfa000L), valueOf(0x383d9170b85ff80bL),
3776 valueOf(0x5a3c23e39c000000L), valueOf( 0x4e900abb53e6b71L),
3777 valueOf( 0x7600ec618141000L), valueOf( 0xaee5720ee830681L),
3778 valueOf(0x1000000000000000L), valueOf(0x172588ad4f5f0981L),
3779 valueOf(0x211e44f7d02c1000L), valueOf(0x2ee56725f06e5c71L),
3780 valueOf(0x41c21cb8e1000000L)};
3781
3782 /*
3783 * These two arrays are the integer analogue of above.
3784 */
3785 private static int digitsPerInt[] = {0, 0, 30, 19, 15, 13, 11,
3786 11, 10, 9, 9, 8, 8, 8, 8, 7, 7, 7, 7, 7, 7, 7, 6, 6, 6, 6,
3787 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 5};
3788
3789 private static int intRadix[] = {0, 0,
3790 0x40000000, 0x4546b3db, 0x40000000, 0x48c27395, 0x159fd800,
3791 0x75db9c97, 0x40000000, 0x17179149, 0x3b9aca00, 0xcc6db61,
3792 0x19a10000, 0x309f1021, 0x57f6c100, 0xa2f1b6f, 0x10000000,
3793 0x18754571, 0x247dbc80, 0x3547667b, 0x4c4b4000, 0x6b5a6e1d,
3794 0x6c20a40, 0x8d2d931, 0xb640000, 0xe8d4a51, 0x1269ae40,
3795 0x17179149, 0x1cb91000, 0x23744899, 0x2b73a840, 0x34e63b41,
3796 0x40000000, 0x4cfa3cc1, 0x5c13d840, 0x6d91b519, 0x39aa400
3797 };
3798
3799 /**
3800 * These routines provide access to the two's complement representation
3801 * of BigIntegers.
3802 */
3803
3804 /**
3805 * Returns the length of the two's complement representation in ints,
3806 * including space for at least one sign bit.
3807 */
3808 private int intLength() {
3809 return (bitLength() >>> 5) + 1;
3810 }
3811
3812 /* Returns sign bit */
3813 private int signBit() {
3814 return signum < 0 ? 1 : 0;
3815 }
3816
3817 /* Returns an int of sign bits */
3818 private int signInt() {
3819 return signum < 0 ? -1 : 0;
3820 }
3821
3822 /**
3823 * Returns the specified int of the little-endian two's complement
3824 * representation (int 0 is the least significant). The int number can
3825 * be arbitrarily high (values are logically preceded by infinitely many
3826 * sign ints).
3827 */
3828 private int getInt(int n) {
3829 if (n < 0)
3830 return 0;
3831 if (n >= mag.length)
3832 return signInt();
3833
3834 int magInt = mag[mag.length-n-1];
3835
3836 return (signum >= 0 ? magInt :
3837 (n <= firstNonzeroIntNum() ? -magInt : ~magInt));
3838 }
3839
3840 /**
3841 * Returns the index of the int that contains the first nonzero int in the
3842 * little-endian binary representation of the magnitude (int 0 is the
3843 * least significant). If the magnitude is zero, return value is undefined.
3844 */
3845 private int firstNonzeroIntNum() {
3846 int fn = firstNonzeroIntNum - 2;
3847 if (fn == -2) { // firstNonzeroIntNum not initialized yet
3848 fn = 0;
3849
3850 // Search for the first nonzero int
3851 int i;
3852 int mlen = mag.length;
3853 for (i = mlen - 1; i >= 0 && mag[i] == 0; i--)
3854 ;
3855 fn = mlen - i - 1;
3856 firstNonzeroIntNum = fn + 2; // offset by two to initialize
3857 }
3858 return fn;
3859 }
3860
3861 /** use serialVersionUID from JDK 1.1. for interoperability */
3862 private static final long serialVersionUID = -8287574255936472291L;
3863
3864 /**
3865 * Serializable fields for BigInteger.
3866 *
3867 * @serialField signum int
3868 * signum of this BigInteger.
3869 * @serialField magnitude int[]
3870 * magnitude array of this BigInteger.
3871 * @serialField bitCount int
3872 * number of bits in this BigInteger
3873 * @serialField bitLength int
3874 * the number of bits in the minimal two's-complement
3875 * representation of this BigInteger
3876 * @serialField lowestSetBit int
3877 * lowest set bit in the twos complement representation
3878 */
3879 private static final ObjectStreamField[] serialPersistentFields = {
3880 new ObjectStreamField("signum", Integer.TYPE),
3881 new ObjectStreamField("magnitude", byte[].class),
3882 new ObjectStreamField("bitCount", Integer.TYPE),
3883 new ObjectStreamField("bitLength", Integer.TYPE),
3884 new ObjectStreamField("firstNonzeroByteNum", Integer.TYPE),
3885 new ObjectStreamField("lowestSetBit", Integer.TYPE)
3886 };
3887
3888 /**
3889 * Reconstitute the {@code BigInteger} instance from a stream (that is,
3890 * deserialize it). The magnitude is read in as an array of bytes
3891 * for historical reasons, but it is converted to an array of ints
3892 * and the byte array is discarded.
3893 * Note:
3894 * The current convention is to initialize the cache fields, bitCount,
3895 * bitLength and lowestSetBit, to 0 rather than some other marker value.
3896 * Therefore, no explicit action to set these fields needs to be taken in
3897 * readObject because those fields already have a 0 value be default since
3898 * defaultReadObject is not being used.
3899 */
3900 private void readObject(java.io.ObjectInputStream s)
3901 throws java.io.IOException, ClassNotFoundException {
3902 /*
3903 * In order to maintain compatibility with previous serialized forms,
3904 * the magnitude of a BigInteger is serialized as an array of bytes.
3905 * The magnitude field is used as a temporary store for the byte array
3906 * that is deserialized. The cached computation fields should be
3907 * transient but are serialized for compatibility reasons.
3908 */
3909
3910 // prepare to read the alternate persistent fields
3911 ObjectInputStream.GetField fields = s.readFields();
3912
3913 // Read the alternate persistent fields that we care about
3914 int sign = fields.get("signum", -2);
3915 byte[] magnitude = (byte[])fields.get("magnitude", null);
3916
3917 // Validate signum
3918 if (sign < -1 || sign > 1) {
3919 String message = "BigInteger: Invalid signum value";
3920 if (fields.defaulted("signum"))
3921 message = "BigInteger: Signum not present in stream";
3922 throw new java.io.StreamCorruptedException(message);
3923 }
3924 if ((magnitude.length == 0) != (sign == 0)) {
3925 String message = "BigInteger: signum-magnitude mismatch";
3926 if (fields.defaulted("magnitude"))
3927 message = "BigInteger: Magnitude not present in stream";
3928 throw new java.io.StreamCorruptedException(message);
3929 }
3930
3931 // Commit final fields via Unsafe
3932 UnsafeHolder.putSign(this, sign);
3933
3934 // Calculate mag field from magnitude and discard magnitude
3935 UnsafeHolder.putMag(this, stripLeadingZeroBytes(magnitude));
3936 }
3937
3938 // Support for resetting final fields while deserializing
3939 private static class UnsafeHolder {
3940 private static final sun.misc.Unsafe unsafe;
3941 private static final long signumOffset;
3942 private static final long magOffset;
3943 static {
3944 try {
3945 unsafe = sun.misc.Unsafe.getUnsafe();
3946 signumOffset = unsafe.objectFieldOffset
3947 (BigInteger.class.getDeclaredField("signum"));
3948 magOffset = unsafe.objectFieldOffset
3949 (BigInteger.class.getDeclaredField("mag"));
3950 } catch (Exception ex) {
3951 throw new ExceptionInInitializerError(ex);
3952 }
3953 }
3954
3955 static void putSign(BigInteger bi, int sign) {
3956 unsafe.putIntVolatile(bi, signumOffset, sign);
3957 }
3958
3959 static void putMag(BigInteger bi, int[] magnitude) {
3960 unsafe.putObjectVolatile(bi, magOffset, magnitude);
3961 }
3962 }
3963
3964 /**
3965 * Save the {@code BigInteger} instance to a stream.
3966 * The magnitude of a BigInteger is serialized as a byte array for
3967 * historical reasons.
3968 *
3969 * @serialData two necessary fields are written as well as obsolete
3970 * fields for compatibility with older versions.
3971 */
3972 private void writeObject(ObjectOutputStream s) throws IOException {
3973 // set the values of the Serializable fields
3974 ObjectOutputStream.PutField fields = s.putFields();
3975 fields.put("signum", signum);
3976 fields.put("magnitude", magSerializedForm());
3977 // The values written for cached fields are compatible with older
3978 // versions, but are ignored in readObject so don't otherwise matter.
3979 fields.put("bitCount", -1);
3980 fields.put("bitLength", -1);
3981 fields.put("lowestSetBit", -2);
3982 fields.put("firstNonzeroByteNum", -2);
3983
3984 // save them
3985 s.writeFields();
3986 }
3987
3988 /**
3989 * Returns the mag array as an array of bytes.
3990 */
3991 private byte[] magSerializedForm() {
3992 int len = mag.length;
3993
3994 int bitLen = (len == 0 ? 0 : ((len - 1) << 5) + bitLengthForInt(mag[0]));
3995 int byteLen = (bitLen + 7) >>> 3;
3996 byte[] result = new byte[byteLen];
3997
3998 for (int i = byteLen - 1, bytesCopied = 4, intIndex = len - 1, nextInt = 0;
3999 i>=0; i--) {
4000 if (bytesCopied == 4) {
4001 nextInt = mag[intIndex--];
4002 bytesCopied = 1;
4003 } else {
4004 nextInt >>>= 8;
4005 bytesCopied++;
4006 }
4007 result[i] = (byte)nextInt;
4008 }
4009 return result;
4010 }
4011
4012 /**
4013 * Converts this {@code BigInteger} to a {@code long}, checking
4014 * for lost information. If the value of this {@code BigInteger}
4015 * is out of the range of the {@code long} type, then an
4016 * {@code ArithmeticException} is thrown.
4017 *
4018 * @return this {@code BigInteger} converted to a {@code long}.
4019 * @throws ArithmeticException if the value of {@code this} will
4020 * not exactly fit in a {@code long}.
4021 * @see BigInteger#longValue
4022 * @since 1.8
4023 */
4024 public long longValueExact() {
4025 if (mag.length <= 2 && bitLength() <= 63)
4026 return longValue();
4027 else
4028 throw new ArithmeticException("BigInteger out of long range");
4029 }
4030
4031 /**
4032 * Converts this {@code BigInteger} to an {@code int}, checking
4033 * for lost information. If the value of this {@code BigInteger}
4034 * is out of the range of the {@code int} type, then an
4035 * {@code ArithmeticException} is thrown.
4036 *
4037 * @return this {@code BigInteger} converted to an {@code int}.
4038 * @throws ArithmeticException if the value of {@code this} will
4039 * not exactly fit in a {@code int}.
4040 * @see BigInteger#intValue
4041 * @since 1.8
4042 */
4043 public int intValueExact() {
4044 if (mag.length <= 1 && bitLength() <= 31)
4045 return intValue();
4046 else
4047 throw new ArithmeticException("BigInteger out of int range");
4048 }
4049
4050 /**
4051 * Converts this {@code BigInteger} to a {@code short}, checking
4052 * for lost information. If the value of this {@code BigInteger}
4053 * is out of the range of the {@code short} type, then an
4054 * {@code ArithmeticException} is thrown.
4055 *
4056 * @return this {@code BigInteger} converted to a {@code short}.
4057 * @throws ArithmeticException if the value of {@code this} will
4058 * not exactly fit in a {@code short}.
4059 * @see BigInteger#shortValue
4060 * @since 1.8
4061 */
4062 public short shortValueExact() {
4063 if (mag.length <= 1 && bitLength() <= 31) {
4064 int value = intValue();
4065 if (value >= Short.MIN_VALUE && value <= Short.MAX_VALUE)
4066 return shortValue();
4067 }
4068 throw new ArithmeticException("BigInteger out of short range");
4069 }
4070
4071 /**
4072 * Converts this {@code BigInteger} to a {@code byte}, checking
4073 * for lost information. If the value of this {@code BigInteger}
4074 * is out of the range of the {@code byte} type, then an
4075 * {@code ArithmeticException} is thrown.
4076 *
4077 * @return this {@code BigInteger} converted to a {@code byte}.
4078 * @throws ArithmeticException if the value of {@code this} will
4079 * not exactly fit in a {@code byte}.
4080 * @see BigInteger#byteValue
4081 * @since 1.8
4082 */
4083 public byte byteValueExact() {
4084 if (mag.length <= 1 && bitLength() <= 31) {
4085 int value = intValue();
4086 if (value >= Byte.MIN_VALUE && value <= Byte.MAX_VALUE)
4087 return byteValue();
4088 }
4089 throw new ArithmeticException("BigInteger out of byte range");
4090 }
4091 }