1 /* 2 * Copyright (c) 1996, 2014, Oracle and/or its affiliates. All rights reserved. 3 * DO NOT ALTER OR REMOVE COPYRIGHT NOTICES OR THIS FILE HEADER. 4 * 5 * This code is free software; you can redistribute it and/or modify it 6 * under the terms of the GNU General Public License version 2 only, as 7 * published by the Free Software Foundation. Oracle designates this 8 * particular file as subject to the "Classpath" exception as provided 9 * by Oracle in the LICENSE file that accompanied this code. 10 * 11 * This code is distributed in the hope that it will be useful, but WITHOUT 12 * ANY WARRANTY; without even the implied warranty of MERCHANTABILITY or 13 * FITNESS FOR A PARTICULAR PURPOSE. See the GNU General Public License 14 * version 2 for more details (a copy is included in the LICENSE file that 15 * accompanied this code). 16 * 17 * You should have received a copy of the GNU General Public License version 18 * 2 along with this work; if not, write to the Free Software Foundation, 19 * Inc., 51 Franklin St, Fifth Floor, Boston, MA 02110-1301 USA. 20 * 21 * Please contact Oracle, 500 Oracle Parkway, Redwood Shores, CA 94065 USA 22 * or visit www.oracle.com if you need additional information or have any 23 * questions. 24 */ 25 26 /* 27 * Portions Copyright (c) 1995 Colin Plumb. All rights reserved. 28 */ 29 30 package java.math; 31 32 import java.io.IOException; 33 import java.io.ObjectInputStream; 34 import java.io.ObjectOutputStream; 35 import java.io.ObjectStreamField; 36 import java.util.Arrays; 37 import java.util.Random; 38 import java.util.concurrent.ThreadLocalRandom; 39 import sun.misc.DoubleConsts; 40 import sun.misc.FloatConsts; 41 42 /** 43 * Immutable arbitrary-precision integers. All operations behave as if 44 * BigIntegers were represented in two's-complement notation (like Java's 45 * primitive integer types). BigInteger provides analogues to all of Java's 46 * primitive integer operators, and all relevant methods from java.lang.Math. 47 * Additionally, BigInteger provides operations for modular arithmetic, GCD 48 * calculation, primality testing, prime generation, bit manipulation, 49 * and a few other miscellaneous operations. 50 * 51 * <p>Semantics of arithmetic operations exactly mimic those of Java's integer 52 * arithmetic operators, as defined in <i>The Java Language Specification</i>. 53 * For example, division by zero throws an {@code ArithmeticException}, and 54 * division of a negative by a positive yields a negative (or zero) remainder. 55 * All of the details in the Spec concerning overflow are ignored, as 56 * BigIntegers are made as large as necessary to accommodate the results of an 57 * operation. 58 * 59 * <p>Semantics of shift operations extend those of Java's shift operators 60 * to allow for negative shift distances. A right-shift with a negative 61 * shift distance results in a left shift, and vice-versa. The unsigned 62 * right shift operator ({@code >>>}) is omitted, as this operation makes 63 * little sense in combination with the "infinite word size" abstraction 64 * provided by this class. 65 * 66 * <p>Semantics of bitwise logical operations exactly mimic those of Java's 67 * bitwise integer operators. The binary operators ({@code and}, 68 * {@code or}, {@code xor}) implicitly perform sign extension on the shorter 69 * of the two operands prior to performing the operation. 70 * 71 * <p>Comparison operations perform signed integer comparisons, analogous to 72 * those performed by Java's relational and equality operators. 73 * 74 * <p>Modular arithmetic operations are provided to compute residues, perform 75 * exponentiation, and compute multiplicative inverses. These methods always 76 * return a non-negative result, between {@code 0} and {@code (modulus - 1)}, 77 * inclusive. 78 * 79 * <p>Bit operations operate on a single bit of the two's-complement 80 * representation of their operand. If necessary, the operand is sign- 81 * extended so that it contains the designated bit. None of the single-bit 82 * operations can produce a BigInteger with a different sign from the 83 * BigInteger being operated on, as they affect only a single bit, and the 84 * "infinite word size" abstraction provided by this class ensures that there 85 * are infinitely many "virtual sign bits" preceding each BigInteger. 86 * 87 * <p>For the sake of brevity and clarity, pseudo-code is used throughout the 88 * descriptions of BigInteger methods. The pseudo-code expression 89 * {@code (i + j)} is shorthand for "a BigInteger whose value is 90 * that of the BigInteger {@code i} plus that of the BigInteger {@code j}." 91 * The pseudo-code expression {@code (i == j)} is shorthand for 92 * "{@code true} if and only if the BigInteger {@code i} represents the same 93 * value as the BigInteger {@code j}." Other pseudo-code expressions are 94 * interpreted similarly. 95 * 96 * <p>All methods and constructors in this class throw 97 * {@code NullPointerException} when passed 98 * a null object reference for any input parameter. 99 * 100 * BigInteger must support values in the range 101 * -2<sup>{@code Integer.MAX_VALUE}</sup> (exclusive) to 102 * +2<sup>{@code Integer.MAX_VALUE}</sup> (exclusive) 103 * and may support values outside of that range. 104 * 105 * The range of probable prime values is limited and may be less than 106 * the full supported positive range of {@code BigInteger}. 107 * The range must be at least 1 to 2<sup>500000000</sup>. 108 * 109 * @implNote 110 * BigInteger constructors and operations throw {@code ArithmeticException} when 111 * the result is out of the supported range of 112 * -2<sup>{@code Integer.MAX_VALUE}</sup> (exclusive) to 113 * +2<sup>{@code Integer.MAX_VALUE}</sup> (exclusive). 114 * 115 * @see BigDecimal 116 * @author Josh Bloch 117 * @author Michael McCloskey 118 * @author Alan Eliasen 119 * @author Timothy Buktu 120 * @since 1.1 121 */ 122 123 public class BigInteger extends Number implements Comparable<BigInteger> { 124 /** 125 * The signum of this BigInteger: -1 for negative, 0 for zero, or 126 * 1 for positive. Note that the BigInteger zero <i>must</i> have 127 * a signum of 0. This is necessary to ensures that there is exactly one 128 * representation for each BigInteger value. 129 */ 130 final int signum; 131 132 /** 133 * The magnitude of this BigInteger, in <i>big-endian</i> order: the 134 * zeroth element of this array is the most-significant int of the 135 * magnitude. The magnitude must be "minimal" in that the most-significant 136 * int ({@code mag[0]}) must be non-zero. This is necessary to 137 * ensure that there is exactly one representation for each BigInteger 138 * value. Note that this implies that the BigInteger zero has a 139 * zero-length mag array. 140 */ 141 final int[] mag; 142 143 // The following fields are stable variables. A stable variable's value 144 // changes at most once from the default zero value to a non-zero stable 145 // value. A stable value is calculated lazily on demand. 146 147 /** 148 * One plus the bitCount of this BigInteger. This is a stable variable. 149 * 150 * @see #bitCount 151 */ 152 private int bitCountPlusOne; 153 154 /** 155 * One plus the bitLength of this BigInteger. This is a stable variable. 156 * (either value is acceptable). 157 * 158 * @see #bitLength() 159 */ 160 private int bitLengthPlusOne; 161 162 /** 163 * Two plus the lowest set bit of this BigInteger. This is a stable variable. 164 * 165 * @see #getLowestSetBit 166 */ 167 private int lowestSetBitPlusTwo; 168 169 /** 170 * Two plus the index of the lowest-order int in the magnitude of this 171 * BigInteger that contains a nonzero int. This is a stable variable. The 172 * least significant int has int-number 0, the next int in order of 173 * increasing significance has int-number 1, and so forth. 174 * 175 * <p>Note: never used for a BigInteger with a magnitude of zero. 176 * 177 * @see #firstNonzeroIntNum() 178 */ 179 private int firstNonzeroIntNumPlusTwo; 180 181 /** 182 * This mask is used to obtain the value of an int as if it were unsigned. 183 */ 184 final static long LONG_MASK = 0xffffffffL; 185 186 /** 187 * This constant limits {@code mag.length} of BigIntegers to the supported 188 * range. 189 */ 190 private static final int MAX_MAG_LENGTH = Integer.MAX_VALUE / Integer.SIZE + 1; // (1 << 26) 191 192 /** 193 * Bit lengths larger than this constant can cause overflow in searchLen 194 * calculation and in BitSieve.singleSearch method. 195 */ 196 private static final int PRIME_SEARCH_BIT_LENGTH_LIMIT = 500000000; 197 198 /** 199 * The threshold value for using Karatsuba multiplication. If the number 200 * of ints in both mag arrays are greater than this number, then 201 * Karatsuba multiplication will be used. This value is found 202 * experimentally to work well. 203 */ 204 private static final int KARATSUBA_THRESHOLD = 80; 205 206 /** 207 * The threshold value for using 3-way Toom-Cook multiplication. 208 * If the number of ints in each mag array is greater than the 209 * Karatsuba threshold, and the number of ints in at least one of 210 * the mag arrays is greater than this threshold, then Toom-Cook 211 * multiplication will be used. 212 */ 213 private static final int TOOM_COOK_THRESHOLD = 240; 214 215 /** 216 * The threshold value for using Karatsuba squaring. If the number 217 * of ints in the number are larger than this value, 218 * Karatsuba squaring will be used. This value is found 219 * experimentally to work well. 220 */ 221 private static final int KARATSUBA_SQUARE_THRESHOLD = 128; 222 223 /** 224 * The threshold value for using Toom-Cook squaring. If the number 225 * of ints in the number are larger than this value, 226 * Toom-Cook squaring will be used. This value is found 227 * experimentally to work well. 228 */ 229 private static final int TOOM_COOK_SQUARE_THRESHOLD = 216; 230 231 /** 232 * The threshold value for using Burnikel-Ziegler division. If the number 233 * of ints in the divisor are larger than this value, Burnikel-Ziegler 234 * division may be used. This value is found experimentally to work well. 235 */ 236 static final int BURNIKEL_ZIEGLER_THRESHOLD = 80; 237 238 /** 239 * The offset value for using Burnikel-Ziegler division. If the number 240 * of ints in the divisor exceeds the Burnikel-Ziegler threshold, and the 241 * number of ints in the dividend is greater than the number of ints in the 242 * divisor plus this value, Burnikel-Ziegler division will be used. This 243 * value is found experimentally to work well. 244 */ 245 static final int BURNIKEL_ZIEGLER_OFFSET = 40; 246 247 /** 248 * The threshold value for using Schoenhage recursive base conversion. If 249 * the number of ints in the number are larger than this value, 250 * the Schoenhage algorithm will be used. In practice, it appears that the 251 * Schoenhage routine is faster for any threshold down to 2, and is 252 * relatively flat for thresholds between 2-25, so this choice may be 253 * varied within this range for very small effect. 254 */ 255 private static final int SCHOENHAGE_BASE_CONVERSION_THRESHOLD = 20; 256 257 /** 258 * The threshold value for using squaring code to perform multiplication 259 * of a {@code BigInteger} instance by itself. If the number of ints in 260 * the number are larger than this value, {@code multiply(this)} will 261 * return {@code square()}. 262 */ 263 private static final int MULTIPLY_SQUARE_THRESHOLD = 20; 264 265 // Constructors 266 267 /** 268 * Translates a byte sub-array containing the two's-complement binary 269 * representation of a BigInteger into a BigInteger. The sub-array is 270 * specified via an offset into the array and a length. The sub-array is 271 * assumed to be in <i>big-endian</i> byte-order: the most significant 272 * byte is the element at index {@code off}. The {@code val} array is 273 * assumed to be unchanged for the duration of the constructor call. 274 * 275 * An {@code IndexOutOfBoundsException} is thrown if the length of the array 276 * {@code val} is non-zero and either {@code off} is negative, {@code len} 277 * is negative, or {@code off+len} is greater than the length of 278 * {@code val}. 279 * 280 * @param val byte array containing a sub-array which is the big-endian 281 * two's-complement binary representation of a BigInteger. 282 * @param off the start offset of the binary representation. 283 * @param len the number of bytes to use. 284 * @throws NumberFormatException {@code val} is zero bytes long. 285 * @throws IndexOutOfBoundsException if the provided array offset and 286 * length would cause an index into the byte array to be 287 * negative or greater than or equal to the array length. 288 * @since 1.9 289 */ 290 public BigInteger(byte[] val, int off, int len) { 291 if (val.length == 0) { 292 throw new NumberFormatException("Zero length BigInteger"); 293 } else if ((off < 0) || (off >= val.length) || (len < 0) || 294 (len > val.length - off)) { // 0 <= off < val.length 295 throw new IndexOutOfBoundsException(); 296 } 297 298 if (val[off] < 0) { 299 mag = makePositive(val, off, len); 300 signum = -1; 301 } else { 302 mag = stripLeadingZeroBytes(val, off, len); 303 signum = (mag.length == 0 ? 0 : 1); 304 } 305 if (mag.length >= MAX_MAG_LENGTH) { 306 checkRange(); 307 } 308 } 309 310 /** 311 * Translates a byte array containing the two's-complement binary 312 * representation of a BigInteger into a BigInteger. The input array is 313 * assumed to be in <i>big-endian</i> byte-order: the most significant 314 * byte is in the zeroth element. The {@code val} array is assumed to be 315 * unchanged for the duration of the constructor call. 316 * 317 * @param val big-endian two's-complement binary representation of a 318 * BigInteger. 319 * @throws NumberFormatException {@code val} is zero bytes long. 320 */ 321 public BigInteger(byte[] val) { 322 this(val, 0, val.length); 323 } 324 325 /** 326 * This private constructor translates an int array containing the 327 * two's-complement binary representation of a BigInteger into a 328 * BigInteger. The input array is assumed to be in <i>big-endian</i> 329 * int-order: the most significant int is in the zeroth element. The 330 * {@code val} array is assumed to be unchanged for the duration of 331 * the constructor call. 332 */ 333 private BigInteger(int[] val) { 334 if (val.length == 0) 335 throw new NumberFormatException("Zero length BigInteger"); 336 337 if (val[0] < 0) { 338 mag = makePositive(val); 339 signum = -1; 340 } else { 341 mag = trustedStripLeadingZeroInts(val); 342 signum = (mag.length == 0 ? 0 : 1); 343 } 344 if (mag.length >= MAX_MAG_LENGTH) { 345 checkRange(); 346 } 347 } 348 349 /** 350 * Translates the sign-magnitude representation of a BigInteger into a 351 * BigInteger. The sign is represented as an integer signum value: -1 for 352 * negative, 0 for zero, or 1 for positive. The magnitude is a sub-array of 353 * a byte array in <i>big-endian</i> byte-order: the most significant byte 354 * is the element at index {@code off}. A zero value of the length 355 * {@code len} is permissible, and will result in a BigInteger value of 0, 356 * whether signum is -1, 0 or 1. The {@code magnitude} array is assumed to 357 * be unchanged for the duration of the constructor call. 358 * 359 * An {@code IndexOutOfBoundsException} is thrown if the length of the array 360 * {@code magnitude} is non-zero and either {@code off} is negative, 361 * {@code len} is negative, or {@code off+len} is greater than the length of 362 * {@code magnitude}. 363 * 364 * @param signum signum of the number (-1 for negative, 0 for zero, 1 365 * for positive). 366 * @param magnitude big-endian binary representation of the magnitude of 367 * the number. 368 * @param off the start offset of the binary representation. 369 * @param len the number of bytes to use. 370 * @throws NumberFormatException {@code signum} is not one of the three 371 * legal values (-1, 0, and 1), or {@code signum} is 0 and 372 * {@code magnitude} contains one or more non-zero bytes. 373 * @throws IndexOutOfBoundsException if the provided array offset and 374 * length would cause an index into the byte array to be 375 * negative or greater than or equal to the array length. 376 * @since 1.9 377 */ 378 public BigInteger(int signum, byte[] magnitude, int off, int len) { 379 if (signum < -1 || signum > 1) { 380 throw(new NumberFormatException("Invalid signum value")); 381 } else if ((off < 0) || (len < 0) || 382 (len > 0 && 383 ((off >= magnitude.length) || 384 (len > magnitude.length - off)))) { // 0 <= off < magnitude.length 385 throw new IndexOutOfBoundsException(); 386 } 387 388 // stripLeadingZeroBytes() returns a zero length array if len == 0 389 this.mag = stripLeadingZeroBytes(magnitude, off, len); 390 391 if (this.mag.length == 0) { 392 this.signum = 0; 393 } else { 394 if (signum == 0) 395 throw(new NumberFormatException("signum-magnitude mismatch")); 396 this.signum = signum; 397 } 398 if (mag.length >= MAX_MAG_LENGTH) { 399 checkRange(); 400 } 401 } 402 403 /** 404 * Translates the sign-magnitude representation of a BigInteger into a 405 * BigInteger. The sign is represented as an integer signum value: -1 for 406 * negative, 0 for zero, or 1 for positive. The magnitude is a byte array 407 * in <i>big-endian</i> byte-order: the most significant byte is the 408 * zeroth element. A zero-length magnitude array is permissible, and will 409 * result in a BigInteger value of 0, whether signum is -1, 0 or 1. The 410 * {@code magnitude} array is assumed to be unchanged for the duration of 411 * the constructor call. 412 * 413 * @param signum signum of the number (-1 for negative, 0 for zero, 1 414 * for positive). 415 * @param magnitude big-endian binary representation of the magnitude of 416 * the number. 417 * @throws NumberFormatException {@code signum} is not one of the three 418 * legal values (-1, 0, and 1), or {@code signum} is 0 and 419 * {@code magnitude} contains one or more non-zero bytes. 420 */ 421 public BigInteger(int signum, byte[] magnitude) { 422 this(signum, magnitude, 0, magnitude.length); 423 } 424 425 /** 426 * A constructor for internal use that translates the sign-magnitude 427 * representation of a BigInteger into a BigInteger. It checks the 428 * arguments and copies the magnitude so this constructor would be 429 * safe for external use. The {@code magnitude} array is assumed to be 430 * unchanged for the duration of the constructor call. 431 */ 432 private BigInteger(int signum, int[] magnitude) { 433 this.mag = stripLeadingZeroInts(magnitude); 434 435 if (signum < -1 || signum > 1) 436 throw(new NumberFormatException("Invalid signum value")); 437 438 if (this.mag.length == 0) { 439 this.signum = 0; 440 } else { 441 if (signum == 0) 442 throw(new NumberFormatException("signum-magnitude mismatch")); 443 this.signum = signum; 444 } 445 if (mag.length >= MAX_MAG_LENGTH) { 446 checkRange(); 447 } 448 } 449 450 /** 451 * Translates the String representation of a BigInteger in the 452 * specified radix into a BigInteger. The String representation 453 * consists of an optional minus or plus sign followed by a 454 * sequence of one or more digits in the specified radix. The 455 * character-to-digit mapping is provided by {@code 456 * Character.digit}. The String may not contain any extraneous 457 * characters (whitespace, for example). 458 * 459 * @param val String representation of BigInteger. 460 * @param radix radix to be used in interpreting {@code val}. 461 * @throws NumberFormatException {@code val} is not a valid representation 462 * of a BigInteger in the specified radix, or {@code radix} is 463 * outside the range from {@link Character#MIN_RADIX} to 464 * {@link Character#MAX_RADIX}, inclusive. 465 * @see Character#digit 466 */ 467 public BigInteger(String val, int radix) { 468 int cursor = 0, numDigits; 469 final int len = val.length(); 470 471 if (radix < Character.MIN_RADIX || radix > Character.MAX_RADIX) 472 throw new NumberFormatException("Radix out of range"); 473 if (len == 0) 474 throw new NumberFormatException("Zero length BigInteger"); 475 476 // Check for at most one leading sign 477 int sign = 1; 478 int index1 = val.lastIndexOf('-'); 479 int index2 = val.lastIndexOf('+'); 480 if (index1 >= 0) { 481 if (index1 != 0 || index2 >= 0) { 482 throw new NumberFormatException("Illegal embedded sign character"); 483 } 484 sign = -1; 485 cursor = 1; 486 } else if (index2 >= 0) { 487 if (index2 != 0) { 488 throw new NumberFormatException("Illegal embedded sign character"); 489 } 490 cursor = 1; 491 } 492 if (cursor == len) 493 throw new NumberFormatException("Zero length BigInteger"); 494 495 // Skip leading zeros and compute number of digits in magnitude 496 while (cursor < len && 497 Character.digit(val.charAt(cursor), radix) == 0) { 498 cursor++; 499 } 500 501 if (cursor == len) { 502 signum = 0; 503 mag = ZERO.mag; 504 return; 505 } 506 507 numDigits = len - cursor; 508 signum = sign; 509 510 // Pre-allocate array of expected size. May be too large but can 511 // never be too small. Typically exact. 512 long numBits = ((numDigits * bitsPerDigit[radix]) >>> 10) + 1; 513 if (numBits + 31 >= (1L << 32)) { 514 reportOverflow(); 515 } 516 int numWords = (int) (numBits + 31) >>> 5; 517 int[] magnitude = new int[numWords]; 518 519 // Process first (potentially short) digit group 520 int firstGroupLen = numDigits % digitsPerInt[radix]; 521 if (firstGroupLen == 0) 522 firstGroupLen = digitsPerInt[radix]; 523 String group = val.substring(cursor, cursor += firstGroupLen); 524 magnitude[numWords - 1] = Integer.parseInt(group, radix); 525 if (magnitude[numWords - 1] < 0) 526 throw new NumberFormatException("Illegal digit"); 527 528 // Process remaining digit groups 529 int superRadix = intRadix[radix]; 530 int groupVal = 0; 531 while (cursor < len) { 532 group = val.substring(cursor, cursor += digitsPerInt[radix]); 533 groupVal = Integer.parseInt(group, radix); 534 if (groupVal < 0) 535 throw new NumberFormatException("Illegal digit"); 536 destructiveMulAdd(magnitude, superRadix, groupVal); 537 } 538 // Required for cases where the array was overallocated. 539 mag = trustedStripLeadingZeroInts(magnitude); 540 if (mag.length >= MAX_MAG_LENGTH) { 541 checkRange(); 542 } 543 } 544 545 /* 546 * Constructs a new BigInteger using a char array with radix=10. 547 * Sign is precalculated outside and not allowed in the val. The {@code val} 548 * array is assumed to be unchanged for the duration of the constructor 549 * call. 550 */ 551 BigInteger(char[] val, int sign, int len) { 552 int cursor = 0, numDigits; 553 554 // Skip leading zeros and compute number of digits in magnitude 555 while (cursor < len && Character.digit(val[cursor], 10) == 0) { 556 cursor++; 557 } 558 if (cursor == len) { 559 signum = 0; 560 mag = ZERO.mag; 561 return; 562 } 563 564 numDigits = len - cursor; 565 signum = sign; 566 // Pre-allocate array of expected size 567 int numWords; 568 if (len < 10) { 569 numWords = 1; 570 } else { 571 long numBits = ((numDigits * bitsPerDigit[10]) >>> 10) + 1; 572 if (numBits + 31 >= (1L << 32)) { 573 reportOverflow(); 574 } 575 numWords = (int) (numBits + 31) >>> 5; 576 } 577 int[] magnitude = new int[numWords]; 578 579 // Process first (potentially short) digit group 580 int firstGroupLen = numDigits % digitsPerInt[10]; 581 if (firstGroupLen == 0) 582 firstGroupLen = digitsPerInt[10]; 583 magnitude[numWords - 1] = parseInt(val, cursor, cursor += firstGroupLen); 584 585 // Process remaining digit groups 586 while (cursor < len) { 587 int groupVal = parseInt(val, cursor, cursor += digitsPerInt[10]); 588 destructiveMulAdd(magnitude, intRadix[10], groupVal); 589 } 590 mag = trustedStripLeadingZeroInts(magnitude); 591 if (mag.length >= MAX_MAG_LENGTH) { 592 checkRange(); 593 } 594 } 595 596 // Create an integer with the digits between the two indexes 597 // Assumes start < end. The result may be negative, but it 598 // is to be treated as an unsigned value. 599 private int parseInt(char[] source, int start, int end) { 600 int result = Character.digit(source[start++], 10); 601 if (result == -1) 602 throw new NumberFormatException(new String(source)); 603 604 for (int index = start; index < end; index++) { 605 int nextVal = Character.digit(source[index], 10); 606 if (nextVal == -1) 607 throw new NumberFormatException(new String(source)); 608 result = 10*result + nextVal; 609 } 610 611 return result; 612 } 613 614 // bitsPerDigit in the given radix times 1024 615 // Rounded up to avoid underallocation. 616 private static long bitsPerDigit[] = { 0, 0, 617 1024, 1624, 2048, 2378, 2648, 2875, 3072, 3247, 3402, 3543, 3672, 618 3790, 3899, 4001, 4096, 4186, 4271, 4350, 4426, 4498, 4567, 4633, 619 4696, 4756, 4814, 4870, 4923, 4975, 5025, 5074, 5120, 5166, 5210, 620 5253, 5295}; 621 622 // Multiply x array times word y in place, and add word z 623 private static void destructiveMulAdd(int[] x, int y, int z) { 624 // Perform the multiplication word by word 625 long ylong = y & LONG_MASK; 626 long zlong = z & LONG_MASK; 627 int len = x.length; 628 629 long product = 0; 630 long carry = 0; 631 for (int i = len-1; i >= 0; i--) { 632 product = ylong * (x[i] & LONG_MASK) + carry; 633 x[i] = (int)product; 634 carry = product >>> 32; 635 } 636 637 // Perform the addition 638 long sum = (x[len-1] & LONG_MASK) + zlong; 639 x[len-1] = (int)sum; 640 carry = sum >>> 32; 641 for (int i = len-2; i >= 0; i--) { 642 sum = (x[i] & LONG_MASK) + carry; 643 x[i] = (int)sum; 644 carry = sum >>> 32; 645 } 646 } 647 648 /** 649 * Translates the decimal String representation of a BigInteger into a 650 * BigInteger. The String representation consists of an optional minus 651 * sign followed by a sequence of one or more decimal digits. The 652 * character-to-digit mapping is provided by {@code Character.digit}. 653 * The String may not contain any extraneous characters (whitespace, for 654 * example). 655 * 656 * @param val decimal String representation of BigInteger. 657 * @throws NumberFormatException {@code val} is not a valid representation 658 * of a BigInteger. 659 * @see Character#digit 660 */ 661 public BigInteger(String val) { 662 this(val, 10); 663 } 664 665 /** 666 * Constructs a randomly generated BigInteger, uniformly distributed over 667 * the range 0 to (2<sup>{@code numBits}</sup> - 1), inclusive. 668 * The uniformity of the distribution assumes that a fair source of random 669 * bits is provided in {@code rnd}. Note that this constructor always 670 * constructs a non-negative BigInteger. 671 * 672 * @param numBits maximum bitLength of the new BigInteger. 673 * @param rnd source of randomness to be used in computing the new 674 * BigInteger. 675 * @throws IllegalArgumentException {@code numBits} is negative. 676 * @see #bitLength() 677 */ 678 public BigInteger(int numBits, Random rnd) { 679 this(1, randomBits(numBits, rnd)); 680 } 681 682 private static byte[] randomBits(int numBits, Random rnd) { 683 if (numBits < 0) 684 throw new IllegalArgumentException("numBits must be non-negative"); 685 int numBytes = (int)(((long)numBits+7)/8); // avoid overflow 686 byte[] randomBits = new byte[numBytes]; 687 688 // Generate random bytes and mask out any excess bits 689 if (numBytes > 0) { 690 rnd.nextBytes(randomBits); 691 int excessBits = 8*numBytes - numBits; 692 randomBits[0] &= (1 << (8-excessBits)) - 1; 693 } 694 return randomBits; 695 } 696 697 /** 698 * Constructs a randomly generated positive BigInteger that is probably 699 * prime, with the specified bitLength. 700 * 701 * <p>It is recommended that the {@link #probablePrime probablePrime} 702 * method be used in preference to this constructor unless there 703 * is a compelling need to specify a certainty. 704 * 705 * @param bitLength bitLength of the returned BigInteger. 706 * @param certainty a measure of the uncertainty that the caller is 707 * willing to tolerate. The probability that the new BigInteger 708 * represents a prime number will exceed 709 * (1 - 1/2<sup>{@code certainty}</sup>). The execution time of 710 * this constructor is proportional to the value of this parameter. 711 * @param rnd source of random bits used to select candidates to be 712 * tested for primality. 713 * @throws ArithmeticException {@code bitLength < 2} or {@code bitLength} is too large. 714 * @see #bitLength() 715 */ 716 public BigInteger(int bitLength, int certainty, Random rnd) { 717 BigInteger prime; 718 719 if (bitLength < 2) 720 throw new ArithmeticException("bitLength < 2"); 721 prime = (bitLength < SMALL_PRIME_THRESHOLD 722 ? smallPrime(bitLength, certainty, rnd) 723 : largePrime(bitLength, certainty, rnd)); 724 signum = 1; 725 mag = prime.mag; 726 } 727 728 // Minimum size in bits that the requested prime number has 729 // before we use the large prime number generating algorithms. 730 // The cutoff of 95 was chosen empirically for best performance. 731 private static final int SMALL_PRIME_THRESHOLD = 95; 732 733 // Certainty required to meet the spec of probablePrime 734 private static final int DEFAULT_PRIME_CERTAINTY = 100; 735 736 /** 737 * Returns a positive BigInteger that is probably prime, with the 738 * specified bitLength. The probability that a BigInteger returned 739 * by this method is composite does not exceed 2<sup>-100</sup>. 740 * 741 * @param bitLength bitLength of the returned BigInteger. 742 * @param rnd source of random bits used to select candidates to be 743 * tested for primality. 744 * @return a BigInteger of {@code bitLength} bits that is probably prime 745 * @throws ArithmeticException {@code bitLength < 2} or {@code bitLength} is too large. 746 * @see #bitLength() 747 * @since 1.4 748 */ 749 public static BigInteger probablePrime(int bitLength, Random rnd) { 750 if (bitLength < 2) 751 throw new ArithmeticException("bitLength < 2"); 752 753 return (bitLength < SMALL_PRIME_THRESHOLD ? 754 smallPrime(bitLength, DEFAULT_PRIME_CERTAINTY, rnd) : 755 largePrime(bitLength, DEFAULT_PRIME_CERTAINTY, rnd)); 756 } 757 758 /** 759 * Find a random number of the specified bitLength that is probably prime. 760 * This method is used for smaller primes, its performance degrades on 761 * larger bitlengths. 762 * 763 * This method assumes bitLength > 1. 764 */ 765 private static BigInteger smallPrime(int bitLength, int certainty, Random rnd) { 766 int magLen = (bitLength + 31) >>> 5; 767 int temp[] = new int[magLen]; 768 int highBit = 1 << ((bitLength+31) & 0x1f); // High bit of high int 769 int highMask = (highBit << 1) - 1; // Bits to keep in high int 770 771 while (true) { 772 // Construct a candidate 773 for (int i=0; i < magLen; i++) 774 temp[i] = rnd.nextInt(); 775 temp[0] = (temp[0] & highMask) | highBit; // Ensure exact length 776 if (bitLength > 2) 777 temp[magLen-1] |= 1; // Make odd if bitlen > 2 778 779 BigInteger p = new BigInteger(temp, 1); 780 781 // Do cheap "pre-test" if applicable 782 if (bitLength > 6) { 783 long r = p.remainder(SMALL_PRIME_PRODUCT).longValue(); 784 if ((r%3==0) || (r%5==0) || (r%7==0) || (r%11==0) || 785 (r%13==0) || (r%17==0) || (r%19==0) || (r%23==0) || 786 (r%29==0) || (r%31==0) || (r%37==0) || (r%41==0)) 787 continue; // Candidate is composite; try another 788 } 789 790 // All candidates of bitLength 2 and 3 are prime by this point 791 if (bitLength < 4) 792 return p; 793 794 // Do expensive test if we survive pre-test (or it's inapplicable) 795 if (p.primeToCertainty(certainty, rnd)) 796 return p; 797 } 798 } 799 800 private static final BigInteger SMALL_PRIME_PRODUCT 801 = valueOf(3L*5*7*11*13*17*19*23*29*31*37*41); 802 803 /** 804 * Find a random number of the specified bitLength that is probably prime. 805 * This method is more appropriate for larger bitlengths since it uses 806 * a sieve to eliminate most composites before using a more expensive 807 * test. 808 */ 809 private static BigInteger largePrime(int bitLength, int certainty, Random rnd) { 810 BigInteger p; 811 p = new BigInteger(bitLength, rnd).setBit(bitLength-1); 812 p.mag[p.mag.length-1] &= 0xfffffffe; 813 814 // Use a sieve length likely to contain the next prime number 815 int searchLen = getPrimeSearchLen(bitLength); 816 BitSieve searchSieve = new BitSieve(p, searchLen); 817 BigInteger candidate = searchSieve.retrieve(p, certainty, rnd); 818 819 while ((candidate == null) || (candidate.bitLength() != bitLength)) { 820 p = p.add(BigInteger.valueOf(2*searchLen)); 821 if (p.bitLength() != bitLength) 822 p = new BigInteger(bitLength, rnd).setBit(bitLength-1); 823 p.mag[p.mag.length-1] &= 0xfffffffe; 824 searchSieve = new BitSieve(p, searchLen); 825 candidate = searchSieve.retrieve(p, certainty, rnd); 826 } 827 return candidate; 828 } 829 830 /** 831 * Returns the first integer greater than this {@code BigInteger} that 832 * is probably prime. The probability that the number returned by this 833 * method is composite does not exceed 2<sup>-100</sup>. This method will 834 * never skip over a prime when searching: if it returns {@code p}, there 835 * is no prime {@code q} such that {@code this < q < p}. 836 * 837 * @return the first integer greater than this {@code BigInteger} that 838 * is probably prime. 839 * @throws ArithmeticException {@code this < 0} or {@code this} is too large. 840 * @since 1.5 841 */ 842 public BigInteger nextProbablePrime() { 843 if (this.signum < 0) 844 throw new ArithmeticException("start < 0: " + this); 845 846 // Handle trivial cases 847 if ((this.signum == 0) || this.equals(ONE)) 848 return TWO; 849 850 BigInteger result = this.add(ONE); 851 852 // Fastpath for small numbers 853 if (result.bitLength() < SMALL_PRIME_THRESHOLD) { 854 855 // Ensure an odd number 856 if (!result.testBit(0)) 857 result = result.add(ONE); 858 859 while (true) { 860 // Do cheap "pre-test" if applicable 861 if (result.bitLength() > 6) { 862 long r = result.remainder(SMALL_PRIME_PRODUCT).longValue(); 863 if ((r%3==0) || (r%5==0) || (r%7==0) || (r%11==0) || 864 (r%13==0) || (r%17==0) || (r%19==0) || (r%23==0) || 865 (r%29==0) || (r%31==0) || (r%37==0) || (r%41==0)) { 866 result = result.add(TWO); 867 continue; // Candidate is composite; try another 868 } 869 } 870 871 // All candidates of bitLength 2 and 3 are prime by this point 872 if (result.bitLength() < 4) 873 return result; 874 875 // The expensive test 876 if (result.primeToCertainty(DEFAULT_PRIME_CERTAINTY, null)) 877 return result; 878 879 result = result.add(TWO); 880 } 881 } 882 883 // Start at previous even number 884 if (result.testBit(0)) 885 result = result.subtract(ONE); 886 887 // Looking for the next large prime 888 int searchLen = getPrimeSearchLen(result.bitLength()); 889 890 while (true) { 891 BitSieve searchSieve = new BitSieve(result, searchLen); 892 BigInteger candidate = searchSieve.retrieve(result, 893 DEFAULT_PRIME_CERTAINTY, null); 894 if (candidate != null) 895 return candidate; 896 result = result.add(BigInteger.valueOf(2 * searchLen)); 897 } 898 } 899 900 private static int getPrimeSearchLen(int bitLength) { 901 if (bitLength > PRIME_SEARCH_BIT_LENGTH_LIMIT + 1) { 902 throw new ArithmeticException("Prime search implementation restriction on bitLength"); 903 } 904 return bitLength / 20 * 64; 905 } 906 907 /** 908 * Returns {@code true} if this BigInteger is probably prime, 909 * {@code false} if it's definitely composite. 910 * 911 * This method assumes bitLength > 2. 912 * 913 * @param certainty a measure of the uncertainty that the caller is 914 * willing to tolerate: if the call returns {@code true} 915 * the probability that this BigInteger is prime exceeds 916 * {@code (1 - 1/2<sup>certainty</sup>)}. The execution time of 917 * this method is proportional to the value of this parameter. 918 * @return {@code true} if this BigInteger is probably prime, 919 * {@code false} if it's definitely composite. 920 */ 921 boolean primeToCertainty(int certainty, Random random) { 922 int rounds = 0; 923 int n = (Math.min(certainty, Integer.MAX_VALUE-1)+1)/2; 924 925 // The relationship between the certainty and the number of rounds 926 // we perform is given in the draft standard ANSI X9.80, "PRIME 927 // NUMBER GENERATION, PRIMALITY TESTING, AND PRIMALITY CERTIFICATES". 928 int sizeInBits = this.bitLength(); 929 if (sizeInBits < 100) { 930 rounds = 50; 931 rounds = n < rounds ? n : rounds; 932 return passesMillerRabin(rounds, random); 933 } 934 935 if (sizeInBits < 256) { 936 rounds = 27; 937 } else if (sizeInBits < 512) { 938 rounds = 15; 939 } else if (sizeInBits < 768) { 940 rounds = 8; 941 } else if (sizeInBits < 1024) { 942 rounds = 4; 943 } else { 944 rounds = 2; 945 } 946 rounds = n < rounds ? n : rounds; 947 948 return passesMillerRabin(rounds, random) && passesLucasLehmer(); 949 } 950 951 /** 952 * Returns true iff this BigInteger is a Lucas-Lehmer probable prime. 953 * 954 * The following assumptions are made: 955 * This BigInteger is a positive, odd number. 956 */ 957 private boolean passesLucasLehmer() { 958 BigInteger thisPlusOne = this.add(ONE); 959 960 // Step 1 961 int d = 5; 962 while (jacobiSymbol(d, this) != -1) { 963 // 5, -7, 9, -11, ... 964 d = (d < 0) ? Math.abs(d)+2 : -(d+2); 965 } 966 967 // Step 2 968 BigInteger u = lucasLehmerSequence(d, thisPlusOne, this); 969 970 // Step 3 971 return u.mod(this).equals(ZERO); 972 } 973 974 /** 975 * Computes Jacobi(p,n). 976 * Assumes n positive, odd, n>=3. 977 */ 978 private static int jacobiSymbol(int p, BigInteger n) { 979 if (p == 0) 980 return 0; 981 982 // Algorithm and comments adapted from Colin Plumb's C library. 983 int j = 1; 984 int u = n.mag[n.mag.length-1]; 985 986 // Make p positive 987 if (p < 0) { 988 p = -p; 989 int n8 = u & 7; 990 if ((n8 == 3) || (n8 == 7)) 991 j = -j; // 3 (011) or 7 (111) mod 8 992 } 993 994 // Get rid of factors of 2 in p 995 while ((p & 3) == 0) 996 p >>= 2; 997 if ((p & 1) == 0) { 998 p >>= 1; 999 if (((u ^ (u>>1)) & 2) != 0) 1000 j = -j; // 3 (011) or 5 (101) mod 8 1001 } 1002 if (p == 1) 1003 return j; 1004 // Then, apply quadratic reciprocity 1005 if ((p & u & 2) != 0) // p = u = 3 (mod 4)? 1006 j = -j; 1007 // And reduce u mod p 1008 u = n.mod(BigInteger.valueOf(p)).intValue(); 1009 1010 // Now compute Jacobi(u,p), u < p 1011 while (u != 0) { 1012 while ((u & 3) == 0) 1013 u >>= 2; 1014 if ((u & 1) == 0) { 1015 u >>= 1; 1016 if (((p ^ (p>>1)) & 2) != 0) 1017 j = -j; // 3 (011) or 5 (101) mod 8 1018 } 1019 if (u == 1) 1020 return j; 1021 // Now both u and p are odd, so use quadratic reciprocity 1022 assert (u < p); 1023 int t = u; u = p; p = t; 1024 if ((u & p & 2) != 0) // u = p = 3 (mod 4)? 1025 j = -j; 1026 // Now u >= p, so it can be reduced 1027 u %= p; 1028 } 1029 return 0; 1030 } 1031 1032 private static BigInteger lucasLehmerSequence(int z, BigInteger k, BigInteger n) { 1033 BigInteger d = BigInteger.valueOf(z); 1034 BigInteger u = ONE; BigInteger u2; 1035 BigInteger v = ONE; BigInteger v2; 1036 1037 for (int i=k.bitLength()-2; i >= 0; i--) { 1038 u2 = u.multiply(v).mod(n); 1039 1040 v2 = v.square().add(d.multiply(u.square())).mod(n); 1041 if (v2.testBit(0)) 1042 v2 = v2.subtract(n); 1043 1044 v2 = v2.shiftRight(1); 1045 1046 u = u2; v = v2; 1047 if (k.testBit(i)) { 1048 u2 = u.add(v).mod(n); 1049 if (u2.testBit(0)) 1050 u2 = u2.subtract(n); 1051 1052 u2 = u2.shiftRight(1); 1053 v2 = v.add(d.multiply(u)).mod(n); 1054 if (v2.testBit(0)) 1055 v2 = v2.subtract(n); 1056 v2 = v2.shiftRight(1); 1057 1058 u = u2; v = v2; 1059 } 1060 } 1061 return u; 1062 } 1063 1064 /** 1065 * Returns true iff this BigInteger passes the specified number of 1066 * Miller-Rabin tests. This test is taken from the DSA spec (NIST FIPS 1067 * 186-2). 1068 * 1069 * The following assumptions are made: 1070 * This BigInteger is a positive, odd number greater than 2. 1071 * iterations<=50. 1072 */ 1073 private boolean passesMillerRabin(int iterations, Random rnd) { 1074 // Find a and m such that m is odd and this == 1 + 2**a * m 1075 BigInteger thisMinusOne = this.subtract(ONE); 1076 BigInteger m = thisMinusOne; 1077 int a = m.getLowestSetBit(); 1078 m = m.shiftRight(a); 1079 1080 // Do the tests 1081 if (rnd == null) { 1082 rnd = ThreadLocalRandom.current(); 1083 } 1084 for (int i=0; i < iterations; i++) { 1085 // Generate a uniform random on (1, this) 1086 BigInteger b; 1087 do { 1088 b = new BigInteger(this.bitLength(), rnd); 1089 } while (b.compareTo(ONE) <= 0 || b.compareTo(this) >= 0); 1090 1091 int j = 0; 1092 BigInteger z = b.modPow(m, this); 1093 while (!((j == 0 && z.equals(ONE)) || z.equals(thisMinusOne))) { 1094 if (j > 0 && z.equals(ONE) || ++j == a) 1095 return false; 1096 z = z.modPow(TWO, this); 1097 } 1098 } 1099 return true; 1100 } 1101 1102 /** 1103 * This internal constructor differs from its public cousin 1104 * with the arguments reversed in two ways: it assumes that its 1105 * arguments are correct, and it doesn't copy the magnitude array. 1106 */ 1107 BigInteger(int[] magnitude, int signum) { 1108 this.signum = (magnitude.length == 0 ? 0 : signum); 1109 this.mag = magnitude; 1110 if (mag.length >= MAX_MAG_LENGTH) { 1111 checkRange(); 1112 } 1113 } 1114 1115 /** 1116 * This private constructor is for internal use and assumes that its 1117 * arguments are correct. The {@code magnitude} array is assumed to be 1118 * unchanged for the duration of the constructor call. 1119 */ 1120 private BigInteger(byte[] magnitude, int signum) { 1121 this.signum = (magnitude.length == 0 ? 0 : signum); 1122 this.mag = stripLeadingZeroBytes(magnitude, 0, magnitude.length); 1123 if (mag.length >= MAX_MAG_LENGTH) { 1124 checkRange(); 1125 } 1126 } 1127 1128 /** 1129 * Throws an {@code ArithmeticException} if the {@code BigInteger} would be 1130 * out of the supported range. 1131 * 1132 * @throws ArithmeticException if {@code this} exceeds the supported range. 1133 */ 1134 private void checkRange() { 1135 if (mag.length > MAX_MAG_LENGTH || mag.length == MAX_MAG_LENGTH && mag[0] < 0) { 1136 reportOverflow(); 1137 } 1138 } 1139 1140 private static void reportOverflow() { 1141 throw new ArithmeticException("BigInteger would overflow supported range"); 1142 } 1143 1144 //Static Factory Methods 1145 1146 /** 1147 * Returns a BigInteger whose value is equal to that of the 1148 * specified {@code long}. This "static factory method" is 1149 * provided in preference to a ({@code long}) constructor 1150 * because it allows for reuse of frequently used BigIntegers. 1151 * 1152 * @param val value of the BigInteger to return. 1153 * @return a BigInteger with the specified value. 1154 */ 1155 public static BigInteger valueOf(long val) { 1156 // If -MAX_CONSTANT < val < MAX_CONSTANT, return stashed constant 1157 if (val == 0) 1158 return ZERO; 1159 if (val > 0 && val <= MAX_CONSTANT) 1160 return posConst[(int) val]; 1161 else if (val < 0 && val >= -MAX_CONSTANT) 1162 return negConst[(int) -val]; 1163 1164 return new BigInteger(val); 1165 } 1166 1167 /** 1168 * Constructs a BigInteger with the specified value, which may not be zero. 1169 */ 1170 private BigInteger(long val) { 1171 if (val < 0) { 1172 val = -val; 1173 signum = -1; 1174 } else { 1175 signum = 1; 1176 } 1177 1178 int highWord = (int)(val >>> 32); 1179 if (highWord == 0) { 1180 mag = new int[1]; 1181 mag[0] = (int)val; 1182 } else { 1183 mag = new int[2]; 1184 mag[0] = highWord; 1185 mag[1] = (int)val; 1186 } 1187 } 1188 1189 /** 1190 * Returns a BigInteger with the given two's complement representation. 1191 * Assumes that the input array will not be modified (the returned 1192 * BigInteger will reference the input array if feasible). 1193 */ 1194 private static BigInteger valueOf(int val[]) { 1195 return (val[0] > 0 ? new BigInteger(val, 1) : new BigInteger(val)); 1196 } 1197 1198 // Constants 1199 1200 /** 1201 * Initialize static constant array when class is loaded. 1202 */ 1203 private final static int MAX_CONSTANT = 16; 1204 private static BigInteger posConst[] = new BigInteger[MAX_CONSTANT+1]; 1205 private static BigInteger negConst[] = new BigInteger[MAX_CONSTANT+1]; 1206 1207 /** 1208 * The cache of powers of each radix. This allows us to not have to 1209 * recalculate powers of radix^(2^n) more than once. This speeds 1210 * Schoenhage recursive base conversion significantly. 1211 */ 1212 private static volatile BigInteger[][] powerCache; 1213 1214 /** The cache of logarithms of radices for base conversion. */ 1215 private static final double[] logCache; 1216 1217 /** The natural log of 2. This is used in computing cache indices. */ 1218 private static final double LOG_TWO = Math.log(2.0); 1219 1220 static { 1221 for (int i = 1; i <= MAX_CONSTANT; i++) { 1222 int[] magnitude = new int[1]; 1223 magnitude[0] = i; 1224 posConst[i] = new BigInteger(magnitude, 1); 1225 negConst[i] = new BigInteger(magnitude, -1); 1226 } 1227 1228 /* 1229 * Initialize the cache of radix^(2^x) values used for base conversion 1230 * with just the very first value. Additional values will be created 1231 * on demand. 1232 */ 1233 powerCache = new BigInteger[Character.MAX_RADIX+1][]; 1234 logCache = new double[Character.MAX_RADIX+1]; 1235 1236 for (int i=Character.MIN_RADIX; i <= Character.MAX_RADIX; i++) { 1237 powerCache[i] = new BigInteger[] { BigInteger.valueOf(i) }; 1238 logCache[i] = Math.log(i); 1239 } 1240 } 1241 1242 /** 1243 * The BigInteger constant zero. 1244 * 1245 * @since 1.2 1246 */ 1247 public static final BigInteger ZERO = new BigInteger(new int[0], 0); 1248 1249 /** 1250 * The BigInteger constant one. 1251 * 1252 * @since 1.2 1253 */ 1254 public static final BigInteger ONE = valueOf(1); 1255 1256 /** 1257 * The BigInteger constant two. (Not exported.) 1258 */ 1259 private static final BigInteger TWO = valueOf(2); 1260 1261 /** 1262 * The BigInteger constant -1. (Not exported.) 1263 */ 1264 private static final BigInteger NEGATIVE_ONE = valueOf(-1); 1265 1266 /** 1267 * The BigInteger constant ten. 1268 * 1269 * @since 1.5 1270 */ 1271 public static final BigInteger TEN = valueOf(10); 1272 1273 // Arithmetic Operations 1274 1275 /** 1276 * Returns a BigInteger whose value is {@code (this + val)}. 1277 * 1278 * @param val value to be added to this BigInteger. 1279 * @return {@code this + val} 1280 */ 1281 public BigInteger add(BigInteger val) { 1282 if (val.signum == 0) 1283 return this; 1284 if (signum == 0) 1285 return val; 1286 if (val.signum == signum) 1287 return new BigInteger(add(mag, val.mag), signum); 1288 1289 int cmp = compareMagnitude(val); 1290 if (cmp == 0) 1291 return ZERO; 1292 int[] resultMag = (cmp > 0 ? subtract(mag, val.mag) 1293 : subtract(val.mag, mag)); 1294 resultMag = trustedStripLeadingZeroInts(resultMag); 1295 1296 return new BigInteger(resultMag, cmp == signum ? 1 : -1); 1297 } 1298 1299 /** 1300 * Package private methods used by BigDecimal code to add a BigInteger 1301 * with a long. Assumes val is not equal to INFLATED. 1302 */ 1303 BigInteger add(long val) { 1304 if (val == 0) 1305 return this; 1306 if (signum == 0) 1307 return valueOf(val); 1308 if (Long.signum(val) == signum) 1309 return new BigInteger(add(mag, Math.abs(val)), signum); 1310 int cmp = compareMagnitude(val); 1311 if (cmp == 0) 1312 return ZERO; 1313 int[] resultMag = (cmp > 0 ? subtract(mag, Math.abs(val)) : subtract(Math.abs(val), mag)); 1314 resultMag = trustedStripLeadingZeroInts(resultMag); 1315 return new BigInteger(resultMag, cmp == signum ? 1 : -1); 1316 } 1317 1318 /** 1319 * Adds the contents of the int array x and long value val. This 1320 * method allocates a new int array to hold the answer and returns 1321 * a reference to that array. Assumes x.length > 0 and val is 1322 * non-negative 1323 */ 1324 private static int[] add(int[] x, long val) { 1325 int[] y; 1326 long sum = 0; 1327 int xIndex = x.length; 1328 int[] result; 1329 int highWord = (int)(val >>> 32); 1330 if (highWord == 0) { 1331 result = new int[xIndex]; 1332 sum = (x[--xIndex] & LONG_MASK) + val; 1333 result[xIndex] = (int)sum; 1334 } else { 1335 if (xIndex == 1) { 1336 result = new int[2]; 1337 sum = val + (x[0] & LONG_MASK); 1338 result[1] = (int)sum; 1339 result[0] = (int)(sum >>> 32); 1340 return result; 1341 } else { 1342 result = new int[xIndex]; 1343 sum = (x[--xIndex] & LONG_MASK) + (val & LONG_MASK); 1344 result[xIndex] = (int)sum; 1345 sum = (x[--xIndex] & LONG_MASK) + (highWord & LONG_MASK) + (sum >>> 32); 1346 result[xIndex] = (int)sum; 1347 } 1348 } 1349 // Copy remainder of longer number while carry propagation is required 1350 boolean carry = (sum >>> 32 != 0); 1351 while (xIndex > 0 && carry) 1352 carry = ((result[--xIndex] = x[xIndex] + 1) == 0); 1353 // Copy remainder of longer number 1354 while (xIndex > 0) 1355 result[--xIndex] = x[xIndex]; 1356 // Grow result if necessary 1357 if (carry) { 1358 int bigger[] = new int[result.length + 1]; 1359 System.arraycopy(result, 0, bigger, 1, result.length); 1360 bigger[0] = 0x01; 1361 return bigger; 1362 } 1363 return result; 1364 } 1365 1366 /** 1367 * Adds the contents of the int arrays x and y. This method allocates 1368 * a new int array to hold the answer and returns a reference to that 1369 * array. 1370 */ 1371 private static int[] add(int[] x, int[] y) { 1372 // If x is shorter, swap the two arrays 1373 if (x.length < y.length) { 1374 int[] tmp = x; 1375 x = y; 1376 y = tmp; 1377 } 1378 1379 int xIndex = x.length; 1380 int yIndex = y.length; 1381 int result[] = new int[xIndex]; 1382 long sum = 0; 1383 if (yIndex == 1) { 1384 sum = (x[--xIndex] & LONG_MASK) + (y[0] & LONG_MASK) ; 1385 result[xIndex] = (int)sum; 1386 } else { 1387 // Add common parts of both numbers 1388 while (yIndex > 0) { 1389 sum = (x[--xIndex] & LONG_MASK) + 1390 (y[--yIndex] & LONG_MASK) + (sum >>> 32); 1391 result[xIndex] = (int)sum; 1392 } 1393 } 1394 // Copy remainder of longer number while carry propagation is required 1395 boolean carry = (sum >>> 32 != 0); 1396 while (xIndex > 0 && carry) 1397 carry = ((result[--xIndex] = x[xIndex] + 1) == 0); 1398 1399 // Copy remainder of longer number 1400 while (xIndex > 0) 1401 result[--xIndex] = x[xIndex]; 1402 1403 // Grow result if necessary 1404 if (carry) { 1405 int bigger[] = new int[result.length + 1]; 1406 System.arraycopy(result, 0, bigger, 1, result.length); 1407 bigger[0] = 0x01; 1408 return bigger; 1409 } 1410 return result; 1411 } 1412 1413 private static int[] subtract(long val, int[] little) { 1414 int highWord = (int)(val >>> 32); 1415 if (highWord == 0) { 1416 int result[] = new int[1]; 1417 result[0] = (int)(val - (little[0] & LONG_MASK)); 1418 return result; 1419 } else { 1420 int result[] = new int[2]; 1421 if (little.length == 1) { 1422 long difference = ((int)val & LONG_MASK) - (little[0] & LONG_MASK); 1423 result[1] = (int)difference; 1424 // Subtract remainder of longer number while borrow propagates 1425 boolean borrow = (difference >> 32 != 0); 1426 if (borrow) { 1427 result[0] = highWord - 1; 1428 } else { // Copy remainder of longer number 1429 result[0] = highWord; 1430 } 1431 return result; 1432 } else { // little.length == 2 1433 long difference = ((int)val & LONG_MASK) - (little[1] & LONG_MASK); 1434 result[1] = (int)difference; 1435 difference = (highWord & LONG_MASK) - (little[0] & LONG_MASK) + (difference >> 32); 1436 result[0] = (int)difference; 1437 return result; 1438 } 1439 } 1440 } 1441 1442 /** 1443 * Subtracts the contents of the second argument (val) from the 1444 * first (big). The first int array (big) must represent a larger number 1445 * than the second. This method allocates the space necessary to hold the 1446 * answer. 1447 * assumes val >= 0 1448 */ 1449 private static int[] subtract(int[] big, long val) { 1450 int highWord = (int)(val >>> 32); 1451 int bigIndex = big.length; 1452 int result[] = new int[bigIndex]; 1453 long difference = 0; 1454 1455 if (highWord == 0) { 1456 difference = (big[--bigIndex] & LONG_MASK) - val; 1457 result[bigIndex] = (int)difference; 1458 } else { 1459 difference = (big[--bigIndex] & LONG_MASK) - (val & LONG_MASK); 1460 result[bigIndex] = (int)difference; 1461 difference = (big[--bigIndex] & LONG_MASK) - (highWord & LONG_MASK) + (difference >> 32); 1462 result[bigIndex] = (int)difference; 1463 } 1464 1465 // Subtract remainder of longer number while borrow propagates 1466 boolean borrow = (difference >> 32 != 0); 1467 while (bigIndex > 0 && borrow) 1468 borrow = ((result[--bigIndex] = big[bigIndex] - 1) == -1); 1469 1470 // Copy remainder of longer number 1471 while (bigIndex > 0) 1472 result[--bigIndex] = big[bigIndex]; 1473 1474 return result; 1475 } 1476 1477 /** 1478 * Returns a BigInteger whose value is {@code (this - val)}. 1479 * 1480 * @param val value to be subtracted from this BigInteger. 1481 * @return {@code this - val} 1482 */ 1483 public BigInteger subtract(BigInteger val) { 1484 if (val.signum == 0) 1485 return this; 1486 if (signum == 0) 1487 return val.negate(); 1488 if (val.signum != signum) 1489 return new BigInteger(add(mag, val.mag), signum); 1490 1491 int cmp = compareMagnitude(val); 1492 if (cmp == 0) 1493 return ZERO; 1494 int[] resultMag = (cmp > 0 ? subtract(mag, val.mag) 1495 : subtract(val.mag, mag)); 1496 resultMag = trustedStripLeadingZeroInts(resultMag); 1497 return new BigInteger(resultMag, cmp == signum ? 1 : -1); 1498 } 1499 1500 /** 1501 * Subtracts the contents of the second int arrays (little) from the 1502 * first (big). The first int array (big) must represent a larger number 1503 * than the second. This method allocates the space necessary to hold the 1504 * answer. 1505 */ 1506 private static int[] subtract(int[] big, int[] little) { 1507 int bigIndex = big.length; 1508 int result[] = new int[bigIndex]; 1509 int littleIndex = little.length; 1510 long difference = 0; 1511 1512 // Subtract common parts of both numbers 1513 while (littleIndex > 0) { 1514 difference = (big[--bigIndex] & LONG_MASK) - 1515 (little[--littleIndex] & LONG_MASK) + 1516 (difference >> 32); 1517 result[bigIndex] = (int)difference; 1518 } 1519 1520 // Subtract remainder of longer number while borrow propagates 1521 boolean borrow = (difference >> 32 != 0); 1522 while (bigIndex > 0 && borrow) 1523 borrow = ((result[--bigIndex] = big[bigIndex] - 1) == -1); 1524 1525 // Copy remainder of longer number 1526 while (bigIndex > 0) 1527 result[--bigIndex] = big[bigIndex]; 1528 1529 return result; 1530 } 1531 1532 /** 1533 * Returns a BigInteger whose value is {@code (this * val)}. 1534 * 1535 * @implNote An implementation may offer better algorithmic 1536 * performance when {@code val == this}. 1537 * 1538 * @param val value to be multiplied by this BigInteger. 1539 * @return {@code this * val} 1540 */ 1541 public BigInteger multiply(BigInteger val) { 1542 if (val.signum == 0 || signum == 0) 1543 return ZERO; 1544 1545 int xlen = mag.length; 1546 1547 if (val == this && xlen > MULTIPLY_SQUARE_THRESHOLD) { 1548 return square(); 1549 } 1550 1551 int ylen = val.mag.length; 1552 1553 if ((xlen < KARATSUBA_THRESHOLD) || (ylen < KARATSUBA_THRESHOLD)) { 1554 int resultSign = signum == val.signum ? 1 : -1; 1555 if (val.mag.length == 1) { 1556 return multiplyByInt(mag,val.mag[0], resultSign); 1557 } 1558 if (mag.length == 1) { 1559 return multiplyByInt(val.mag,mag[0], resultSign); 1560 } 1561 int[] result = multiplyToLen(mag, xlen, 1562 val.mag, ylen, null); 1563 result = trustedStripLeadingZeroInts(result); 1564 return new BigInteger(result, resultSign); 1565 } else { 1566 if ((xlen < TOOM_COOK_THRESHOLD) && (ylen < TOOM_COOK_THRESHOLD)) { 1567 return multiplyKaratsuba(this, val); 1568 } else { 1569 return multiplyToomCook3(this, val); 1570 } 1571 } 1572 } 1573 1574 private static BigInteger multiplyByInt(int[] x, int y, int sign) { 1575 if (Integer.bitCount(y) == 1) { 1576 return new BigInteger(shiftLeft(x,Integer.numberOfTrailingZeros(y)), sign); 1577 } 1578 int xlen = x.length; 1579 int[] rmag = new int[xlen + 1]; 1580 long carry = 0; 1581 long yl = y & LONG_MASK; 1582 int rstart = rmag.length - 1; 1583 for (int i = xlen - 1; i >= 0; i--) { 1584 long product = (x[i] & LONG_MASK) * yl + carry; 1585 rmag[rstart--] = (int)product; 1586 carry = product >>> 32; 1587 } 1588 if (carry == 0L) { 1589 rmag = java.util.Arrays.copyOfRange(rmag, 1, rmag.length); 1590 } else { 1591 rmag[rstart] = (int)carry; 1592 } 1593 return new BigInteger(rmag, sign); 1594 } 1595 1596 /** 1597 * Package private methods used by BigDecimal code to multiply a BigInteger 1598 * with a long. Assumes v is not equal to INFLATED. 1599 */ 1600 BigInteger multiply(long v) { 1601 if (v == 0 || signum == 0) 1602 return ZERO; 1603 if (v == BigDecimal.INFLATED) 1604 return multiply(BigInteger.valueOf(v)); 1605 int rsign = (v > 0 ? signum : -signum); 1606 if (v < 0) 1607 v = -v; 1608 long dh = v >>> 32; // higher order bits 1609 long dl = v & LONG_MASK; // lower order bits 1610 1611 int xlen = mag.length; 1612 int[] value = mag; 1613 int[] rmag = (dh == 0L) ? (new int[xlen + 1]) : (new int[xlen + 2]); 1614 long carry = 0; 1615 int rstart = rmag.length - 1; 1616 for (int i = xlen - 1; i >= 0; i--) { 1617 long product = (value[i] & LONG_MASK) * dl + carry; 1618 rmag[rstart--] = (int)product; 1619 carry = product >>> 32; 1620 } 1621 rmag[rstart] = (int)carry; 1622 if (dh != 0L) { 1623 carry = 0; 1624 rstart = rmag.length - 2; 1625 for (int i = xlen - 1; i >= 0; i--) { 1626 long product = (value[i] & LONG_MASK) * dh + 1627 (rmag[rstart] & LONG_MASK) + carry; 1628 rmag[rstart--] = (int)product; 1629 carry = product >>> 32; 1630 } 1631 rmag[0] = (int)carry; 1632 } 1633 if (carry == 0L) 1634 rmag = java.util.Arrays.copyOfRange(rmag, 1, rmag.length); 1635 return new BigInteger(rmag, rsign); 1636 } 1637 1638 /** 1639 * Multiplies int arrays x and y to the specified lengths and places 1640 * the result into z. There will be no leading zeros in the resultant array. 1641 */ 1642 private int[] multiplyToLen(int[] x, int xlen, int[] y, int ylen, int[] z) { 1643 int xstart = xlen - 1; 1644 int ystart = ylen - 1; 1645 1646 if (z == null || z.length < (xlen+ ylen)) 1647 z = new int[xlen+ylen]; 1648 1649 long carry = 0; 1650 for (int j=ystart, k=ystart+1+xstart; j >= 0; j--, k--) { 1651 long product = (y[j] & LONG_MASK) * 1652 (x[xstart] & LONG_MASK) + carry; 1653 z[k] = (int)product; 1654 carry = product >>> 32; 1655 } 1656 z[xstart] = (int)carry; 1657 1658 for (int i = xstart-1; i >= 0; i--) { 1659 carry = 0; 1660 for (int j=ystart, k=ystart+1+i; j >= 0; j--, k--) { 1661 long product = (y[j] & LONG_MASK) * 1662 (x[i] & LONG_MASK) + 1663 (z[k] & LONG_MASK) + carry; 1664 z[k] = (int)product; 1665 carry = product >>> 32; 1666 } 1667 z[i] = (int)carry; 1668 } 1669 return z; 1670 } 1671 1672 /** 1673 * Multiplies two BigIntegers using the Karatsuba multiplication 1674 * algorithm. This is a recursive divide-and-conquer algorithm which is 1675 * more efficient for large numbers than what is commonly called the 1676 * "grade-school" algorithm used in multiplyToLen. If the numbers to be 1677 * multiplied have length n, the "grade-school" algorithm has an 1678 * asymptotic complexity of O(n^2). In contrast, the Karatsuba algorithm 1679 * has complexity of O(n^(log2(3))), or O(n^1.585). It achieves this 1680 * increased performance by doing 3 multiplies instead of 4 when 1681 * evaluating the product. As it has some overhead, should be used when 1682 * both numbers are larger than a certain threshold (found 1683 * experimentally). 1684 * 1685 * See: http://en.wikipedia.org/wiki/Karatsuba_algorithm 1686 */ 1687 private static BigInteger multiplyKaratsuba(BigInteger x, BigInteger y) { 1688 int xlen = x.mag.length; 1689 int ylen = y.mag.length; 1690 1691 // The number of ints in each half of the number. 1692 int half = (Math.max(xlen, ylen)+1) / 2; 1693 1694 // xl and yl are the lower halves of x and y respectively, 1695 // xh and yh are the upper halves. 1696 BigInteger xl = x.getLower(half); 1697 BigInteger xh = x.getUpper(half); 1698 BigInteger yl = y.getLower(half); 1699 BigInteger yh = y.getUpper(half); 1700 1701 BigInteger p1 = xh.multiply(yh); // p1 = xh*yh 1702 BigInteger p2 = xl.multiply(yl); // p2 = xl*yl 1703 1704 // p3=(xh+xl)*(yh+yl) 1705 BigInteger p3 = xh.add(xl).multiply(yh.add(yl)); 1706 1707 // result = p1 * 2^(32*2*half) + (p3 - p1 - p2) * 2^(32*half) + p2 1708 BigInteger result = p1.shiftLeft(32*half).add(p3.subtract(p1).subtract(p2)).shiftLeft(32*half).add(p2); 1709 1710 if (x.signum != y.signum) { 1711 return result.negate(); 1712 } else { 1713 return result; 1714 } 1715 } 1716 1717 /** 1718 * Multiplies two BigIntegers using a 3-way Toom-Cook multiplication 1719 * algorithm. This is a recursive divide-and-conquer algorithm which is 1720 * more efficient for large numbers than what is commonly called the 1721 * "grade-school" algorithm used in multiplyToLen. If the numbers to be 1722 * multiplied have length n, the "grade-school" algorithm has an 1723 * asymptotic complexity of O(n^2). In contrast, 3-way Toom-Cook has a 1724 * complexity of about O(n^1.465). It achieves this increased asymptotic 1725 * performance by breaking each number into three parts and by doing 5 1726 * multiplies instead of 9 when evaluating the product. Due to overhead 1727 * (additions, shifts, and one division) in the Toom-Cook algorithm, it 1728 * should only be used when both numbers are larger than a certain 1729 * threshold (found experimentally). This threshold is generally larger 1730 * than that for Karatsuba multiplication, so this algorithm is generally 1731 * only used when numbers become significantly larger. 1732 * 1733 * The algorithm used is the "optimal" 3-way Toom-Cook algorithm outlined 1734 * by Marco Bodrato. 1735 * 1736 * See: http://bodrato.it/toom-cook/ 1737 * http://bodrato.it/papers/#WAIFI2007 1738 * 1739 * "Towards Optimal Toom-Cook Multiplication for Univariate and 1740 * Multivariate Polynomials in Characteristic 2 and 0." by Marco BODRATO; 1741 * In C.Carlet and B.Sunar, Eds., "WAIFI'07 proceedings", p. 116-133, 1742 * LNCS #4547. Springer, Madrid, Spain, June 21-22, 2007. 1743 * 1744 */ 1745 private static BigInteger multiplyToomCook3(BigInteger a, BigInteger b) { 1746 int alen = a.mag.length; 1747 int blen = b.mag.length; 1748 1749 int largest = Math.max(alen, blen); 1750 1751 // k is the size (in ints) of the lower-order slices. 1752 int k = (largest+2)/3; // Equal to ceil(largest/3) 1753 1754 // r is the size (in ints) of the highest-order slice. 1755 int r = largest - 2*k; 1756 1757 // Obtain slices of the numbers. a2 and b2 are the most significant 1758 // bits of the numbers a and b, and a0 and b0 the least significant. 1759 BigInteger a0, a1, a2, b0, b1, b2; 1760 a2 = a.getToomSlice(k, r, 0, largest); 1761 a1 = a.getToomSlice(k, r, 1, largest); 1762 a0 = a.getToomSlice(k, r, 2, largest); 1763 b2 = b.getToomSlice(k, r, 0, largest); 1764 b1 = b.getToomSlice(k, r, 1, largest); 1765 b0 = b.getToomSlice(k, r, 2, largest); 1766 1767 BigInteger v0, v1, v2, vm1, vinf, t1, t2, tm1, da1, db1; 1768 1769 v0 = a0.multiply(b0); 1770 da1 = a2.add(a0); 1771 db1 = b2.add(b0); 1772 vm1 = da1.subtract(a1).multiply(db1.subtract(b1)); 1773 da1 = da1.add(a1); 1774 db1 = db1.add(b1); 1775 v1 = da1.multiply(db1); 1776 v2 = da1.add(a2).shiftLeft(1).subtract(a0).multiply( 1777 db1.add(b2).shiftLeft(1).subtract(b0)); 1778 vinf = a2.multiply(b2); 1779 1780 // The algorithm requires two divisions by 2 and one by 3. 1781 // All divisions are known to be exact, that is, they do not produce 1782 // remainders, and all results are positive. The divisions by 2 are 1783 // implemented as right shifts which are relatively efficient, leaving 1784 // only an exact division by 3, which is done by a specialized 1785 // linear-time algorithm. 1786 t2 = v2.subtract(vm1).exactDivideBy3(); 1787 tm1 = v1.subtract(vm1).shiftRight(1); 1788 t1 = v1.subtract(v0); 1789 t2 = t2.subtract(t1).shiftRight(1); 1790 t1 = t1.subtract(tm1).subtract(vinf); 1791 t2 = t2.subtract(vinf.shiftLeft(1)); 1792 tm1 = tm1.subtract(t2); 1793 1794 // Number of bits to shift left. 1795 int ss = k*32; 1796 1797 BigInteger result = vinf.shiftLeft(ss).add(t2).shiftLeft(ss).add(t1).shiftLeft(ss).add(tm1).shiftLeft(ss).add(v0); 1798 1799 if (a.signum != b.signum) { 1800 return result.negate(); 1801 } else { 1802 return result; 1803 } 1804 } 1805 1806 1807 /** 1808 * Returns a slice of a BigInteger for use in Toom-Cook multiplication. 1809 * 1810 * @param lowerSize The size of the lower-order bit slices. 1811 * @param upperSize The size of the higher-order bit slices. 1812 * @param slice The index of which slice is requested, which must be a 1813 * number from 0 to size-1. Slice 0 is the highest-order bits, and slice 1814 * size-1 are the lowest-order bits. Slice 0 may be of different size than 1815 * the other slices. 1816 * @param fullsize The size of the larger integer array, used to align 1817 * slices to the appropriate position when multiplying different-sized 1818 * numbers. 1819 */ 1820 private BigInteger getToomSlice(int lowerSize, int upperSize, int slice, 1821 int fullsize) { 1822 int start, end, sliceSize, len, offset; 1823 1824 len = mag.length; 1825 offset = fullsize - len; 1826 1827 if (slice == 0) { 1828 start = 0 - offset; 1829 end = upperSize - 1 - offset; 1830 } else { 1831 start = upperSize + (slice-1)*lowerSize - offset; 1832 end = start + lowerSize - 1; 1833 } 1834 1835 if (start < 0) { 1836 start = 0; 1837 } 1838 if (end < 0) { 1839 return ZERO; 1840 } 1841 1842 sliceSize = (end-start) + 1; 1843 1844 if (sliceSize <= 0) { 1845 return ZERO; 1846 } 1847 1848 // While performing Toom-Cook, all slices are positive and 1849 // the sign is adjusted when the final number is composed. 1850 if (start == 0 && sliceSize >= len) { 1851 return this.abs(); 1852 } 1853 1854 int intSlice[] = new int[sliceSize]; 1855 System.arraycopy(mag, start, intSlice, 0, sliceSize); 1856 1857 return new BigInteger(trustedStripLeadingZeroInts(intSlice), 1); 1858 } 1859 1860 /** 1861 * Does an exact division (that is, the remainder is known to be zero) 1862 * of the specified number by 3. This is used in Toom-Cook 1863 * multiplication. This is an efficient algorithm that runs in linear 1864 * time. If the argument is not exactly divisible by 3, results are 1865 * undefined. Note that this is expected to be called with positive 1866 * arguments only. 1867 */ 1868 private BigInteger exactDivideBy3() { 1869 int len = mag.length; 1870 int[] result = new int[len]; 1871 long x, w, q, borrow; 1872 borrow = 0L; 1873 for (int i=len-1; i >= 0; i--) { 1874 x = (mag[i] & LONG_MASK); 1875 w = x - borrow; 1876 if (borrow > x) { // Did we make the number go negative? 1877 borrow = 1L; 1878 } else { 1879 borrow = 0L; 1880 } 1881 1882 // 0xAAAAAAAB is the modular inverse of 3 (mod 2^32). Thus, 1883 // the effect of this is to divide by 3 (mod 2^32). 1884 // This is much faster than division on most architectures. 1885 q = (w * 0xAAAAAAABL) & LONG_MASK; 1886 result[i] = (int) q; 1887 1888 // Now check the borrow. The second check can of course be 1889 // eliminated if the first fails. 1890 if (q >= 0x55555556L) { 1891 borrow++; 1892 if (q >= 0xAAAAAAABL) 1893 borrow++; 1894 } 1895 } 1896 result = trustedStripLeadingZeroInts(result); 1897 return new BigInteger(result, signum); 1898 } 1899 1900 /** 1901 * Returns a new BigInteger representing n lower ints of the number. 1902 * This is used by Karatsuba multiplication and Karatsuba squaring. 1903 */ 1904 private BigInteger getLower(int n) { 1905 int len = mag.length; 1906 1907 if (len <= n) { 1908 return abs(); 1909 } 1910 1911 int lowerInts[] = new int[n]; 1912 System.arraycopy(mag, len-n, lowerInts, 0, n); 1913 1914 return new BigInteger(trustedStripLeadingZeroInts(lowerInts), 1); 1915 } 1916 1917 /** 1918 * Returns a new BigInteger representing mag.length-n upper 1919 * ints of the number. This is used by Karatsuba multiplication and 1920 * Karatsuba squaring. 1921 */ 1922 private BigInteger getUpper(int n) { 1923 int len = mag.length; 1924 1925 if (len <= n) { 1926 return ZERO; 1927 } 1928 1929 int upperLen = len - n; 1930 int upperInts[] = new int[upperLen]; 1931 System.arraycopy(mag, 0, upperInts, 0, upperLen); 1932 1933 return new BigInteger(trustedStripLeadingZeroInts(upperInts), 1); 1934 } 1935 1936 // Squaring 1937 1938 /** 1939 * Returns a BigInteger whose value is {@code (this<sup>2</sup>)}. 1940 * 1941 * @return {@code this<sup>2</sup>} 1942 */ 1943 private BigInteger square() { 1944 if (signum == 0) { 1945 return ZERO; 1946 } 1947 int len = mag.length; 1948 1949 if (len < KARATSUBA_SQUARE_THRESHOLD) { 1950 int[] z = squareToLen(mag, len, null); 1951 return new BigInteger(trustedStripLeadingZeroInts(z), 1); 1952 } else { 1953 if (len < TOOM_COOK_SQUARE_THRESHOLD) { 1954 return squareKaratsuba(); 1955 } else { 1956 return squareToomCook3(); 1957 } 1958 } 1959 } 1960 1961 /** 1962 * Squares the contents of the int array x. The result is placed into the 1963 * int array z. The contents of x are not changed. 1964 */ 1965 private static final int[] squareToLen(int[] x, int len, int[] z) { 1966 int zlen = len << 1; 1967 if (z == null || z.length < zlen) 1968 z = new int[zlen]; 1969 1970 // Execute checks before calling intrinsified method. 1971 implSquareToLenChecks(x, len, z, zlen); 1972 return implSquareToLen(x, len, z, zlen); 1973 } 1974 1975 /** 1976 * Parameters validation. 1977 */ 1978 private static void implSquareToLenChecks(int[] x, int len, int[] z, int zlen) throws RuntimeException { 1979 if (len < 1) { 1980 throw new IllegalArgumentException("invalid input length: " + len); 1981 } 1982 if (len > x.length) { 1983 throw new IllegalArgumentException("input length out of bound: " + 1984 len + " > " + x.length); 1985 } 1986 if (len * 2 > z.length) { 1987 throw new IllegalArgumentException("input length out of bound: " + 1988 (len * 2) + " > " + z.length); 1989 } 1990 if (zlen < 1) { 1991 throw new IllegalArgumentException("invalid input length: " + zlen); 1992 } 1993 if (zlen > z.length) { 1994 throw new IllegalArgumentException("input length out of bound: " + 1995 len + " > " + z.length); 1996 } 1997 } 1998 1999 /** 2000 * Java Runtime may use intrinsic for this method. 2001 */ 2002 private static final int[] implSquareToLen(int[] x, int len, int[] z, int zlen) { 2003 /* 2004 * The algorithm used here is adapted from Colin Plumb's C library. 2005 * Technique: Consider the partial products in the multiplication 2006 * of "abcde" by itself: 2007 * 2008 * a b c d e 2009 * * a b c d e 2010 * ================== 2011 * ae be ce de ee 2012 * ad bd cd dd de 2013 * ac bc cc cd ce 2014 * ab bb bc bd be 2015 * aa ab ac ad ae 2016 * 2017 * Note that everything above the main diagonal: 2018 * ae be ce de = (abcd) * e 2019 * ad bd cd = (abc) * d 2020 * ac bc = (ab) * c 2021 * ab = (a) * b 2022 * 2023 * is a copy of everything below the main diagonal: 2024 * de 2025 * cd ce 2026 * bc bd be 2027 * ab ac ad ae 2028 * 2029 * Thus, the sum is 2 * (off the diagonal) + diagonal. 2030 * 2031 * This is accumulated beginning with the diagonal (which 2032 * consist of the squares of the digits of the input), which is then 2033 * divided by two, the off-diagonal added, and multiplied by two 2034 * again. The low bit is simply a copy of the low bit of the 2035 * input, so it doesn't need special care. 2036 */ 2037 2038 // Store the squares, right shifted one bit (i.e., divided by 2) 2039 int lastProductLowWord = 0; 2040 for (int j=0, i=0; j < len; j++) { 2041 long piece = (x[j] & LONG_MASK); 2042 long product = piece * piece; 2043 z[i++] = (lastProductLowWord << 31) | (int)(product >>> 33); 2044 z[i++] = (int)(product >>> 1); 2045 lastProductLowWord = (int)product; 2046 } 2047 2048 // Add in off-diagonal sums 2049 for (int i=len, offset=1; i > 0; i--, offset+=2) { 2050 int t = x[i-1]; 2051 t = mulAdd(z, x, offset, i-1, t); 2052 addOne(z, offset-1, i, t); 2053 } 2054 2055 // Shift back up and set low bit 2056 primitiveLeftShift(z, zlen, 1); 2057 z[zlen-1] |= x[len-1] & 1; 2058 2059 return z; 2060 } 2061 2062 /** 2063 * Squares a BigInteger using the Karatsuba squaring algorithm. It should 2064 * be used when both numbers are larger than a certain threshold (found 2065 * experimentally). It is a recursive divide-and-conquer algorithm that 2066 * has better asymptotic performance than the algorithm used in 2067 * squareToLen. 2068 */ 2069 private BigInteger squareKaratsuba() { 2070 int half = (mag.length+1) / 2; 2071 2072 BigInteger xl = getLower(half); 2073 BigInteger xh = getUpper(half); 2074 2075 BigInteger xhs = xh.square(); // xhs = xh^2 2076 BigInteger xls = xl.square(); // xls = xl^2 2077 2078 // xh^2 << 64 + (((xl+xh)^2 - (xh^2 + xl^2)) << 32) + xl^2 2079 return xhs.shiftLeft(half*32).add(xl.add(xh).square().subtract(xhs.add(xls))).shiftLeft(half*32).add(xls); 2080 } 2081 2082 /** 2083 * Squares a BigInteger using the 3-way Toom-Cook squaring algorithm. It 2084 * should be used when both numbers are larger than a certain threshold 2085 * (found experimentally). It is a recursive divide-and-conquer algorithm 2086 * that has better asymptotic performance than the algorithm used in 2087 * squareToLen or squareKaratsuba. 2088 */ 2089 private BigInteger squareToomCook3() { 2090 int len = mag.length; 2091 2092 // k is the size (in ints) of the lower-order slices. 2093 int k = (len+2)/3; // Equal to ceil(largest/3) 2094 2095 // r is the size (in ints) of the highest-order slice. 2096 int r = len - 2*k; 2097 2098 // Obtain slices of the numbers. a2 is the most significant 2099 // bits of the number, and a0 the least significant. 2100 BigInteger a0, a1, a2; 2101 a2 = getToomSlice(k, r, 0, len); 2102 a1 = getToomSlice(k, r, 1, len); 2103 a0 = getToomSlice(k, r, 2, len); 2104 BigInteger v0, v1, v2, vm1, vinf, t1, t2, tm1, da1; 2105 2106 v0 = a0.square(); 2107 da1 = a2.add(a0); 2108 vm1 = da1.subtract(a1).square(); 2109 da1 = da1.add(a1); 2110 v1 = da1.square(); 2111 vinf = a2.square(); 2112 v2 = da1.add(a2).shiftLeft(1).subtract(a0).square(); 2113 2114 // The algorithm requires two divisions by 2 and one by 3. 2115 // All divisions are known to be exact, that is, they do not produce 2116 // remainders, and all results are positive. The divisions by 2 are 2117 // implemented as right shifts which are relatively efficient, leaving 2118 // only a division by 3. 2119 // The division by 3 is done by an optimized algorithm for this case. 2120 t2 = v2.subtract(vm1).exactDivideBy3(); 2121 tm1 = v1.subtract(vm1).shiftRight(1); 2122 t1 = v1.subtract(v0); 2123 t2 = t2.subtract(t1).shiftRight(1); 2124 t1 = t1.subtract(tm1).subtract(vinf); 2125 t2 = t2.subtract(vinf.shiftLeft(1)); 2126 tm1 = tm1.subtract(t2); 2127 2128 // Number of bits to shift left. 2129 int ss = k*32; 2130 2131 return vinf.shiftLeft(ss).add(t2).shiftLeft(ss).add(t1).shiftLeft(ss).add(tm1).shiftLeft(ss).add(v0); 2132 } 2133 2134 // Division 2135 2136 /** 2137 * Returns a BigInteger whose value is {@code (this / val)}. 2138 * 2139 * @param val value by which this BigInteger is to be divided. 2140 * @return {@code this / val} 2141 * @throws ArithmeticException if {@code val} is zero. 2142 */ 2143 public BigInteger divide(BigInteger val) { 2144 if (val.mag.length < BURNIKEL_ZIEGLER_THRESHOLD || 2145 mag.length - val.mag.length < BURNIKEL_ZIEGLER_OFFSET) { 2146 return divideKnuth(val); 2147 } else { 2148 return divideBurnikelZiegler(val); 2149 } 2150 } 2151 2152 /** 2153 * Returns a BigInteger whose value is {@code (this / val)} using an O(n^2) algorithm from Knuth. 2154 * 2155 * @param val value by which this BigInteger is to be divided. 2156 * @return {@code this / val} 2157 * @throws ArithmeticException if {@code val} is zero. 2158 * @see MutableBigInteger#divideKnuth(MutableBigInteger, MutableBigInteger, boolean) 2159 */ 2160 private BigInteger divideKnuth(BigInteger val) { 2161 MutableBigInteger q = new MutableBigInteger(), 2162 a = new MutableBigInteger(this.mag), 2163 b = new MutableBigInteger(val.mag); 2164 2165 a.divideKnuth(b, q, false); 2166 return q.toBigInteger(this.signum * val.signum); 2167 } 2168 2169 /** 2170 * Returns an array of two BigIntegers containing {@code (this / val)} 2171 * followed by {@code (this % val)}. 2172 * 2173 * @param val value by which this BigInteger is to be divided, and the 2174 * remainder computed. 2175 * @return an array of two BigIntegers: the quotient {@code (this / val)} 2176 * is the initial element, and the remainder {@code (this % val)} 2177 * is the final element. 2178 * @throws ArithmeticException if {@code val} is zero. 2179 */ 2180 public BigInteger[] divideAndRemainder(BigInteger val) { 2181 if (val.mag.length < BURNIKEL_ZIEGLER_THRESHOLD || 2182 mag.length - val.mag.length < BURNIKEL_ZIEGLER_OFFSET) { 2183 return divideAndRemainderKnuth(val); 2184 } else { 2185 return divideAndRemainderBurnikelZiegler(val); 2186 } 2187 } 2188 2189 /** Long division */ 2190 private BigInteger[] divideAndRemainderKnuth(BigInteger val) { 2191 BigInteger[] result = new BigInteger[2]; 2192 MutableBigInteger q = new MutableBigInteger(), 2193 a = new MutableBigInteger(this.mag), 2194 b = new MutableBigInteger(val.mag); 2195 MutableBigInteger r = a.divideKnuth(b, q); 2196 result[0] = q.toBigInteger(this.signum == val.signum ? 1 : -1); 2197 result[1] = r.toBigInteger(this.signum); 2198 return result; 2199 } 2200 2201 /** 2202 * Returns a BigInteger whose value is {@code (this % val)}. 2203 * 2204 * @param val value by which this BigInteger is to be divided, and the 2205 * remainder computed. 2206 * @return {@code this % val} 2207 * @throws ArithmeticException if {@code val} is zero. 2208 */ 2209 public BigInteger remainder(BigInteger val) { 2210 if (val.mag.length < BURNIKEL_ZIEGLER_THRESHOLD || 2211 mag.length - val.mag.length < BURNIKEL_ZIEGLER_OFFSET) { 2212 return remainderKnuth(val); 2213 } else { 2214 return remainderBurnikelZiegler(val); 2215 } 2216 } 2217 2218 /** Long division */ 2219 private BigInteger remainderKnuth(BigInteger val) { 2220 MutableBigInteger q = new MutableBigInteger(), 2221 a = new MutableBigInteger(this.mag), 2222 b = new MutableBigInteger(val.mag); 2223 2224 return a.divideKnuth(b, q).toBigInteger(this.signum); 2225 } 2226 2227 /** 2228 * Calculates {@code this / val} using the Burnikel-Ziegler algorithm. 2229 * @param val the divisor 2230 * @return {@code this / val} 2231 */ 2232 private BigInteger divideBurnikelZiegler(BigInteger val) { 2233 return divideAndRemainderBurnikelZiegler(val)[0]; 2234 } 2235 2236 /** 2237 * Calculates {@code this % val} using the Burnikel-Ziegler algorithm. 2238 * @param val the divisor 2239 * @return {@code this % val} 2240 */ 2241 private BigInteger remainderBurnikelZiegler(BigInteger val) { 2242 return divideAndRemainderBurnikelZiegler(val)[1]; 2243 } 2244 2245 /** 2246 * Computes {@code this / val} and {@code this % val} using the 2247 * Burnikel-Ziegler algorithm. 2248 * @param val the divisor 2249 * @return an array containing the quotient and remainder 2250 */ 2251 private BigInteger[] divideAndRemainderBurnikelZiegler(BigInteger val) { 2252 MutableBigInteger q = new MutableBigInteger(); 2253 MutableBigInteger r = new MutableBigInteger(this).divideAndRemainderBurnikelZiegler(new MutableBigInteger(val), q); 2254 BigInteger qBigInt = q.isZero() ? ZERO : q.toBigInteger(signum*val.signum); 2255 BigInteger rBigInt = r.isZero() ? ZERO : r.toBigInteger(signum); 2256 return new BigInteger[] {qBigInt, rBigInt}; 2257 } 2258 2259 /** 2260 * Returns a BigInteger whose value is <tt>(this<sup>exponent</sup>)</tt>. 2261 * Note that {@code exponent} is an integer rather than a BigInteger. 2262 * 2263 * @param exponent exponent to which this BigInteger is to be raised. 2264 * @return <tt>this<sup>exponent</sup></tt> 2265 * @throws ArithmeticException {@code exponent} is negative. (This would 2266 * cause the operation to yield a non-integer value.) 2267 */ 2268 public BigInteger pow(int exponent) { 2269 if (exponent < 0) { 2270 throw new ArithmeticException("Negative exponent"); 2271 } 2272 if (signum == 0) { 2273 return (exponent == 0 ? ONE : this); 2274 } 2275 2276 BigInteger partToSquare = this.abs(); 2277 2278 // Factor out powers of two from the base, as the exponentiation of 2279 // these can be done by left shifts only. 2280 // The remaining part can then be exponentiated faster. The 2281 // powers of two will be multiplied back at the end. 2282 int powersOfTwo = partToSquare.getLowestSetBit(); 2283 long bitsToShift = (long)powersOfTwo * exponent; 2284 if (bitsToShift > Integer.MAX_VALUE) { 2285 reportOverflow(); 2286 } 2287 2288 int remainingBits; 2289 2290 // Factor the powers of two out quickly by shifting right, if needed. 2291 if (powersOfTwo > 0) { 2292 partToSquare = partToSquare.shiftRight(powersOfTwo); 2293 remainingBits = partToSquare.bitLength(); 2294 if (remainingBits == 1) { // Nothing left but +/- 1? 2295 if (signum < 0 && (exponent&1) == 1) { 2296 return NEGATIVE_ONE.shiftLeft(powersOfTwo*exponent); 2297 } else { 2298 return ONE.shiftLeft(powersOfTwo*exponent); 2299 } 2300 } 2301 } else { 2302 remainingBits = partToSquare.bitLength(); 2303 if (remainingBits == 1) { // Nothing left but +/- 1? 2304 if (signum < 0 && (exponent&1) == 1) { 2305 return NEGATIVE_ONE; 2306 } else { 2307 return ONE; 2308 } 2309 } 2310 } 2311 2312 // This is a quick way to approximate the size of the result, 2313 // similar to doing log2[n] * exponent. This will give an upper bound 2314 // of how big the result can be, and which algorithm to use. 2315 long scaleFactor = (long)remainingBits * exponent; 2316 2317 // Use slightly different algorithms for small and large operands. 2318 // See if the result will safely fit into a long. (Largest 2^63-1) 2319 if (partToSquare.mag.length == 1 && scaleFactor <= 62) { 2320 // Small number algorithm. Everything fits into a long. 2321 int newSign = (signum <0 && (exponent&1) == 1 ? -1 : 1); 2322 long result = 1; 2323 long baseToPow2 = partToSquare.mag[0] & LONG_MASK; 2324 2325 int workingExponent = exponent; 2326 2327 // Perform exponentiation using repeated squaring trick 2328 while (workingExponent != 0) { 2329 if ((workingExponent & 1) == 1) { 2330 result = result * baseToPow2; 2331 } 2332 2333 if ((workingExponent >>>= 1) != 0) { 2334 baseToPow2 = baseToPow2 * baseToPow2; 2335 } 2336 } 2337 2338 // Multiply back the powers of two (quickly, by shifting left) 2339 if (powersOfTwo > 0) { 2340 if (bitsToShift + scaleFactor <= 62) { // Fits in long? 2341 return valueOf((result << bitsToShift) * newSign); 2342 } else { 2343 return valueOf(result*newSign).shiftLeft((int) bitsToShift); 2344 } 2345 } 2346 else { 2347 return valueOf(result*newSign); 2348 } 2349 } else { 2350 // Large number algorithm. This is basically identical to 2351 // the algorithm above, but calls multiply() and square() 2352 // which may use more efficient algorithms for large numbers. 2353 BigInteger answer = ONE; 2354 2355 int workingExponent = exponent; 2356 // Perform exponentiation using repeated squaring trick 2357 while (workingExponent != 0) { 2358 if ((workingExponent & 1) == 1) { 2359 answer = answer.multiply(partToSquare); 2360 } 2361 2362 if ((workingExponent >>>= 1) != 0) { 2363 partToSquare = partToSquare.square(); 2364 } 2365 } 2366 // Multiply back the (exponentiated) powers of two (quickly, 2367 // by shifting left) 2368 if (powersOfTwo > 0) { 2369 answer = answer.shiftLeft(powersOfTwo*exponent); 2370 } 2371 2372 if (signum < 0 && (exponent&1) == 1) { 2373 return answer.negate(); 2374 } else { 2375 return answer; 2376 } 2377 } 2378 } 2379 2380 /** 2381 * Returns a BigInteger whose value is the greatest common divisor of 2382 * {@code abs(this)} and {@code abs(val)}. Returns 0 if 2383 * {@code this == 0 && val == 0}. 2384 * 2385 * @param val value with which the GCD is to be computed. 2386 * @return {@code GCD(abs(this), abs(val))} 2387 */ 2388 public BigInteger gcd(BigInteger val) { 2389 if (val.signum == 0) 2390 return this.abs(); 2391 else if (this.signum == 0) 2392 return val.abs(); 2393 2394 MutableBigInteger a = new MutableBigInteger(this); 2395 MutableBigInteger b = new MutableBigInteger(val); 2396 2397 MutableBigInteger result = a.hybridGCD(b); 2398 2399 return result.toBigInteger(1); 2400 } 2401 2402 /** 2403 * Package private method to return bit length for an integer. 2404 */ 2405 static int bitLengthForInt(int n) { 2406 return 32 - Integer.numberOfLeadingZeros(n); 2407 } 2408 2409 /** 2410 * Left shift int array a up to len by n bits. Returns the array that 2411 * results from the shift since space may have to be reallocated. 2412 */ 2413 private static int[] leftShift(int[] a, int len, int n) { 2414 int nInts = n >>> 5; 2415 int nBits = n&0x1F; 2416 int bitsInHighWord = bitLengthForInt(a[0]); 2417 2418 // If shift can be done without recopy, do so 2419 if (n <= (32-bitsInHighWord)) { 2420 primitiveLeftShift(a, len, nBits); 2421 return a; 2422 } else { // Array must be resized 2423 if (nBits <= (32-bitsInHighWord)) { 2424 int result[] = new int[nInts+len]; 2425 System.arraycopy(a, 0, result, 0, len); 2426 primitiveLeftShift(result, result.length, nBits); 2427 return result; 2428 } else { 2429 int result[] = new int[nInts+len+1]; 2430 System.arraycopy(a, 0, result, 0, len); 2431 primitiveRightShift(result, result.length, 32 - nBits); 2432 return result; 2433 } 2434 } 2435 } 2436 2437 // shifts a up to len right n bits assumes no leading zeros, 0<n<32 2438 static void primitiveRightShift(int[] a, int len, int n) { 2439 int n2 = 32 - n; 2440 for (int i=len-1, c=a[i]; i > 0; i--) { 2441 int b = c; 2442 c = a[i-1]; 2443 a[i] = (c << n2) | (b >>> n); 2444 } 2445 a[0] >>>= n; 2446 } 2447 2448 // shifts a up to len left n bits assumes no leading zeros, 0<=n<32 2449 static void primitiveLeftShift(int[] a, int len, int n) { 2450 if (len == 0 || n == 0) 2451 return; 2452 2453 int n2 = 32 - n; 2454 for (int i=0, c=a[i], m=i+len-1; i < m; i++) { 2455 int b = c; 2456 c = a[i+1]; 2457 a[i] = (b << n) | (c >>> n2); 2458 } 2459 a[len-1] <<= n; 2460 } 2461 2462 /** 2463 * Calculate bitlength of contents of the first len elements an int array, 2464 * assuming there are no leading zero ints. 2465 */ 2466 private static int bitLength(int[] val, int len) { 2467 if (len == 0) 2468 return 0; 2469 return ((len - 1) << 5) + bitLengthForInt(val[0]); 2470 } 2471 2472 /** 2473 * Returns a BigInteger whose value is the absolute value of this 2474 * BigInteger. 2475 * 2476 * @return {@code abs(this)} 2477 */ 2478 public BigInteger abs() { 2479 return (signum >= 0 ? this : this.negate()); 2480 } 2481 2482 /** 2483 * Returns a BigInteger whose value is {@code (-this)}. 2484 * 2485 * @return {@code -this} 2486 */ 2487 public BigInteger negate() { 2488 return new BigInteger(this.mag, -this.signum); 2489 } 2490 2491 /** 2492 * Returns the signum function of this BigInteger. 2493 * 2494 * @return -1, 0 or 1 as the value of this BigInteger is negative, zero or 2495 * positive. 2496 */ 2497 public int signum() { 2498 return this.signum; 2499 } 2500 2501 // Modular Arithmetic Operations 2502 2503 /** 2504 * Returns a BigInteger whose value is {@code (this mod m}). This method 2505 * differs from {@code remainder} in that it always returns a 2506 * <i>non-negative</i> BigInteger. 2507 * 2508 * @param m the modulus. 2509 * @return {@code this mod m} 2510 * @throws ArithmeticException {@code m} ≤ 0 2511 * @see #remainder 2512 */ 2513 public BigInteger mod(BigInteger m) { 2514 if (m.signum <= 0) 2515 throw new ArithmeticException("BigInteger: modulus not positive"); 2516 2517 BigInteger result = this.remainder(m); 2518 return (result.signum >= 0 ? result : result.add(m)); 2519 } 2520 2521 /** 2522 * Returns a BigInteger whose value is 2523 * <tt>(this<sup>exponent</sup> mod m)</tt>. (Unlike {@code pow}, this 2524 * method permits negative exponents.) 2525 * 2526 * @param exponent the exponent. 2527 * @param m the modulus. 2528 * @return <tt>this<sup>exponent</sup> mod m</tt> 2529 * @throws ArithmeticException {@code m} ≤ 0 or the exponent is 2530 * negative and this BigInteger is not <i>relatively 2531 * prime</i> to {@code m}. 2532 * @see #modInverse 2533 */ 2534 public BigInteger modPow(BigInteger exponent, BigInteger m) { 2535 if (m.signum <= 0) 2536 throw new ArithmeticException("BigInteger: modulus not positive"); 2537 2538 // Trivial cases 2539 if (exponent.signum == 0) 2540 return (m.equals(ONE) ? ZERO : ONE); 2541 2542 if (this.equals(ONE)) 2543 return (m.equals(ONE) ? ZERO : ONE); 2544 2545 if (this.equals(ZERO) && exponent.signum >= 0) 2546 return ZERO; 2547 2548 if (this.equals(negConst[1]) && (!exponent.testBit(0))) 2549 return (m.equals(ONE) ? ZERO : ONE); 2550 2551 boolean invertResult; 2552 if ((invertResult = (exponent.signum < 0))) 2553 exponent = exponent.negate(); 2554 2555 BigInteger base = (this.signum < 0 || this.compareTo(m) >= 0 2556 ? this.mod(m) : this); 2557 BigInteger result; 2558 if (m.testBit(0)) { // odd modulus 2559 result = base.oddModPow(exponent, m); 2560 } else { 2561 /* 2562 * Even modulus. Tear it into an "odd part" (m1) and power of two 2563 * (m2), exponentiate mod m1, manually exponentiate mod m2, and 2564 * use Chinese Remainder Theorem to combine results. 2565 */ 2566 2567 // Tear m apart into odd part (m1) and power of 2 (m2) 2568 int p = m.getLowestSetBit(); // Max pow of 2 that divides m 2569 2570 BigInteger m1 = m.shiftRight(p); // m/2**p 2571 BigInteger m2 = ONE.shiftLeft(p); // 2**p 2572 2573 // Calculate new base from m1 2574 BigInteger base2 = (this.signum < 0 || this.compareTo(m1) >= 0 2575 ? this.mod(m1) : this); 2576 2577 // Caculate (base ** exponent) mod m1. 2578 BigInteger a1 = (m1.equals(ONE) ? ZERO : 2579 base2.oddModPow(exponent, m1)); 2580 2581 // Calculate (this ** exponent) mod m2 2582 BigInteger a2 = base.modPow2(exponent, p); 2583 2584 // Combine results using Chinese Remainder Theorem 2585 BigInteger y1 = m2.modInverse(m1); 2586 BigInteger y2 = m1.modInverse(m2); 2587 2588 if (m.mag.length < MAX_MAG_LENGTH / 2) { 2589 result = a1.multiply(m2).multiply(y1).add(a2.multiply(m1).multiply(y2)).mod(m); 2590 } else { 2591 MutableBigInteger t1 = new MutableBigInteger(); 2592 new MutableBigInteger(a1.multiply(m2)).multiply(new MutableBigInteger(y1), t1); 2593 MutableBigInteger t2 = new MutableBigInteger(); 2594 new MutableBigInteger(a2.multiply(m1)).multiply(new MutableBigInteger(y2), t2); 2595 t1.add(t2); 2596 MutableBigInteger q = new MutableBigInteger(); 2597 result = t1.divide(new MutableBigInteger(m), q).toBigInteger(); 2598 } 2599 } 2600 2601 return (invertResult ? result.modInverse(m) : result); 2602 } 2603 2604 static int[] bnExpModThreshTable = {7, 25, 81, 241, 673, 1793, 2605 Integer.MAX_VALUE}; // Sentinel 2606 2607 /** 2608 * Returns a BigInteger whose value is x to the power of y mod z. 2609 * Assumes: z is odd && x < z. 2610 */ 2611 private BigInteger oddModPow(BigInteger y, BigInteger z) { 2612 /* 2613 * The algorithm is adapted from Colin Plumb's C library. 2614 * 2615 * The window algorithm: 2616 * The idea is to keep a running product of b1 = n^(high-order bits of exp) 2617 * and then keep appending exponent bits to it. The following patterns 2618 * apply to a 3-bit window (k = 3): 2619 * To append 0: square 2620 * To append 1: square, multiply by n^1 2621 * To append 10: square, multiply by n^1, square 2622 * To append 11: square, square, multiply by n^3 2623 * To append 100: square, multiply by n^1, square, square 2624 * To append 101: square, square, square, multiply by n^5 2625 * To append 110: square, square, multiply by n^3, square 2626 * To append 111: square, square, square, multiply by n^7 2627 * 2628 * Since each pattern involves only one multiply, the longer the pattern 2629 * the better, except that a 0 (no multiplies) can be appended directly. 2630 * We precompute a table of odd powers of n, up to 2^k, and can then 2631 * multiply k bits of exponent at a time. Actually, assuming random 2632 * exponents, there is on average one zero bit between needs to 2633 * multiply (1/2 of the time there's none, 1/4 of the time there's 1, 2634 * 1/8 of the time, there's 2, 1/32 of the time, there's 3, etc.), so 2635 * you have to do one multiply per k+1 bits of exponent. 2636 * 2637 * The loop walks down the exponent, squaring the result buffer as 2638 * it goes. There is a wbits+1 bit lookahead buffer, buf, that is 2639 * filled with the upcoming exponent bits. (What is read after the 2640 * end of the exponent is unimportant, but it is filled with zero here.) 2641 * When the most-significant bit of this buffer becomes set, i.e. 2642 * (buf & tblmask) != 0, we have to decide what pattern to multiply 2643 * by, and when to do it. We decide, remember to do it in future 2644 * after a suitable number of squarings have passed (e.g. a pattern 2645 * of "100" in the buffer requires that we multiply by n^1 immediately; 2646 * a pattern of "110" calls for multiplying by n^3 after one more 2647 * squaring), clear the buffer, and continue. 2648 * 2649 * When we start, there is one more optimization: the result buffer 2650 * is implcitly one, so squaring it or multiplying by it can be 2651 * optimized away. Further, if we start with a pattern like "100" 2652 * in the lookahead window, rather than placing n into the buffer 2653 * and then starting to square it, we have already computed n^2 2654 * to compute the odd-powers table, so we can place that into 2655 * the buffer and save a squaring. 2656 * 2657 * This means that if you have a k-bit window, to compute n^z, 2658 * where z is the high k bits of the exponent, 1/2 of the time 2659 * it requires no squarings. 1/4 of the time, it requires 1 2660 * squaring, ... 1/2^(k-1) of the time, it reqires k-2 squarings. 2661 * And the remaining 1/2^(k-1) of the time, the top k bits are a 2662 * 1 followed by k-1 0 bits, so it again only requires k-2 2663 * squarings, not k-1. The average of these is 1. Add that 2664 * to the one squaring we have to do to compute the table, 2665 * and you'll see that a k-bit window saves k-2 squarings 2666 * as well as reducing the multiplies. (It actually doesn't 2667 * hurt in the case k = 1, either.) 2668 */ 2669 // Special case for exponent of one 2670 if (y.equals(ONE)) 2671 return this; 2672 2673 // Special case for base of zero 2674 if (signum == 0) 2675 return ZERO; 2676 2677 int[] base = mag.clone(); 2678 int[] exp = y.mag; 2679 int[] mod = z.mag; 2680 int modLen = mod.length; 2681 2682 // Select an appropriate window size 2683 int wbits = 0; 2684 int ebits = bitLength(exp, exp.length); 2685 // if exponent is 65537 (0x10001), use minimum window size 2686 if ((ebits != 17) || (exp[0] != 65537)) { 2687 while (ebits > bnExpModThreshTable[wbits]) { 2688 wbits++; 2689 } 2690 } 2691 2692 // Calculate appropriate table size 2693 int tblmask = 1 << wbits; 2694 2695 // Allocate table for precomputed odd powers of base in Montgomery form 2696 int[][] table = new int[tblmask][]; 2697 for (int i=0; i < tblmask; i++) 2698 table[i] = new int[modLen]; 2699 2700 // Compute the modular inverse 2701 int inv = -MutableBigInteger.inverseMod32(mod[modLen-1]); 2702 2703 // Convert base to Montgomery form 2704 int[] a = leftShift(base, base.length, modLen << 5); 2705 2706 MutableBigInteger q = new MutableBigInteger(), 2707 a2 = new MutableBigInteger(a), 2708 b2 = new MutableBigInteger(mod); 2709 2710 MutableBigInteger r= a2.divide(b2, q); 2711 table[0] = r.toIntArray(); 2712 2713 // Pad table[0] with leading zeros so its length is at least modLen 2714 if (table[0].length < modLen) { 2715 int offset = modLen - table[0].length; 2716 int[] t2 = new int[modLen]; 2717 for (int i=0; i < table[0].length; i++) 2718 t2[i+offset] = table[0][i]; 2719 table[0] = t2; 2720 } 2721 2722 // Set b to the square of the base 2723 int[] b = squareToLen(table[0], modLen, null); 2724 b = montReduce(b, mod, modLen, inv); 2725 2726 // Set t to high half of b 2727 int[] t = Arrays.copyOf(b, modLen); 2728 2729 // Fill in the table with odd powers of the base 2730 for (int i=1; i < tblmask; i++) { 2731 int[] prod = multiplyToLen(t, modLen, table[i-1], modLen, null); 2732 table[i] = montReduce(prod, mod, modLen, inv); 2733 } 2734 2735 // Pre load the window that slides over the exponent 2736 int bitpos = 1 << ((ebits-1) & (32-1)); 2737 2738 int buf = 0; 2739 int elen = exp.length; 2740 int eIndex = 0; 2741 for (int i = 0; i <= wbits; i++) { 2742 buf = (buf << 1) | (((exp[eIndex] & bitpos) != 0)?1:0); 2743 bitpos >>>= 1; 2744 if (bitpos == 0) { 2745 eIndex++; 2746 bitpos = 1 << (32-1); 2747 elen--; 2748 } 2749 } 2750 2751 int multpos = ebits; 2752 2753 // The first iteration, which is hoisted out of the main loop 2754 ebits--; 2755 boolean isone = true; 2756 2757 multpos = ebits - wbits; 2758 while ((buf & 1) == 0) { 2759 buf >>>= 1; 2760 multpos++; 2761 } 2762 2763 int[] mult = table[buf >>> 1]; 2764 2765 buf = 0; 2766 if (multpos == ebits) 2767 isone = false; 2768 2769 // The main loop 2770 while (true) { 2771 ebits--; 2772 // Advance the window 2773 buf <<= 1; 2774 2775 if (elen != 0) { 2776 buf |= ((exp[eIndex] & bitpos) != 0) ? 1 : 0; 2777 bitpos >>>= 1; 2778 if (bitpos == 0) { 2779 eIndex++; 2780 bitpos = 1 << (32-1); 2781 elen--; 2782 } 2783 } 2784 2785 // Examine the window for pending multiplies 2786 if ((buf & tblmask) != 0) { 2787 multpos = ebits - wbits; 2788 while ((buf & 1) == 0) { 2789 buf >>>= 1; 2790 multpos++; 2791 } 2792 mult = table[buf >>> 1]; 2793 buf = 0; 2794 } 2795 2796 // Perform multiply 2797 if (ebits == multpos) { 2798 if (isone) { 2799 b = mult.clone(); 2800 isone = false; 2801 } else { 2802 t = b; 2803 a = multiplyToLen(t, modLen, mult, modLen, a); 2804 a = montReduce(a, mod, modLen, inv); 2805 t = a; a = b; b = t; 2806 } 2807 } 2808 2809 // Check if done 2810 if (ebits == 0) 2811 break; 2812 2813 // Square the input 2814 if (!isone) { 2815 t = b; 2816 a = squareToLen(t, modLen, a); 2817 a = montReduce(a, mod, modLen, inv); 2818 t = a; a = b; b = t; 2819 } 2820 } 2821 2822 // Convert result out of Montgomery form and return 2823 int[] t2 = new int[2*modLen]; 2824 System.arraycopy(b, 0, t2, modLen, modLen); 2825 2826 b = montReduce(t2, mod, modLen, inv); 2827 2828 t2 = Arrays.copyOf(b, modLen); 2829 2830 return new BigInteger(1, t2); 2831 } 2832 2833 /** 2834 * Montgomery reduce n, modulo mod. This reduces modulo mod and divides 2835 * by 2^(32*mlen). Adapted from Colin Plumb's C library. 2836 */ 2837 private static int[] montReduce(int[] n, int[] mod, int mlen, int inv) { 2838 int c=0; 2839 int len = mlen; 2840 int offset=0; 2841 2842 do { 2843 int nEnd = n[n.length-1-offset]; 2844 int carry = mulAdd(n, mod, offset, mlen, inv * nEnd); 2845 c += addOne(n, offset, mlen, carry); 2846 offset++; 2847 } while (--len > 0); 2848 2849 while (c > 0) 2850 c += subN(n, mod, mlen); 2851 2852 while (intArrayCmpToLen(n, mod, mlen) >= 0) 2853 subN(n, mod, mlen); 2854 2855 return n; 2856 } 2857 2858 2859 /* 2860 * Returns -1, 0 or +1 as big-endian unsigned int array arg1 is less than, 2861 * equal to, or greater than arg2 up to length len. 2862 */ 2863 private static int intArrayCmpToLen(int[] arg1, int[] arg2, int len) { 2864 for (int i=0; i < len; i++) { 2865 long b1 = arg1[i] & LONG_MASK; 2866 long b2 = arg2[i] & LONG_MASK; 2867 if (b1 < b2) 2868 return -1; 2869 if (b1 > b2) 2870 return 1; 2871 } 2872 return 0; 2873 } 2874 2875 /** 2876 * Subtracts two numbers of same length, returning borrow. 2877 */ 2878 private static int subN(int[] a, int[] b, int len) { 2879 long sum = 0; 2880 2881 while (--len >= 0) { 2882 sum = (a[len] & LONG_MASK) - 2883 (b[len] & LONG_MASK) + (sum >> 32); 2884 a[len] = (int)sum; 2885 } 2886 2887 return (int)(sum >> 32); 2888 } 2889 2890 /** 2891 * Multiply an array by one word k and add to result, return the carry 2892 */ 2893 static int mulAdd(int[] out, int[] in, int offset, int len, int k) { 2894 implMulAddCheck(out, in, offset, len, k); 2895 return implMulAdd(out, in, offset, len, k); 2896 } 2897 2898 /** 2899 * Parameters validation. 2900 */ 2901 private static void implMulAddCheck(int[] out, int[] in, int offset, int len, int k) { 2902 if (len > in.length) { 2903 throw new IllegalArgumentException("input length is out of bound: " + len + " > " + in.length); 2904 } 2905 if (offset < 0) { 2906 throw new IllegalArgumentException("input offset is invalid: " + offset); 2907 } 2908 if (offset > (out.length - 1)) { 2909 throw new IllegalArgumentException("input offset is out of bound: " + offset + " > " + (out.length - 1)); 2910 } 2911 if (len > (out.length - offset)) { 2912 throw new IllegalArgumentException("input len is out of bound: " + len + " > " + (out.length - offset)); 2913 } 2914 } 2915 2916 /** 2917 * Java Runtime may use intrinsic for this method. 2918 */ 2919 private static int implMulAdd(int[] out, int[] in, int offset, int len, int k) { 2920 long kLong = k & LONG_MASK; 2921 long carry = 0; 2922 2923 offset = out.length-offset - 1; 2924 for (int j=len-1; j >= 0; j--) { 2925 long product = (in[j] & LONG_MASK) * kLong + 2926 (out[offset] & LONG_MASK) + carry; 2927 out[offset--] = (int)product; 2928 carry = product >>> 32; 2929 } 2930 return (int)carry; 2931 } 2932 2933 /** 2934 * Add one word to the number a mlen words into a. Return the resulting 2935 * carry. 2936 */ 2937 static int addOne(int[] a, int offset, int mlen, int carry) { 2938 offset = a.length-1-mlen-offset; 2939 long t = (a[offset] & LONG_MASK) + (carry & LONG_MASK); 2940 2941 a[offset] = (int)t; 2942 if ((t >>> 32) == 0) 2943 return 0; 2944 while (--mlen >= 0) { 2945 if (--offset < 0) { // Carry out of number 2946 return 1; 2947 } else { 2948 a[offset]++; 2949 if (a[offset] != 0) 2950 return 0; 2951 } 2952 } 2953 return 1; 2954 } 2955 2956 /** 2957 * Returns a BigInteger whose value is (this ** exponent) mod (2**p) 2958 */ 2959 private BigInteger modPow2(BigInteger exponent, int p) { 2960 /* 2961 * Perform exponentiation using repeated squaring trick, chopping off 2962 * high order bits as indicated by modulus. 2963 */ 2964 BigInteger result = ONE; 2965 BigInteger baseToPow2 = this.mod2(p); 2966 int expOffset = 0; 2967 2968 int limit = exponent.bitLength(); 2969 2970 if (this.testBit(0)) 2971 limit = (p-1) < limit ? (p-1) : limit; 2972 2973 while (expOffset < limit) { 2974 if (exponent.testBit(expOffset)) 2975 result = result.multiply(baseToPow2).mod2(p); 2976 expOffset++; 2977 if (expOffset < limit) 2978 baseToPow2 = baseToPow2.square().mod2(p); 2979 } 2980 2981 return result; 2982 } 2983 2984 /** 2985 * Returns a BigInteger whose value is this mod(2**p). 2986 * Assumes that this {@code BigInteger >= 0} and {@code p > 0}. 2987 */ 2988 private BigInteger mod2(int p) { 2989 if (bitLength() <= p) 2990 return this; 2991 2992 // Copy remaining ints of mag 2993 int numInts = (p + 31) >>> 5; 2994 int[] mag = new int[numInts]; 2995 System.arraycopy(this.mag, (this.mag.length - numInts), mag, 0, numInts); 2996 2997 // Mask out any excess bits 2998 int excessBits = (numInts << 5) - p; 2999 mag[0] &= (1L << (32-excessBits)) - 1; 3000 3001 return (mag[0] == 0 ? new BigInteger(1, mag) : new BigInteger(mag, 1)); 3002 } 3003 3004 /** 3005 * Returns a BigInteger whose value is {@code (this}<sup>-1</sup> {@code mod m)}. 3006 * 3007 * @param m the modulus. 3008 * @return {@code this}<sup>-1</sup> {@code mod m}. 3009 * @throws ArithmeticException {@code m} ≤ 0, or this BigInteger 3010 * has no multiplicative inverse mod m (that is, this BigInteger 3011 * is not <i>relatively prime</i> to m). 3012 */ 3013 public BigInteger modInverse(BigInteger m) { 3014 if (m.signum != 1) 3015 throw new ArithmeticException("BigInteger: modulus not positive"); 3016 3017 if (m.equals(ONE)) 3018 return ZERO; 3019 3020 // Calculate (this mod m) 3021 BigInteger modVal = this; 3022 if (signum < 0 || (this.compareMagnitude(m) >= 0)) 3023 modVal = this.mod(m); 3024 3025 if (modVal.equals(ONE)) 3026 return ONE; 3027 3028 MutableBigInteger a = new MutableBigInteger(modVal); 3029 MutableBigInteger b = new MutableBigInteger(m); 3030 3031 MutableBigInteger result = a.mutableModInverse(b); 3032 return result.toBigInteger(1); 3033 } 3034 3035 // Shift Operations 3036 3037 /** 3038 * Returns a BigInteger whose value is {@code (this << n)}. 3039 * The shift distance, {@code n}, may be negative, in which case 3040 * this method performs a right shift. 3041 * (Computes <tt>floor(this * 2<sup>n</sup>)</tt>.) 3042 * 3043 * @param n shift distance, in bits. 3044 * @return {@code this << n} 3045 * @see #shiftRight 3046 */ 3047 public BigInteger shiftLeft(int n) { 3048 if (signum == 0) 3049 return ZERO; 3050 if (n > 0) { 3051 return new BigInteger(shiftLeft(mag, n), signum); 3052 } else if (n == 0) { 3053 return this; 3054 } else { 3055 // Possible int overflow in (-n) is not a trouble, 3056 // because shiftRightImpl considers its argument unsigned 3057 return shiftRightImpl(-n); 3058 } 3059 } 3060 3061 /** 3062 * Returns a magnitude array whose value is {@code (mag << n)}. 3063 * The shift distance, {@code n}, is considered unnsigned. 3064 * (Computes <tt>this * 2<sup>n</sup></tt>.) 3065 * 3066 * @param mag magnitude, the most-significant int ({@code mag[0]}) must be non-zero. 3067 * @param n unsigned shift distance, in bits. 3068 * @return {@code mag << n} 3069 */ 3070 private static int[] shiftLeft(int[] mag, int n) { 3071 int nInts = n >>> 5; 3072 int nBits = n & 0x1f; 3073 int magLen = mag.length; 3074 int newMag[] = null; 3075 3076 if (nBits == 0) { 3077 newMag = new int[magLen + nInts]; 3078 System.arraycopy(mag, 0, newMag, 0, magLen); 3079 } else { 3080 int i = 0; 3081 int nBits2 = 32 - nBits; 3082 int highBits = mag[0] >>> nBits2; 3083 if (highBits != 0) { 3084 newMag = new int[magLen + nInts + 1]; 3085 newMag[i++] = highBits; 3086 } else { 3087 newMag = new int[magLen + nInts]; 3088 } 3089 int j=0; 3090 while (j < magLen-1) 3091 newMag[i++] = mag[j++] << nBits | mag[j] >>> nBits2; 3092 newMag[i] = mag[j] << nBits; 3093 } 3094 return newMag; 3095 } 3096 3097 /** 3098 * Returns a BigInteger whose value is {@code (this >> n)}. Sign 3099 * extension is performed. The shift distance, {@code n}, may be 3100 * negative, in which case this method performs a left shift. 3101 * (Computes <tt>floor(this / 2<sup>n</sup>)</tt>.) 3102 * 3103 * @param n shift distance, in bits. 3104 * @return {@code this >> n} 3105 * @see #shiftLeft 3106 */ 3107 public BigInteger shiftRight(int n) { 3108 if (signum == 0) 3109 return ZERO; 3110 if (n > 0) { 3111 return shiftRightImpl(n); 3112 } else if (n == 0) { 3113 return this; 3114 } else { 3115 // Possible int overflow in {@code -n} is not a trouble, 3116 // because shiftLeft considers its argument unsigned 3117 return new BigInteger(shiftLeft(mag, -n), signum); 3118 } 3119 } 3120 3121 /** 3122 * Returns a BigInteger whose value is {@code (this >> n)}. The shift 3123 * distance, {@code n}, is considered unsigned. 3124 * (Computes <tt>floor(this * 2<sup>-n</sup>)</tt>.) 3125 * 3126 * @param n unsigned shift distance, in bits. 3127 * @return {@code this >> n} 3128 */ 3129 private BigInteger shiftRightImpl(int n) { 3130 int nInts = n >>> 5; 3131 int nBits = n & 0x1f; 3132 int magLen = mag.length; 3133 int newMag[] = null; 3134 3135 // Special case: entire contents shifted off the end 3136 if (nInts >= magLen) 3137 return (signum >= 0 ? ZERO : negConst[1]); 3138 3139 if (nBits == 0) { 3140 int newMagLen = magLen - nInts; 3141 newMag = Arrays.copyOf(mag, newMagLen); 3142 } else { 3143 int i = 0; 3144 int highBits = mag[0] >>> nBits; 3145 if (highBits != 0) { 3146 newMag = new int[magLen - nInts]; 3147 newMag[i++] = highBits; 3148 } else { 3149 newMag = new int[magLen - nInts -1]; 3150 } 3151 3152 int nBits2 = 32 - nBits; 3153 int j=0; 3154 while (j < magLen - nInts - 1) 3155 newMag[i++] = (mag[j++] << nBits2) | (mag[j] >>> nBits); 3156 } 3157 3158 if (signum < 0) { 3159 // Find out whether any one-bits were shifted off the end. 3160 boolean onesLost = false; 3161 for (int i=magLen-1, j=magLen-nInts; i >= j && !onesLost; i--) 3162 onesLost = (mag[i] != 0); 3163 if (!onesLost && nBits != 0) 3164 onesLost = (mag[magLen - nInts - 1] << (32 - nBits) != 0); 3165 3166 if (onesLost) 3167 newMag = javaIncrement(newMag); 3168 } 3169 3170 return new BigInteger(newMag, signum); 3171 } 3172 3173 int[] javaIncrement(int[] val) { 3174 int lastSum = 0; 3175 for (int i=val.length-1; i >= 0 && lastSum == 0; i--) 3176 lastSum = (val[i] += 1); 3177 if (lastSum == 0) { 3178 val = new int[val.length+1]; 3179 val[0] = 1; 3180 } 3181 return val; 3182 } 3183 3184 // Bitwise Operations 3185 3186 /** 3187 * Returns a BigInteger whose value is {@code (this & val)}. (This 3188 * method returns a negative BigInteger if and only if this and val are 3189 * both negative.) 3190 * 3191 * @param val value to be AND'ed with this BigInteger. 3192 * @return {@code this & val} 3193 */ 3194 public BigInteger and(BigInteger val) { 3195 int[] result = new int[Math.max(intLength(), val.intLength())]; 3196 for (int i=0; i < result.length; i++) 3197 result[i] = (getInt(result.length-i-1) 3198 & val.getInt(result.length-i-1)); 3199 3200 return valueOf(result); 3201 } 3202 3203 /** 3204 * Returns a BigInteger whose value is {@code (this | val)}. (This method 3205 * returns a negative BigInteger if and only if either this or val is 3206 * negative.) 3207 * 3208 * @param val value to be OR'ed with this BigInteger. 3209 * @return {@code this | val} 3210 */ 3211 public BigInteger or(BigInteger val) { 3212 int[] result = new int[Math.max(intLength(), val.intLength())]; 3213 for (int i=0; i < result.length; i++) 3214 result[i] = (getInt(result.length-i-1) 3215 | val.getInt(result.length-i-1)); 3216 3217 return valueOf(result); 3218 } 3219 3220 /** 3221 * Returns a BigInteger whose value is {@code (this ^ val)}. (This method 3222 * returns a negative BigInteger if and only if exactly one of this and 3223 * val are negative.) 3224 * 3225 * @param val value to be XOR'ed with this BigInteger. 3226 * @return {@code this ^ val} 3227 */ 3228 public BigInteger xor(BigInteger val) { 3229 int[] result = new int[Math.max(intLength(), val.intLength())]; 3230 for (int i=0; i < result.length; i++) 3231 result[i] = (getInt(result.length-i-1) 3232 ^ val.getInt(result.length-i-1)); 3233 3234 return valueOf(result); 3235 } 3236 3237 /** 3238 * Returns a BigInteger whose value is {@code (~this)}. (This method 3239 * returns a negative value if and only if this BigInteger is 3240 * non-negative.) 3241 * 3242 * @return {@code ~this} 3243 */ 3244 public BigInteger not() { 3245 int[] result = new int[intLength()]; 3246 for (int i=0; i < result.length; i++) 3247 result[i] = ~getInt(result.length-i-1); 3248 3249 return valueOf(result); 3250 } 3251 3252 /** 3253 * Returns a BigInteger whose value is {@code (this & ~val)}. This 3254 * method, which is equivalent to {@code and(val.not())}, is provided as 3255 * a convenience for masking operations. (This method returns a negative 3256 * BigInteger if and only if {@code this} is negative and {@code val} is 3257 * positive.) 3258 * 3259 * @param val value to be complemented and AND'ed with this BigInteger. 3260 * @return {@code this & ~val} 3261 */ 3262 public BigInteger andNot(BigInteger val) { 3263 int[] result = new int[Math.max(intLength(), val.intLength())]; 3264 for (int i=0; i < result.length; i++) 3265 result[i] = (getInt(result.length-i-1) 3266 & ~val.getInt(result.length-i-1)); 3267 3268 return valueOf(result); 3269 } 3270 3271 3272 // Single Bit Operations 3273 3274 /** 3275 * Returns {@code true} if and only if the designated bit is set. 3276 * (Computes {@code ((this & (1<<n)) != 0)}.) 3277 * 3278 * @param n index of bit to test. 3279 * @return {@code true} if and only if the designated bit is set. 3280 * @throws ArithmeticException {@code n} is negative. 3281 */ 3282 public boolean testBit(int n) { 3283 if (n < 0) 3284 throw new ArithmeticException("Negative bit address"); 3285 3286 return (getInt(n >>> 5) & (1 << (n & 31))) != 0; 3287 } 3288 3289 /** 3290 * Returns a BigInteger whose value is equivalent to this BigInteger 3291 * with the designated bit set. (Computes {@code (this | (1<<n))}.) 3292 * 3293 * @param n index of bit to set. 3294 * @return {@code this | (1<<n)} 3295 * @throws ArithmeticException {@code n} is negative. 3296 */ 3297 public BigInteger setBit(int n) { 3298 if (n < 0) 3299 throw new ArithmeticException("Negative bit address"); 3300 3301 int intNum = n >>> 5; 3302 int[] result = new int[Math.max(intLength(), intNum+2)]; 3303 3304 for (int i=0; i < result.length; i++) 3305 result[result.length-i-1] = getInt(i); 3306 3307 result[result.length-intNum-1] |= (1 << (n & 31)); 3308 3309 return valueOf(result); 3310 } 3311 3312 /** 3313 * Returns a BigInteger whose value is equivalent to this BigInteger 3314 * with the designated bit cleared. 3315 * (Computes {@code (this & ~(1<<n))}.) 3316 * 3317 * @param n index of bit to clear. 3318 * @return {@code this & ~(1<<n)} 3319 * @throws ArithmeticException {@code n} is negative. 3320 */ 3321 public BigInteger clearBit(int n) { 3322 if (n < 0) 3323 throw new ArithmeticException("Negative bit address"); 3324 3325 int intNum = n >>> 5; 3326 int[] result = new int[Math.max(intLength(), ((n + 1) >>> 5) + 1)]; 3327 3328 for (int i=0; i < result.length; i++) 3329 result[result.length-i-1] = getInt(i); 3330 3331 result[result.length-intNum-1] &= ~(1 << (n & 31)); 3332 3333 return valueOf(result); 3334 } 3335 3336 /** 3337 * Returns a BigInteger whose value is equivalent to this BigInteger 3338 * with the designated bit flipped. 3339 * (Computes {@code (this ^ (1<<n))}.) 3340 * 3341 * @param n index of bit to flip. 3342 * @return {@code this ^ (1<<n)} 3343 * @throws ArithmeticException {@code n} is negative. 3344 */ 3345 public BigInteger flipBit(int n) { 3346 if (n < 0) 3347 throw new ArithmeticException("Negative bit address"); 3348 3349 int intNum = n >>> 5; 3350 int[] result = new int[Math.max(intLength(), intNum+2)]; 3351 3352 for (int i=0; i < result.length; i++) 3353 result[result.length-i-1] = getInt(i); 3354 3355 result[result.length-intNum-1] ^= (1 << (n & 31)); 3356 3357 return valueOf(result); 3358 } 3359 3360 /** 3361 * Returns the index of the rightmost (lowest-order) one bit in this 3362 * BigInteger (the number of zero bits to the right of the rightmost 3363 * one bit). Returns -1 if this BigInteger contains no one bits. 3364 * (Computes {@code (this == 0? -1 : log2(this & -this))}.) 3365 * 3366 * @return index of the rightmost one bit in this BigInteger. 3367 */ 3368 public int getLowestSetBit() { 3369 int lsb = lowestSetBitPlusTwo - 2; 3370 if (lsb == -2) { // lowestSetBit not initialized yet 3371 lsb = 0; 3372 if (signum == 0) { 3373 lsb -= 1; 3374 } else { 3375 // Search for lowest order nonzero int 3376 int i,b; 3377 for (i=0; (b = getInt(i)) == 0; i++) 3378 ; 3379 lsb += (i << 5) + Integer.numberOfTrailingZeros(b); 3380 } 3381 lowestSetBitPlusTwo = lsb + 2; 3382 } 3383 return lsb; 3384 } 3385 3386 3387 // Miscellaneous Bit Operations 3388 3389 /** 3390 * Returns the number of bits in the minimal two's-complement 3391 * representation of this BigInteger, <i>excluding</i> a sign bit. 3392 * For positive BigIntegers, this is equivalent to the number of bits in 3393 * the ordinary binary representation. (Computes 3394 * {@code (ceil(log2(this < 0 ? -this : this+1)))}.) 3395 * 3396 * @return number of bits in the minimal two's-complement 3397 * representation of this BigInteger, <i>excluding</i> a sign bit. 3398 */ 3399 public int bitLength() { 3400 int n = bitLengthPlusOne - 1; 3401 if (n == -1) { // bitLength not initialized yet 3402 int[] m = mag; 3403 int len = m.length; 3404 if (len == 0) { 3405 n = 0; // offset by one to initialize 3406 } else { 3407 // Calculate the bit length of the magnitude 3408 int magBitLength = ((len - 1) << 5) + bitLengthForInt(mag[0]); 3409 if (signum < 0) { 3410 // Check if magnitude is a power of two 3411 boolean pow2 = (Integer.bitCount(mag[0]) == 1); 3412 for (int i=1; i< len && pow2; i++) 3413 pow2 = (mag[i] == 0); 3414 3415 n = (pow2 ? magBitLength -1 : magBitLength); 3416 } else { 3417 n = magBitLength; 3418 } 3419 } 3420 bitLengthPlusOne = n + 1; 3421 } 3422 return n; 3423 } 3424 3425 /** 3426 * Returns the number of bits in the two's complement representation 3427 * of this BigInteger that differ from its sign bit. This method is 3428 * useful when implementing bit-vector style sets atop BigIntegers. 3429 * 3430 * @return number of bits in the two's complement representation 3431 * of this BigInteger that differ from its sign bit. 3432 */ 3433 public int bitCount() { 3434 int bc = bitCountPlusOne - 1; 3435 if (bc == -1) { // bitCount not initialized yet 3436 bc = 0; // offset by one to initialize 3437 // Count the bits in the magnitude 3438 for (int i=0; i < mag.length; i++) 3439 bc += Integer.bitCount(mag[i]); 3440 if (signum < 0) { 3441 // Count the trailing zeros in the magnitude 3442 int magTrailingZeroCount = 0, j; 3443 for (j=mag.length-1; mag[j] == 0; j--) 3444 magTrailingZeroCount += 32; 3445 magTrailingZeroCount += Integer.numberOfTrailingZeros(mag[j]); 3446 bc += magTrailingZeroCount - 1; 3447 } 3448 bitCountPlusOne = bc + 1; 3449 } 3450 return bc; 3451 } 3452 3453 // Primality Testing 3454 3455 /** 3456 * Returns {@code true} if this BigInteger is probably prime, 3457 * {@code false} if it's definitely composite. If 3458 * {@code certainty} is ≤ 0, {@code true} is 3459 * returned. 3460 * 3461 * @param certainty a measure of the uncertainty that the caller is 3462 * willing to tolerate: if the call returns {@code true} 3463 * the probability that this BigInteger is prime exceeds 3464 * (1 - 1/2<sup>{@code certainty}</sup>). The execution time of 3465 * this method is proportional to the value of this parameter. 3466 * @return {@code true} if this BigInteger is probably prime, 3467 * {@code false} if it's definitely composite. 3468 */ 3469 public boolean isProbablePrime(int certainty) { 3470 if (certainty <= 0) 3471 return true; 3472 BigInteger w = this.abs(); 3473 if (w.equals(TWO)) 3474 return true; 3475 if (!w.testBit(0) || w.equals(ONE)) 3476 return false; 3477 3478 return w.primeToCertainty(certainty, null); 3479 } 3480 3481 // Comparison Operations 3482 3483 /** 3484 * Compares this BigInteger with the specified BigInteger. This 3485 * method is provided in preference to individual methods for each 3486 * of the six boolean comparison operators ({@literal <}, ==, 3487 * {@literal >}, {@literal >=}, !=, {@literal <=}). The suggested 3488 * idiom for performing these comparisons is: {@code 3489 * (x.compareTo(y)} <<i>op</i>> {@code 0)}, where 3490 * <<i>op</i>> is one of the six comparison operators. 3491 * 3492 * @param val BigInteger to which this BigInteger is to be compared. 3493 * @return -1, 0 or 1 as this BigInteger is numerically less than, equal 3494 * to, or greater than {@code val}. 3495 */ 3496 public int compareTo(BigInteger val) { 3497 if (signum == val.signum) { 3498 switch (signum) { 3499 case 1: 3500 return compareMagnitude(val); 3501 case -1: 3502 return val.compareMagnitude(this); 3503 default: 3504 return 0; 3505 } 3506 } 3507 return signum > val.signum ? 1 : -1; 3508 } 3509 3510 /** 3511 * Compares the magnitude array of this BigInteger with the specified 3512 * BigInteger's. This is the version of compareTo ignoring sign. 3513 * 3514 * @param val BigInteger whose magnitude array to be compared. 3515 * @return -1, 0 or 1 as this magnitude array is less than, equal to or 3516 * greater than the magnitude aray for the specified BigInteger's. 3517 */ 3518 final int compareMagnitude(BigInteger val) { 3519 int[] m1 = mag; 3520 int len1 = m1.length; 3521 int[] m2 = val.mag; 3522 int len2 = m2.length; 3523 if (len1 < len2) 3524 return -1; 3525 if (len1 > len2) 3526 return 1; 3527 for (int i = 0; i < len1; i++) { 3528 int a = m1[i]; 3529 int b = m2[i]; 3530 if (a != b) 3531 return ((a & LONG_MASK) < (b & LONG_MASK)) ? -1 : 1; 3532 } 3533 return 0; 3534 } 3535 3536 /** 3537 * Version of compareMagnitude that compares magnitude with long value. 3538 * val can't be Long.MIN_VALUE. 3539 */ 3540 final int compareMagnitude(long val) { 3541 assert val != Long.MIN_VALUE; 3542 int[] m1 = mag; 3543 int len = m1.length; 3544 if (len > 2) { 3545 return 1; 3546 } 3547 if (val < 0) { 3548 val = -val; 3549 } 3550 int highWord = (int)(val >>> 32); 3551 if (highWord == 0) { 3552 if (len < 1) 3553 return -1; 3554 if (len > 1) 3555 return 1; 3556 int a = m1[0]; 3557 int b = (int)val; 3558 if (a != b) { 3559 return ((a & LONG_MASK) < (b & LONG_MASK))? -1 : 1; 3560 } 3561 return 0; 3562 } else { 3563 if (len < 2) 3564 return -1; 3565 int a = m1[0]; 3566 int b = highWord; 3567 if (a != b) { 3568 return ((a & LONG_MASK) < (b & LONG_MASK))? -1 : 1; 3569 } 3570 a = m1[1]; 3571 b = (int)val; 3572 if (a != b) { 3573 return ((a & LONG_MASK) < (b & LONG_MASK))? -1 : 1; 3574 } 3575 return 0; 3576 } 3577 } 3578 3579 /** 3580 * Compares this BigInteger with the specified Object for equality. 3581 * 3582 * @param x Object to which this BigInteger is to be compared. 3583 * @return {@code true} if and only if the specified Object is a 3584 * BigInteger whose value is numerically equal to this BigInteger. 3585 */ 3586 public boolean equals(Object x) { 3587 // This test is just an optimization, which may or may not help 3588 if (x == this) 3589 return true; 3590 3591 if (!(x instanceof BigInteger)) 3592 return false; 3593 3594 BigInteger xInt = (BigInteger) x; 3595 if (xInt.signum != signum) 3596 return false; 3597 3598 int[] m = mag; 3599 int len = m.length; 3600 int[] xm = xInt.mag; 3601 if (len != xm.length) 3602 return false; 3603 3604 for (int i = 0; i < len; i++) 3605 if (xm[i] != m[i]) 3606 return false; 3607 3608 return true; 3609 } 3610 3611 /** 3612 * Returns the minimum of this BigInteger and {@code val}. 3613 * 3614 * @param val value with which the minimum is to be computed. 3615 * @return the BigInteger whose value is the lesser of this BigInteger and 3616 * {@code val}. If they are equal, either may be returned. 3617 */ 3618 public BigInteger min(BigInteger val) { 3619 return (compareTo(val) < 0 ? this : val); 3620 } 3621 3622 /** 3623 * Returns the maximum of this BigInteger and {@code val}. 3624 * 3625 * @param val value with which the maximum is to be computed. 3626 * @return the BigInteger whose value is the greater of this and 3627 * {@code val}. If they are equal, either may be returned. 3628 */ 3629 public BigInteger max(BigInteger val) { 3630 return (compareTo(val) > 0 ? this : val); 3631 } 3632 3633 3634 // Hash Function 3635 3636 /** 3637 * Returns the hash code for this BigInteger. 3638 * 3639 * @return hash code for this BigInteger. 3640 */ 3641 public int hashCode() { 3642 int hashCode = 0; 3643 3644 for (int i=0; i < mag.length; i++) 3645 hashCode = (int)(31*hashCode + (mag[i] & LONG_MASK)); 3646 3647 return hashCode * signum; 3648 } 3649 3650 /** 3651 * Returns the String representation of this BigInteger in the 3652 * given radix. If the radix is outside the range from {@link 3653 * Character#MIN_RADIX} to {@link Character#MAX_RADIX} inclusive, 3654 * it will default to 10 (as is the case for 3655 * {@code Integer.toString}). The digit-to-character mapping 3656 * provided by {@code Character.forDigit} is used, and a minus 3657 * sign is prepended if appropriate. (This representation is 3658 * compatible with the {@link #BigInteger(String, int) (String, 3659 * int)} constructor.) 3660 * 3661 * @param radix radix of the String representation. 3662 * @return String representation of this BigInteger in the given radix. 3663 * @see Integer#toString 3664 * @see Character#forDigit 3665 * @see #BigInteger(java.lang.String, int) 3666 */ 3667 public String toString(int radix) { 3668 if (signum == 0) 3669 return "0"; 3670 if (radix < Character.MIN_RADIX || radix > Character.MAX_RADIX) 3671 radix = 10; 3672 3673 // If it's small enough, use smallToString. 3674 if (mag.length <= SCHOENHAGE_BASE_CONVERSION_THRESHOLD) 3675 return smallToString(radix); 3676 3677 // Otherwise use recursive toString, which requires positive arguments. 3678 // The results will be concatenated into this StringBuilder 3679 StringBuilder sb = new StringBuilder(); 3680 if (signum < 0) { 3681 toString(this.negate(), sb, radix, 0); 3682 sb.insert(0, '-'); 3683 } 3684 else 3685 toString(this, sb, radix, 0); 3686 3687 return sb.toString(); 3688 } 3689 3690 /** This method is used to perform toString when arguments are small. */ 3691 private String smallToString(int radix) { 3692 if (signum == 0) { 3693 return "0"; 3694 } 3695 3696 // Compute upper bound on number of digit groups and allocate space 3697 int maxNumDigitGroups = (4*mag.length + 6)/7; 3698 String digitGroup[] = new String[maxNumDigitGroups]; 3699 3700 // Translate number to string, a digit group at a time 3701 BigInteger tmp = this.abs(); 3702 int numGroups = 0; 3703 while (tmp.signum != 0) { 3704 BigInteger d = longRadix[radix]; 3705 3706 MutableBigInteger q = new MutableBigInteger(), 3707 a = new MutableBigInteger(tmp.mag), 3708 b = new MutableBigInteger(d.mag); 3709 MutableBigInteger r = a.divide(b, q); 3710 BigInteger q2 = q.toBigInteger(tmp.signum * d.signum); 3711 BigInteger r2 = r.toBigInteger(tmp.signum * d.signum); 3712 3713 digitGroup[numGroups++] = Long.toString(r2.longValue(), radix); 3714 tmp = q2; 3715 } 3716 3717 // Put sign (if any) and first digit group into result buffer 3718 StringBuilder buf = new StringBuilder(numGroups*digitsPerLong[radix]+1); 3719 if (signum < 0) { 3720 buf.append('-'); 3721 } 3722 buf.append(digitGroup[numGroups-1]); 3723 3724 // Append remaining digit groups padded with leading zeros 3725 for (int i=numGroups-2; i >= 0; i--) { 3726 // Prepend (any) leading zeros for this digit group 3727 int numLeadingZeros = digitsPerLong[radix]-digitGroup[i].length(); 3728 if (numLeadingZeros != 0) { 3729 buf.append(zeros[numLeadingZeros]); 3730 } 3731 buf.append(digitGroup[i]); 3732 } 3733 return buf.toString(); 3734 } 3735 3736 /** 3737 * Converts the specified BigInteger to a string and appends to 3738 * {@code sb}. This implements the recursive Schoenhage algorithm 3739 * for base conversions. 3740 * <p> 3741 * See Knuth, Donald, _The Art of Computer Programming_, Vol. 2, 3742 * Answers to Exercises (4.4) Question 14. 3743 * 3744 * @param u The number to convert to a string. 3745 * @param sb The StringBuilder that will be appended to in place. 3746 * @param radix The base to convert to. 3747 * @param digits The minimum number of digits to pad to. 3748 */ 3749 private static void toString(BigInteger u, StringBuilder sb, int radix, 3750 int digits) { 3751 // If we're smaller than a certain threshold, use the smallToString 3752 // method, padding with leading zeroes when necessary. 3753 if (u.mag.length <= SCHOENHAGE_BASE_CONVERSION_THRESHOLD) { 3754 String s = u.smallToString(radix); 3755 3756 // Pad with internal zeros if necessary. 3757 // Don't pad if we're at the beginning of the string. 3758 if ((s.length() < digits) && (sb.length() > 0)) { 3759 for (int i=s.length(); i < digits; i++) { 3760 sb.append('0'); 3761 } 3762 } 3763 3764 sb.append(s); 3765 return; 3766 } 3767 3768 int b, n; 3769 b = u.bitLength(); 3770 3771 // Calculate a value for n in the equation radix^(2^n) = u 3772 // and subtract 1 from that value. This is used to find the 3773 // cache index that contains the best value to divide u. 3774 n = (int) Math.round(Math.log(b * LOG_TWO / logCache[radix]) / LOG_TWO - 1.0); 3775 BigInteger v = getRadixConversionCache(radix, n); 3776 BigInteger[] results; 3777 results = u.divideAndRemainder(v); 3778 3779 int expectedDigits = 1 << n; 3780 3781 // Now recursively build the two halves of each number. 3782 toString(results[0], sb, radix, digits-expectedDigits); 3783 toString(results[1], sb, radix, expectedDigits); 3784 } 3785 3786 /** 3787 * Returns the value radix^(2^exponent) from the cache. 3788 * If this value doesn't already exist in the cache, it is added. 3789 * <p> 3790 * This could be changed to a more complicated caching method using 3791 * {@code Future}. 3792 */ 3793 private static BigInteger getRadixConversionCache(int radix, int exponent) { 3794 BigInteger[] cacheLine = powerCache[radix]; // volatile read 3795 if (exponent < cacheLine.length) { 3796 return cacheLine[exponent]; 3797 } 3798 3799 int oldLength = cacheLine.length; 3800 cacheLine = Arrays.copyOf(cacheLine, exponent + 1); 3801 for (int i = oldLength; i <= exponent; i++) { 3802 cacheLine[i] = cacheLine[i - 1].pow(2); 3803 } 3804 3805 BigInteger[][] pc = powerCache; // volatile read again 3806 if (exponent >= pc[radix].length) { 3807 pc = pc.clone(); 3808 pc[radix] = cacheLine; 3809 powerCache = pc; // volatile write, publish 3810 } 3811 return cacheLine[exponent]; 3812 } 3813 3814 /* zero[i] is a string of i consecutive zeros. */ 3815 private static String zeros[] = new String[64]; 3816 static { 3817 zeros[63] = 3818 "000000000000000000000000000000000000000000000000000000000000000"; 3819 for (int i=0; i < 63; i++) 3820 zeros[i] = zeros[63].substring(0, i); 3821 } 3822 3823 /** 3824 * Returns the decimal String representation of this BigInteger. 3825 * The digit-to-character mapping provided by 3826 * {@code Character.forDigit} is used, and a minus sign is 3827 * prepended if appropriate. (This representation is compatible 3828 * with the {@link #BigInteger(String) (String)} constructor, and 3829 * allows for String concatenation with Java's + operator.) 3830 * 3831 * @return decimal String representation of this BigInteger. 3832 * @see Character#forDigit 3833 * @see #BigInteger(java.lang.String) 3834 */ 3835 public String toString() { 3836 return toString(10); 3837 } 3838 3839 /** 3840 * Returns a byte array containing the two's-complement 3841 * representation of this BigInteger. The byte array will be in 3842 * <i>big-endian</i> byte-order: the most significant byte is in 3843 * the zeroth element. The array will contain the minimum number 3844 * of bytes required to represent this BigInteger, including at 3845 * least one sign bit, which is {@code (ceil((this.bitLength() + 3846 * 1)/8))}. (This representation is compatible with the 3847 * {@link #BigInteger(byte[]) (byte[])} constructor.) 3848 * 3849 * @return a byte array containing the two's-complement representation of 3850 * this BigInteger. 3851 * @see #BigInteger(byte[]) 3852 */ 3853 public byte[] toByteArray() { 3854 int byteLen = bitLength()/8 + 1; 3855 byte[] byteArray = new byte[byteLen]; 3856 3857 for (int i=byteLen-1, bytesCopied=4, nextInt=0, intIndex=0; i >= 0; i--) { 3858 if (bytesCopied == 4) { 3859 nextInt = getInt(intIndex++); 3860 bytesCopied = 1; 3861 } else { 3862 nextInt >>>= 8; 3863 bytesCopied++; 3864 } 3865 byteArray[i] = (byte)nextInt; 3866 } 3867 return byteArray; 3868 } 3869 3870 /** 3871 * Converts this BigInteger to an {@code int}. This 3872 * conversion is analogous to a 3873 * <i>narrowing primitive conversion</i> from {@code long} to 3874 * {@code int} as defined in section 5.1.3 of 3875 * <cite>The Java™ Language Specification</cite>: 3876 * if this BigInteger is too big to fit in an 3877 * {@code int}, only the low-order 32 bits are returned. 3878 * Note that this conversion can lose information about the 3879 * overall magnitude of the BigInteger value as well as return a 3880 * result with the opposite sign. 3881 * 3882 * @return this BigInteger converted to an {@code int}. 3883 * @see #intValueExact() 3884 */ 3885 public int intValue() { 3886 int result = 0; 3887 result = getInt(0); 3888 return result; 3889 } 3890 3891 /** 3892 * Converts this BigInteger to a {@code long}. This 3893 * conversion is analogous to a 3894 * <i>narrowing primitive conversion</i> from {@code long} to 3895 * {@code int} as defined in section 5.1.3 of 3896 * <cite>The Java™ Language Specification</cite>: 3897 * if this BigInteger is too big to fit in a 3898 * {@code long}, only the low-order 64 bits are returned. 3899 * Note that this conversion can lose information about the 3900 * overall magnitude of the BigInteger value as well as return a 3901 * result with the opposite sign. 3902 * 3903 * @return this BigInteger converted to a {@code long}. 3904 * @see #longValueExact() 3905 */ 3906 public long longValue() { 3907 long result = 0; 3908 3909 for (int i=1; i >= 0; i--) 3910 result = (result << 32) + (getInt(i) & LONG_MASK); 3911 return result; 3912 } 3913 3914 /** 3915 * Converts this BigInteger to a {@code float}. This 3916 * conversion is similar to the 3917 * <i>narrowing primitive conversion</i> from {@code double} to 3918 * {@code float} as defined in section 5.1.3 of 3919 * <cite>The Java™ Language Specification</cite>: 3920 * if this BigInteger has too great a magnitude 3921 * to represent as a {@code float}, it will be converted to 3922 * {@link Float#NEGATIVE_INFINITY} or {@link 3923 * Float#POSITIVE_INFINITY} as appropriate. Note that even when 3924 * the return value is finite, this conversion can lose 3925 * information about the precision of the BigInteger value. 3926 * 3927 * @return this BigInteger converted to a {@code float}. 3928 */ 3929 public float floatValue() { 3930 if (signum == 0) { 3931 return 0.0f; 3932 } 3933 3934 int exponent = ((mag.length - 1) << 5) + bitLengthForInt(mag[0]) - 1; 3935 3936 // exponent == floor(log2(abs(this))) 3937 if (exponent < Long.SIZE - 1) { 3938 return longValue(); 3939 } else if (exponent > Float.MAX_EXPONENT) { 3940 return signum > 0 ? Float.POSITIVE_INFINITY : Float.NEGATIVE_INFINITY; 3941 } 3942 3943 /* 3944 * We need the top SIGNIFICAND_WIDTH bits, including the "implicit" 3945 * one bit. To make rounding easier, we pick out the top 3946 * SIGNIFICAND_WIDTH + 1 bits, so we have one to help us round up or 3947 * down. twiceSignifFloor will contain the top SIGNIFICAND_WIDTH + 1 3948 * bits, and signifFloor the top SIGNIFICAND_WIDTH. 3949 * 3950 * It helps to consider the real number signif = abs(this) * 3951 * 2^(SIGNIFICAND_WIDTH - 1 - exponent). 3952 */ 3953 int shift = exponent - FloatConsts.SIGNIFICAND_WIDTH; 3954 3955 int twiceSignifFloor; 3956 // twiceSignifFloor will be == abs().shiftRight(shift).intValue() 3957 // We do the shift into an int directly to improve performance. 3958 3959 int nBits = shift & 0x1f; 3960 int nBits2 = 32 - nBits; 3961 3962 if (nBits == 0) { 3963 twiceSignifFloor = mag[0]; 3964 } else { 3965 twiceSignifFloor = mag[0] >>> nBits; 3966 if (twiceSignifFloor == 0) { 3967 twiceSignifFloor = (mag[0] << nBits2) | (mag[1] >>> nBits); 3968 } 3969 } 3970 3971 int signifFloor = twiceSignifFloor >> 1; 3972 signifFloor &= FloatConsts.SIGNIF_BIT_MASK; // remove the implied bit 3973 3974 /* 3975 * We round up if either the fractional part of signif is strictly 3976 * greater than 0.5 (which is true if the 0.5 bit is set and any lower 3977 * bit is set), or if the fractional part of signif is >= 0.5 and 3978 * signifFloor is odd (which is true if both the 0.5 bit and the 1 bit 3979 * are set). This is equivalent to the desired HALF_EVEN rounding. 3980 */ 3981 boolean increment = (twiceSignifFloor & 1) != 0 3982 && ((signifFloor & 1) != 0 || abs().getLowestSetBit() < shift); 3983 int signifRounded = increment ? signifFloor + 1 : signifFloor; 3984 int bits = ((exponent + FloatConsts.EXP_BIAS)) 3985 << (FloatConsts.SIGNIFICAND_WIDTH - 1); 3986 bits += signifRounded; 3987 /* 3988 * If signifRounded == 2^24, we'd need to set all of the significand 3989 * bits to zero and add 1 to the exponent. This is exactly the behavior 3990 * we get from just adding signifRounded to bits directly. If the 3991 * exponent is Float.MAX_EXPONENT, we round up (correctly) to 3992 * Float.POSITIVE_INFINITY. 3993 */ 3994 bits |= signum & FloatConsts.SIGN_BIT_MASK; 3995 return Float.intBitsToFloat(bits); 3996 } 3997 3998 /** 3999 * Converts this BigInteger to a {@code double}. This 4000 * conversion is similar to the 4001 * <i>narrowing primitive conversion</i> from {@code double} to 4002 * {@code float} as defined in section 5.1.3 of 4003 * <cite>The Java™ Language Specification</cite>: 4004 * if this BigInteger has too great a magnitude 4005 * to represent as a {@code double}, it will be converted to 4006 * {@link Double#NEGATIVE_INFINITY} or {@link 4007 * Double#POSITIVE_INFINITY} as appropriate. Note that even when 4008 * the return value is finite, this conversion can lose 4009 * information about the precision of the BigInteger value. 4010 * 4011 * @return this BigInteger converted to a {@code double}. 4012 */ 4013 public double doubleValue() { 4014 if (signum == 0) { 4015 return 0.0; 4016 } 4017 4018 int exponent = ((mag.length - 1) << 5) + bitLengthForInt(mag[0]) - 1; 4019 4020 // exponent == floor(log2(abs(this))Double) 4021 if (exponent < Long.SIZE - 1) { 4022 return longValue(); 4023 } else if (exponent > Double.MAX_EXPONENT) { 4024 return signum > 0 ? Double.POSITIVE_INFINITY : Double.NEGATIVE_INFINITY; 4025 } 4026 4027 /* 4028 * We need the top SIGNIFICAND_WIDTH bits, including the "implicit" 4029 * one bit. To make rounding easier, we pick out the top 4030 * SIGNIFICAND_WIDTH + 1 bits, so we have one to help us round up or 4031 * down. twiceSignifFloor will contain the top SIGNIFICAND_WIDTH + 1 4032 * bits, and signifFloor the top SIGNIFICAND_WIDTH. 4033 * 4034 * It helps to consider the real number signif = abs(this) * 4035 * 2^(SIGNIFICAND_WIDTH - 1 - exponent). 4036 */ 4037 int shift = exponent - DoubleConsts.SIGNIFICAND_WIDTH; 4038 4039 long twiceSignifFloor; 4040 // twiceSignifFloor will be == abs().shiftRight(shift).longValue() 4041 // We do the shift into a long directly to improve performance. 4042 4043 int nBits = shift & 0x1f; 4044 int nBits2 = 32 - nBits; 4045 4046 int highBits; 4047 int lowBits; 4048 if (nBits == 0) { 4049 highBits = mag[0]; 4050 lowBits = mag[1]; 4051 } else { 4052 highBits = mag[0] >>> nBits; 4053 lowBits = (mag[0] << nBits2) | (mag[1] >>> nBits); 4054 if (highBits == 0) { 4055 highBits = lowBits; 4056 lowBits = (mag[1] << nBits2) | (mag[2] >>> nBits); 4057 } 4058 } 4059 4060 twiceSignifFloor = ((highBits & LONG_MASK) << 32) 4061 | (lowBits & LONG_MASK); 4062 4063 long signifFloor = twiceSignifFloor >> 1; 4064 signifFloor &= DoubleConsts.SIGNIF_BIT_MASK; // remove the implied bit 4065 4066 /* 4067 * We round up if either the fractional part of signif is strictly 4068 * greater than 0.5 (which is true if the 0.5 bit is set and any lower 4069 * bit is set), or if the fractional part of signif is >= 0.5 and 4070 * signifFloor is odd (which is true if both the 0.5 bit and the 1 bit 4071 * are set). This is equivalent to the desired HALF_EVEN rounding. 4072 */ 4073 boolean increment = (twiceSignifFloor & 1) != 0 4074 && ((signifFloor & 1) != 0 || abs().getLowestSetBit() < shift); 4075 long signifRounded = increment ? signifFloor + 1 : signifFloor; 4076 long bits = (long) ((exponent + DoubleConsts.EXP_BIAS)) 4077 << (DoubleConsts.SIGNIFICAND_WIDTH - 1); 4078 bits += signifRounded; 4079 /* 4080 * If signifRounded == 2^53, we'd need to set all of the significand 4081 * bits to zero and add 1 to the exponent. This is exactly the behavior 4082 * we get from just adding signifRounded to bits directly. If the 4083 * exponent is Double.MAX_EXPONENT, we round up (correctly) to 4084 * Double.POSITIVE_INFINITY. 4085 */ 4086 bits |= signum & DoubleConsts.SIGN_BIT_MASK; 4087 return Double.longBitsToDouble(bits); 4088 } 4089 4090 /** 4091 * Returns a copy of the input array stripped of any leading zero bytes. 4092 */ 4093 private static int[] stripLeadingZeroInts(int val[]) { 4094 int vlen = val.length; 4095 int keep; 4096 4097 // Find first nonzero byte 4098 for (keep = 0; keep < vlen && val[keep] == 0; keep++) 4099 ; 4100 return java.util.Arrays.copyOfRange(val, keep, vlen); 4101 } 4102 4103 /** 4104 * Returns the input array stripped of any leading zero bytes. 4105 * Since the source is trusted the copying may be skipped. 4106 */ 4107 private static int[] trustedStripLeadingZeroInts(int val[]) { 4108 int vlen = val.length; 4109 int keep; 4110 4111 // Find first nonzero byte 4112 for (keep = 0; keep < vlen && val[keep] == 0; keep++) 4113 ; 4114 return keep == 0 ? val : java.util.Arrays.copyOfRange(val, keep, vlen); 4115 } 4116 4117 /** 4118 * Returns a copy of the input array stripped of any leading zero bytes. 4119 */ 4120 private static int[] stripLeadingZeroBytes(byte a[], int off, int len) { 4121 int indexBound = off + len; 4122 int keep; 4123 4124 // Find first nonzero byte 4125 for (keep = off; keep < indexBound && a[keep] == 0; keep++) 4126 ; 4127 4128 // Allocate new array and copy relevant part of input array 4129 int intLength = ((indexBound - keep) + 3) >>> 2; 4130 int[] result = new int[intLength]; 4131 int b = indexBound - 1; 4132 for (int i = intLength-1; i >= 0; i--) { 4133 result[i] = a[b--] & 0xff; 4134 int bytesRemaining = b - keep + 1; 4135 int bytesToTransfer = Math.min(3, bytesRemaining); 4136 for (int j=8; j <= (bytesToTransfer << 3); j += 8) 4137 result[i] |= ((a[b--] & 0xff) << j); 4138 } 4139 return result; 4140 } 4141 4142 /** 4143 * Takes an array a representing a negative 2's-complement number and 4144 * returns the minimal (no leading zero bytes) unsigned whose value is -a. 4145 */ 4146 private static int[] makePositive(byte a[], int off, int len) { 4147 int keep, k; 4148 int indexBound = off + len; 4149 4150 // Find first non-sign (0xff) byte of input 4151 for (keep=off; keep < indexBound && a[keep] == -1; keep++) 4152 ; 4153 4154 4155 /* Allocate output array. If all non-sign bytes are 0x00, we must 4156 * allocate space for one extra output byte. */ 4157 for (k=keep; k < indexBound && a[k] == 0; k++) 4158 ; 4159 4160 int extraByte = (k == indexBound) ? 1 : 0; 4161 int intLength = ((indexBound - keep + extraByte) + 3) >>> 2; 4162 int result[] = new int[intLength]; 4163 4164 /* Copy one's complement of input into output, leaving extra 4165 * byte (if it exists) == 0x00 */ 4166 int b = indexBound - 1; 4167 for (int i = intLength-1; i >= 0; i--) { 4168 result[i] = a[b--] & 0xff; 4169 int numBytesToTransfer = Math.min(3, b-keep+1); 4170 if (numBytesToTransfer < 0) 4171 numBytesToTransfer = 0; 4172 for (int j=8; j <= 8*numBytesToTransfer; j += 8) 4173 result[i] |= ((a[b--] & 0xff) << j); 4174 4175 // Mask indicates which bits must be complemented 4176 int mask = -1 >>> (8*(3-numBytesToTransfer)); 4177 result[i] = ~result[i] & mask; 4178 } 4179 4180 // Add one to one's complement to generate two's complement 4181 for (int i=result.length-1; i >= 0; i--) { 4182 result[i] = (int)((result[i] & LONG_MASK) + 1); 4183 if (result[i] != 0) 4184 break; 4185 } 4186 4187 return result; 4188 } 4189 4190 /** 4191 * Takes an array a representing a negative 2's-complement number and 4192 * returns the minimal (no leading zero ints) unsigned whose value is -a. 4193 */ 4194 private static int[] makePositive(int a[]) { 4195 int keep, j; 4196 4197 // Find first non-sign (0xffffffff) int of input 4198 for (keep=0; keep < a.length && a[keep] == -1; keep++) 4199 ; 4200 4201 /* Allocate output array. If all non-sign ints are 0x00, we must 4202 * allocate space for one extra output int. */ 4203 for (j=keep; j < a.length && a[j] == 0; j++) 4204 ; 4205 int extraInt = (j == a.length ? 1 : 0); 4206 int result[] = new int[a.length - keep + extraInt]; 4207 4208 /* Copy one's complement of input into output, leaving extra 4209 * int (if it exists) == 0x00 */ 4210 for (int i = keep; i < a.length; i++) 4211 result[i - keep + extraInt] = ~a[i]; 4212 4213 // Add one to one's complement to generate two's complement 4214 for (int i=result.length-1; ++result[i] == 0; i--) 4215 ; 4216 4217 return result; 4218 } 4219 4220 /* 4221 * The following two arrays are used for fast String conversions. Both 4222 * are indexed by radix. The first is the number of digits of the given 4223 * radix that can fit in a Java long without "going negative", i.e., the 4224 * highest integer n such that radix**n < 2**63. The second is the 4225 * "long radix" that tears each number into "long digits", each of which 4226 * consists of the number of digits in the corresponding element in 4227 * digitsPerLong (longRadix[i] = i**digitPerLong[i]). Both arrays have 4228 * nonsense values in their 0 and 1 elements, as radixes 0 and 1 are not 4229 * used. 4230 */ 4231 private static int digitsPerLong[] = {0, 0, 4232 62, 39, 31, 27, 24, 22, 20, 19, 18, 18, 17, 17, 16, 16, 15, 15, 15, 14, 4233 14, 14, 14, 13, 13, 13, 13, 13, 13, 12, 12, 12, 12, 12, 12, 12, 12}; 4234 4235 private static BigInteger longRadix[] = {null, null, 4236 valueOf(0x4000000000000000L), valueOf(0x383d9170b85ff80bL), 4237 valueOf(0x4000000000000000L), valueOf(0x6765c793fa10079dL), 4238 valueOf(0x41c21cb8e1000000L), valueOf(0x3642798750226111L), 4239 valueOf(0x1000000000000000L), valueOf(0x12bf307ae81ffd59L), 4240 valueOf( 0xde0b6b3a7640000L), valueOf(0x4d28cb56c33fa539L), 4241 valueOf(0x1eca170c00000000L), valueOf(0x780c7372621bd74dL), 4242 valueOf(0x1e39a5057d810000L), valueOf(0x5b27ac993df97701L), 4243 valueOf(0x1000000000000000L), valueOf(0x27b95e997e21d9f1L), 4244 valueOf(0x5da0e1e53c5c8000L), valueOf( 0xb16a458ef403f19L), 4245 valueOf(0x16bcc41e90000000L), valueOf(0x2d04b7fdd9c0ef49L), 4246 valueOf(0x5658597bcaa24000L), valueOf( 0x6feb266931a75b7L), 4247 valueOf( 0xc29e98000000000L), valueOf(0x14adf4b7320334b9L), 4248 valueOf(0x226ed36478bfa000L), valueOf(0x383d9170b85ff80bL), 4249 valueOf(0x5a3c23e39c000000L), valueOf( 0x4e900abb53e6b71L), 4250 valueOf( 0x7600ec618141000L), valueOf( 0xaee5720ee830681L), 4251 valueOf(0x1000000000000000L), valueOf(0x172588ad4f5f0981L), 4252 valueOf(0x211e44f7d02c1000L), valueOf(0x2ee56725f06e5c71L), 4253 valueOf(0x41c21cb8e1000000L)}; 4254 4255 /* 4256 * These two arrays are the integer analogue of above. 4257 */ 4258 private static int digitsPerInt[] = {0, 0, 30, 19, 15, 13, 11, 4259 11, 10, 9, 9, 8, 8, 8, 8, 7, 7, 7, 7, 7, 7, 7, 6, 6, 6, 6, 4260 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 5}; 4261 4262 private static int intRadix[] = {0, 0, 4263 0x40000000, 0x4546b3db, 0x40000000, 0x48c27395, 0x159fd800, 4264 0x75db9c97, 0x40000000, 0x17179149, 0x3b9aca00, 0xcc6db61, 4265 0x19a10000, 0x309f1021, 0x57f6c100, 0xa2f1b6f, 0x10000000, 4266 0x18754571, 0x247dbc80, 0x3547667b, 0x4c4b4000, 0x6b5a6e1d, 4267 0x6c20a40, 0x8d2d931, 0xb640000, 0xe8d4a51, 0x1269ae40, 4268 0x17179149, 0x1cb91000, 0x23744899, 0x2b73a840, 0x34e63b41, 4269 0x40000000, 0x4cfa3cc1, 0x5c13d840, 0x6d91b519, 0x39aa400 4270 }; 4271 4272 /** 4273 * These routines provide access to the two's complement representation 4274 * of BigIntegers. 4275 */ 4276 4277 /** 4278 * Returns the length of the two's complement representation in ints, 4279 * including space for at least one sign bit. 4280 */ 4281 private int intLength() { 4282 return (bitLength() >>> 5) + 1; 4283 } 4284 4285 /* Returns sign bit */ 4286 private int signBit() { 4287 return signum < 0 ? 1 : 0; 4288 } 4289 4290 /* Returns an int of sign bits */ 4291 private int signInt() { 4292 return signum < 0 ? -1 : 0; 4293 } 4294 4295 /** 4296 * Returns the specified int of the little-endian two's complement 4297 * representation (int 0 is the least significant). The int number can 4298 * be arbitrarily high (values are logically preceded by infinitely many 4299 * sign ints). 4300 */ 4301 private int getInt(int n) { 4302 if (n < 0) 4303 return 0; 4304 if (n >= mag.length) 4305 return signInt(); 4306 4307 int magInt = mag[mag.length-n-1]; 4308 4309 return (signum >= 0 ? magInt : 4310 (n <= firstNonzeroIntNum() ? -magInt : ~magInt)); 4311 } 4312 4313 /** 4314 * Returns the index of the int that contains the first nonzero int in the 4315 * little-endian binary representation of the magnitude (int 0 is the 4316 * least significant). If the magnitude is zero, return value is undefined. 4317 * 4318 * <p>Note: never used for a BigInteger with a magnitude of zero. 4319 * @see #getInt. 4320 */ 4321 private int firstNonzeroIntNum() { 4322 int fn = firstNonzeroIntNumPlusTwo - 2; 4323 if (fn == -2) { // firstNonzeroIntNum not initialized yet 4324 // Search for the first nonzero int 4325 int i; 4326 int mlen = mag.length; 4327 for (i = mlen - 1; i >= 0 && mag[i] == 0; i--) 4328 ; 4329 fn = mlen - i - 1; 4330 firstNonzeroIntNumPlusTwo = fn + 2; // offset by two to initialize 4331 } 4332 return fn; 4333 } 4334 4335 /** use serialVersionUID from JDK 1.1. for interoperability */ 4336 private static final long serialVersionUID = -8287574255936472291L; 4337 4338 /** 4339 * Serializable fields for BigInteger. 4340 * 4341 * @serialField signum int 4342 * signum of this BigInteger 4343 * @serialField magnitude byte[] 4344 * magnitude array of this BigInteger 4345 * @serialField bitCount int 4346 * appears in the serialized form for backward compatibility 4347 * @serialField bitLength int 4348 * appears in the serialized form for backward compatibility 4349 * @serialField firstNonzeroByteNum int 4350 * appears in the serialized form for backward compatibility 4351 * @serialField lowestSetBit int 4352 * appears in the serialized form for backward compatibility 4353 */ 4354 private static final ObjectStreamField[] serialPersistentFields = { 4355 new ObjectStreamField("signum", Integer.TYPE), 4356 new ObjectStreamField("magnitude", byte[].class), 4357 new ObjectStreamField("bitCount", Integer.TYPE), 4358 new ObjectStreamField("bitLength", Integer.TYPE), 4359 new ObjectStreamField("firstNonzeroByteNum", Integer.TYPE), 4360 new ObjectStreamField("lowestSetBit", Integer.TYPE) 4361 }; 4362 4363 /** 4364 * Reconstitute the {@code BigInteger} instance from a stream (that is, 4365 * deserialize it). The magnitude is read in as an array of bytes 4366 * for historical reasons, but it is converted to an array of ints 4367 * and the byte array is discarded. 4368 * Note: 4369 * The current convention is to initialize the cache fields, bitCountPlusOne, 4370 * bitLengthPlusOne and lowestSetBitPlusTwo, to 0 rather than some other 4371 * marker value. Therefore, no explicit action to set these fields needs to 4372 * be taken in readObject because those fields already have a 0 value by 4373 * default since defaultReadObject is not being used. 4374 */ 4375 private void readObject(java.io.ObjectInputStream s) 4376 throws java.io.IOException, ClassNotFoundException { 4377 // prepare to read the alternate persistent fields 4378 ObjectInputStream.GetField fields = s.readFields(); 4379 4380 // Read the alternate persistent fields that we care about 4381 int sign = fields.get("signum", -2); 4382 byte[] magnitude = (byte[])fields.get("magnitude", null); 4383 4384 // Validate signum 4385 if (sign < -1 || sign > 1) { 4386 String message = "BigInteger: Invalid signum value"; 4387 if (fields.defaulted("signum")) 4388 message = "BigInteger: Signum not present in stream"; 4389 throw new java.io.StreamCorruptedException(message); 4390 } 4391 int[] mag = stripLeadingZeroBytes(magnitude, 0, magnitude.length); 4392 if ((mag.length == 0) != (sign == 0)) { 4393 String message = "BigInteger: signum-magnitude mismatch"; 4394 if (fields.defaulted("magnitude")) 4395 message = "BigInteger: Magnitude not present in stream"; 4396 throw new java.io.StreamCorruptedException(message); 4397 } 4398 4399 // Commit final fields via Unsafe 4400 UnsafeHolder.putSign(this, sign); 4401 4402 // Calculate mag field from magnitude and discard magnitude 4403 UnsafeHolder.putMag(this, mag); 4404 if (mag.length >= MAX_MAG_LENGTH) { 4405 try { 4406 checkRange(); 4407 } catch (ArithmeticException e) { 4408 throw new java.io.StreamCorruptedException("BigInteger: Out of the supported range"); 4409 } 4410 } 4411 } 4412 4413 // Support for resetting final fields while deserializing 4414 private static class UnsafeHolder { 4415 private static final sun.misc.Unsafe unsafe; 4416 private static final long signumOffset; 4417 private static final long magOffset; 4418 static { 4419 try { 4420 unsafe = sun.misc.Unsafe.getUnsafe(); 4421 signumOffset = unsafe.objectFieldOffset 4422 (BigInteger.class.getDeclaredField("signum")); 4423 magOffset = unsafe.objectFieldOffset 4424 (BigInteger.class.getDeclaredField("mag")); 4425 } catch (Exception ex) { 4426 throw new ExceptionInInitializerError(ex); 4427 } 4428 } 4429 4430 static void putSign(BigInteger bi, int sign) { 4431 unsafe.putInt(bi, signumOffset, sign); 4432 } 4433 4434 static void putMag(BigInteger bi, int[] magnitude) { 4435 unsafe.putObject(bi, magOffset, magnitude); 4436 } 4437 } 4438 4439 /** 4440 * Save the {@code BigInteger} instance to a stream. The magnitude of a 4441 * {@code BigInteger} is serialized as a byte array for historical reasons. 4442 * To maintain compatibility with older implementations, the integers 4443 * -1, -1, -2, and -2 are written as the values of the obsolete fields 4444 * {@code bitCount}, {@code bitLength}, {@code lowestSetBit}, and 4445 * {@code firstNonzeroByteNum}, respectively. These values are compatible 4446 * with older implementations, but will be ignored by current 4447 * implementations. 4448 */ 4449 private void writeObject(ObjectOutputStream s) throws IOException { 4450 // set the values of the Serializable fields 4451 ObjectOutputStream.PutField fields = s.putFields(); 4452 fields.put("signum", signum); 4453 fields.put("magnitude", magSerializedForm()); 4454 // The values written for cached fields are compatible with older 4455 // versions, but are ignored in readObject so don't otherwise matter. 4456 fields.put("bitCount", -1); 4457 fields.put("bitLength", -1); 4458 fields.put("lowestSetBit", -2); 4459 fields.put("firstNonzeroByteNum", -2); 4460 4461 // save them 4462 s.writeFields(); 4463 } 4464 4465 /** 4466 * Returns the mag array as an array of bytes. 4467 */ 4468 private byte[] magSerializedForm() { 4469 int len = mag.length; 4470 4471 int bitLen = (len == 0 ? 0 : ((len - 1) << 5) + bitLengthForInt(mag[0])); 4472 int byteLen = (bitLen + 7) >>> 3; 4473 byte[] result = new byte[byteLen]; 4474 4475 for (int i = byteLen - 1, bytesCopied = 4, intIndex = len - 1, nextInt = 0; 4476 i >= 0; i--) { 4477 if (bytesCopied == 4) { 4478 nextInt = mag[intIndex--]; 4479 bytesCopied = 1; 4480 } else { 4481 nextInt >>>= 8; 4482 bytesCopied++; 4483 } 4484 result[i] = (byte)nextInt; 4485 } 4486 return result; 4487 } 4488 4489 /** 4490 * Converts this {@code BigInteger} to a {@code long}, checking 4491 * for lost information. If the value of this {@code BigInteger} 4492 * is out of the range of the {@code long} type, then an 4493 * {@code ArithmeticException} is thrown. 4494 * 4495 * @return this {@code BigInteger} converted to a {@code long}. 4496 * @throws ArithmeticException if the value of {@code this} will 4497 * not exactly fit in a {@code long}. 4498 * @see BigInteger#longValue 4499 * @since 1.8 4500 */ 4501 public long longValueExact() { 4502 if (mag.length <= 2 && bitLength() <= 63) 4503 return longValue(); 4504 else 4505 throw new ArithmeticException("BigInteger out of long range"); 4506 } 4507 4508 /** 4509 * Converts this {@code BigInteger} to an {@code int}, checking 4510 * for lost information. If the value of this {@code BigInteger} 4511 * is out of the range of the {@code int} type, then an 4512 * {@code ArithmeticException} is thrown. 4513 * 4514 * @return this {@code BigInteger} converted to an {@code int}. 4515 * @throws ArithmeticException if the value of {@code this} will 4516 * not exactly fit in a {@code int}. 4517 * @see BigInteger#intValue 4518 * @since 1.8 4519 */ 4520 public int intValueExact() { 4521 if (mag.length <= 1 && bitLength() <= 31) 4522 return intValue(); 4523 else 4524 throw new ArithmeticException("BigInteger out of int range"); 4525 } 4526 4527 /** 4528 * Converts this {@code BigInteger} to a {@code short}, checking 4529 * for lost information. If the value of this {@code BigInteger} 4530 * is out of the range of the {@code short} type, then an 4531 * {@code ArithmeticException} is thrown. 4532 * 4533 * @return this {@code BigInteger} converted to a {@code short}. 4534 * @throws ArithmeticException if the value of {@code this} will 4535 * not exactly fit in a {@code short}. 4536 * @see BigInteger#shortValue 4537 * @since 1.8 4538 */ 4539 public short shortValueExact() { 4540 if (mag.length <= 1 && bitLength() <= 31) { 4541 int value = intValue(); 4542 if (value >= Short.MIN_VALUE && value <= Short.MAX_VALUE) 4543 return shortValue(); 4544 } 4545 throw new ArithmeticException("BigInteger out of short range"); 4546 } 4547 4548 /** 4549 * Converts this {@code BigInteger} to a {@code byte}, checking 4550 * for lost information. If the value of this {@code BigInteger} 4551 * is out of the range of the {@code byte} type, then an 4552 * {@code ArithmeticException} is thrown. 4553 * 4554 * @return this {@code BigInteger} converted to a {@code byte}. 4555 * @throws ArithmeticException if the value of {@code this} will 4556 * not exactly fit in a {@code byte}. 4557 * @see BigInteger#byteValue 4558 * @since 1.8 4559 */ 4560 public byte byteValueExact() { 4561 if (mag.length <= 1 && bitLength() <= 31) { 4562 int value = intValue(); 4563 if (value >= Byte.MIN_VALUE && value <= Byte.MAX_VALUE) 4564 return byteValue(); 4565 } 4566 throw new ArithmeticException("BigInteger out of byte range"); 4567 } 4568 }