libgo/go/math/big/int.go

   1 // Copyright 2009 The Go Authors. All rights reserved.
   2 // Use of this source code is governed by a BSD-style
   3 // license that can be found in the LICENSE file.
   4
   5 // This file implements signed multi-precision integers.
   6
   7 package big
   8
   9 import (
  10         "errors"
  11         "fmt"
  12         "io"
  13         "math/rand"
  14         "strings"
  15 )
  16
  17 // An Int represents a signed multi-precision integer.
  18 // The zero value for an Int represents the value 0.
  19 type Int struct {
  20         neg bool // sign
  21         abs nat  // absolute value of the integer
  22 }
  23
  24 var intOne = &Int{false, natOne}
  25
  26 // Sign returns:
  27 //
  28 //      -1 if x <  0
  29 //       0 if x == 0
  30 //      +1 if x >  0
  31 //
  32 func (x *Int) Sign() int {
  33         if len(x.abs) == 0 {
  34                 return 0
  35         }
  36         if x.neg {
  37                 return -1
  38         }
  39         return 1
  40 }
  41
  42 // SetInt64 sets z to x and returns z.
  43 func (z *Int) SetInt64(x int64) *Int {
  44         neg := false
  45         if x < 0 {
  46                 neg = true
  47                 x = -x
  48         }
  49         z.abs = z.abs.setUint64(uint64(x))
  50         z.neg = neg
  51         return z
  52 }
  53
  54 // SetUint64 sets z to x and returns z.
  55 func (z *Int) SetUint64(x uint64) *Int {
  56         z.abs = z.abs.setUint64(x)
  57         z.neg = false
  58         return z
  59 }
  60
  61 // NewInt allocates and returns a new Int set to x.
  62 func NewInt(x int64) *Int {
  63         return new(Int).SetInt64(x)
  64 }
  65
  66 // Set sets z to x and returns z.
  67 func (z *Int) Set(x *Int) *Int {
  68         if z != x {
  69                 z.abs = z.abs.set(x.abs)
  70                 z.neg = x.neg
  71         }
  72         return z
  73 }
  74
  75 // Bits provides raw (unchecked but fast) access to x by returning its
  76 // absolute value as a little-endian Word slice. The result and x share
  77 // the same underlying array.
  78 // Bits is intended to support implementation of missing low-level Int
  79 // functionality outside this package; it should be avoided otherwise.
  80 func (x *Int) Bits() []Word {
  81         return x.abs
  82 }
  83
  84 // SetBits provides raw (unchecked but fast) access to z by setting its
  85 // value to abs, interpreted as a little-endian Word slice, and returning
  86 // z. The result and abs share the same underlying array.
  87 // SetBits is intended to support implementation of missing low-level Int
  88 // functionality outside this package; it should be avoided otherwise.
  89 func (z *Int) SetBits(abs []Word) *Int {
  90         z.abs = nat(abs).norm()
  91         z.neg = false
  92         return z
  93 }
  94
  95 // Abs sets z to |x| (the absolute value of x) and returns z.
  96 func (z *Int) Abs(x *Int) *Int {
  97         z.Set(x)
  98         z.neg = false
  99         return z
 100 }
 101
 102 // Neg sets z to -x and returns z.
 103 func (z *Int) Neg(x *Int) *Int {
 104         z.Set(x)
 105         z.neg = len(z.abs) > 0 && !z.neg // 0 has no sign
 106         return z
 107 }
 108
 109 // Add sets z to the sum x+y and returns z.
 110 func (z *Int) Add(x, y *Int) *Int {
 111         neg := x.neg
 112         if x.neg == y.neg {
 113                 // x + y == x + y
 114                 // (-x) + (-y) == -(x + y)
 115                 z.abs = z.abs.add(x.abs, y.abs)
 116         } else {
 117                 // x + (-y) == x - y == -(y - x)
 118                 // (-x) + y == y - x == -(x - y)
 119                 if x.abs.cmp(y.abs) >= 0 {
 120                         z.abs = z.abs.sub(x.abs, y.abs)
 121                 } else {
 122                         neg = !neg
 123                         z.abs = z.abs.sub(y.abs, x.abs)
 124                 }
 125         }
 126         z.neg = len(z.abs) > 0 && neg // 0 has no sign
 127         return z
 128 }
 129
 130 // Sub sets z to the difference x-y and returns z.
 131 func (z *Int) Sub(x, y *Int) *Int {
 132         neg := x.neg
 133         if x.neg != y.neg {
 134                 // x - (-y) == x + y
 135                 // (-x) - y == -(x + y)
 136                 z.abs = z.abs.add(x.abs, y.abs)
 137         } else {
 138                 // x - y == x - y == -(y - x)
 139                 // (-x) - (-y) == y - x == -(x - y)
 140                 if x.abs.cmp(y.abs) >= 0 {
 141                         z.abs = z.abs.sub(x.abs, y.abs)
 142                 } else {
 143                         neg = !neg
 144                         z.abs = z.abs.sub(y.abs, x.abs)
 145                 }
 146         }
 147         z.neg = len(z.abs) > 0 && neg // 0 has no sign
 148         return z
 149 }
 150
 151 // Mul sets z to the product x*y and returns z.
 152 func (z *Int) Mul(x, y *Int) *Int {
 153         // x * y == x * y
 154         // x * (-y) == -(x * y)
 155         // (-x) * y == -(x * y)
 156         // (-x) * (-y) == x * y
 157         z.abs = z.abs.mul(x.abs, y.abs)
 158         z.neg = len(z.abs) > 0 && x.neg != y.neg // 0 has no sign
 159         return z
 160 }
 161
 162 // MulRange sets z to the product of all integers
 163 // in the range [a, b] inclusively and returns z.
 164 // If a > b (empty range), the result is 1.
 165 func (z *Int) MulRange(a, b int64) *Int {
 166         switch {
 167         case a > b:
 168                 return z.SetInt64(1) // empty range
 169         case a <= 0 && b >= 0:
 170                 return z.SetInt64(0) // range includes 0
 171         }
 172         // a <= b && (b < 0 || a > 0)
 173
 174         neg := false
 175         if a < 0 {
 176                 neg = (b-a)&1 == 0
 177                 a, b = -b, -a
 178         }
 179
 180         z.abs = z.abs.mulRange(uint64(a), uint64(b))
 181         z.neg = neg
 182         return z
 183 }
 184
 185 // Binomial sets z to the binomial coefficient of (n, k) and returns z.
 186 func (z *Int) Binomial(n, k int64) *Int {
 187         var a, b Int
 188         a.MulRange(n-k+1, n)
 189         b.MulRange(1, k)
 190         return z.Quo(&a, &b)
 191 }
 192
 193 // Quo sets z to the quotient x/y for y != 0 and returns z.
 194 // If y == 0, a division-by-zero run-time panic occurs.
 195 // Quo implements truncated division (like Go); see QuoRem for more details.
 196 func (z *Int) Quo(x, y *Int) *Int {
 197         z.abs, _ = z.abs.div(nil, x.abs, y.abs)
 198         z.neg = len(z.abs) > 0 && x.neg != y.neg // 0 has no sign
 199         return z
 200 }
 201
 202 // Rem sets z to the remainder x%y for y != 0 and returns z.
 203 // If y == 0, a division-by-zero run-time panic occurs.
 204 // Rem implements truncated modulus (like Go); see QuoRem for more details.
 205 func (z *Int) Rem(x, y *Int) *Int {
 206         _, z.abs = nat(nil).div(z.abs, x.abs, y.abs)
 207         z.neg = len(z.abs) > 0 && x.neg // 0 has no sign
 208         return z
 209 }
 210
 211 // QuoRem sets z to the quotient x/y and r to the remainder x%y
 212 // and returns the pair (z, r) for y != 0.
 213 // If y == 0, a division-by-zero run-time panic occurs.
 214 //
 215 // QuoRem implements T-division and modulus (like Go):
 216 //
 217 //      q = x/y      with the result truncated to zero
 218 //      r = x - y*q
 219 //
 220 // (See Daan Leijen, ``Division and Modulus for Computer Scientists''.)
 221 // See DivMod for Euclidean division and modulus (unlike Go).
 222 //
 223 func (z *Int) QuoRem(x, y, r *Int) (*Int, *Int) {
 224         z.abs, r.abs = z.abs.div(r.abs, x.abs, y.abs)
 225         z.neg, r.neg = len(z.abs) > 0 && x.neg != y.neg, len(r.abs) > 0 && x.neg // 0 has no sign
 226         return z, r
 227 }
 228
 229 // Div sets z to the quotient x/y for y != 0 and returns z.
 230 // If y == 0, a division-by-zero run-time panic occurs.
 231 // Div implements Euclidean division (unlike Go); see DivMod for more details.
 232 func (z *Int) Div(x, y *Int) *Int {
 233         y_neg := y.neg // z may be an alias for y
 234         var r Int
 235         z.QuoRem(x, y, &r)
 236         if r.neg {
 237                 if y_neg {
 238                         z.Add(z, intOne)
 239                 } else {
 240                         z.Sub(z, intOne)
 241                 }
 242         }
 243         return z
 244 }
 245
 246 // Mod sets z to the modulus x%y for y != 0 and returns z.
 247 // If y == 0, a division-by-zero run-time panic occurs.
 248 // Mod implements Euclidean modulus (unlike Go); see DivMod for more details.
 249 func (z *Int) Mod(x, y *Int) *Int {
 250         y0 := y // save y
 251         if z == y || alias(z.abs, y.abs) {
 252                 y0 = new(Int).Set(y)
 253         }
 254         var q Int
 255         q.QuoRem(x, y, z)
 256         if z.neg {
 257                 if y0.neg {
 258                         z.Sub(z, y0)
 259                 } else {
 260                         z.Add(z, y0)
 261                 }
 262         }
 263         return z
 264 }
 265
 266 // DivMod sets z to the quotient x div y and m to the modulus x mod y
 267 // and returns the pair (z, m) for y != 0.
 268 // If y == 0, a division-by-zero run-time panic occurs.
 269 //
 270 // DivMod implements Euclidean division and modulus (unlike Go):
 271 //
 272 //      q = x div y  such that
 273 //      m = x - y*q  with 0 <= m < |q|
 274 //
 275 // (See Raymond T. Boute, ``The Euclidean definition of the functions
 276 // div and mod''. ACM Transactions on Programming Languages and
 277 // Systems (TOPLAS), 14(2):127-144, New York, NY, USA, 4/1992.
 278 // ACM press.)
 279 // See QuoRem for T-division and modulus (like Go).
 280 //
 281 func (z *Int) DivMod(x, y, m *Int) (*Int, *Int) {
 282         y0 := y // save y
 283         if z == y || alias(z.abs, y.abs) {
 284                 y0 = new(Int).Set(y)
 285         }
 286         z.QuoRem(x, y, m)
 287         if m.neg {
 288                 if y0.neg {
 289                         z.Add(z, intOne)
 290                         m.Sub(m, y0)
 291                 } else {
 292                         z.Sub(z, intOne)
 293                         m.Add(m, y0)
 294                 }
 295         }
 296         return z, m
 297 }
 298
 299 // Cmp compares x and y and returns:
 300 //
 301 //   -1 if x <  y
 302 //    0 if x == y
 303 //   +1 if x >  y
 304 //
 305 func (x *Int) Cmp(y *Int) (r int) {
 306         // x cmp y == x cmp y
 307         // x cmp (-y) == x
 308         // (-x) cmp y == y
 309         // (-x) cmp (-y) == -(x cmp y)
 310         switch {
 311         case x.neg == y.neg:
 312                 r = x.abs.cmp(y.abs)
 313                 if x.neg {
 314                         r = -r
 315                 }
 316         case x.neg:
 317                 r = -1
 318         default:
 319                 r = 1
 320         }
 321         return
 322 }
 323
 324 func (x *Int) String() string {
 325         switch {
 326         case x == nil:
 327                 return "<nil>"
 328         case x.neg:
 329                 return "-" + x.abs.decimalString()
 330         }
 331         return x.abs.decimalString()
 332 }
 333
 334 func charset(ch rune) string {
 335         switch ch {
 336         case 'b':
 337                 return lowercaseDigits[0:2]
 338         case 'o':
 339                 return lowercaseDigits[0:8]
 340         case 'd', 's', 'v':
 341                 return lowercaseDigits[0:10]
 342         case 'x':
 343                 return lowercaseDigits[0:16]
 344         case 'X':
 345                 return uppercaseDigits[0:16]
 346         }
 347         return "" // unknown format
 348 }
 349
 350 // write count copies of text to s
 351 func writeMultiple(s fmt.State, text string, count int) {
 352         if len(text) > 0 {
 353                 b := []byte(text)
 354                 for ; count > 0; count-- {
 355                         s.Write(b)
 356                 }
 357         }
 358 }
 359
 360 // Format is a support routine for fmt.Formatter. It accepts
 361 // the formats 'b' (binary), 'o' (octal), 'd' (decimal), 'x'
 362 // (lowercase hexadecimal), and 'X' (uppercase hexadecimal).
 363 // Also supported are the full suite of package fmt's format
 364 // verbs for integral types, including '+', '-', and ' '
 365 // for sign control, '#' for leading zero in octal and for
 366 // hexadecimal, a leading "0x" or "0X" for "%#x" and "%#X"
 367 // respectively, specification of minimum digits precision,
 368 // output field width, space or zero padding, and left or
 369 // right justification.
 370 //
 371 func (x *Int) Format(s fmt.State, ch rune) {
 372         cs := charset(ch)
 373
 374         // special cases
 375         switch {
 376         case cs == "":
 377                 // unknown format
 378                 fmt.Fprintf(s, "%%!%c(big.Int=%s)", ch, x.String())
 379                 return
 380         case x == nil:
 381                 fmt.Fprint(s, "<nil>")
 382                 return
 383         }
 384
 385         // determine sign character
 386         sign := ""
 387         switch {
 388         case x.neg:
 389                 sign = "-"
 390         case s.Flag('+'): // supersedes ' ' when both specified
 391                 sign = "+"
 392         case s.Flag(' '):
 393                 sign = " "
 394         }
 395
 396         // determine prefix characters for indicating output base
 397         prefix := ""
 398         if s.Flag('#') {
 399                 switch ch {
 400                 case 'o': // octal
 401                         prefix = "0"
 402                 case 'x': // hexadecimal
 403                         prefix = "0x"
 404                 case 'X':
 405                         prefix = "0X"
 406                 }
 407         }
 408
 409         // determine digits with base set by len(cs) and digit characters from cs
 410         digits := x.abs.string(cs)
 411
 412         // number of characters for the three classes of number padding
 413         var left int   // space characters to left of digits for right justification ("%8d")
 414         var zeroes int // zero characters (actually cs[0]) as left-most digits ("%.8d")
 415         var right int  // space characters to right of digits for left justification ("%-8d")
 416
 417         // determine number padding from precision: the least number of digits to output
 418         precision, precisionSet := s.Precision()
 419         if precisionSet {
 420                 switch {
 421                 case len(digits) < precision:
 422                         zeroes = precision - len(digits) // count of zero padding
 423                 case digits == "0" && precision == 0:
 424                         return // print nothing if zero value (x == 0) and zero precision ("." or ".0")
 425                 }
 426         }
 427
 428         // determine field pad from width: the least number of characters to output
 429         length := len(sign) + len(prefix) + zeroes + len(digits)
 430         if width, widthSet := s.Width(); widthSet && length < width { // pad as specified
 431                 switch d := width - length; {
 432                 case s.Flag('-'):
 433                         // pad on the right with spaces; supersedes '0' when both specified
 434                         right = d
 435                 case s.Flag('0') && !precisionSet:
 436                         // pad with zeroes unless precision also specified
 437                         zeroes = d
 438                 default:
 439                         // pad on the left with spaces
 440                         left = d
 441                 }
 442         }
 443
 444         // print number as [left pad][sign][prefix][zero pad][digits][right pad]
 445         writeMultiple(s, " ", left)
 446         writeMultiple(s, sign, 1)
 447         writeMultiple(s, prefix, 1)
 448         writeMultiple(s, "0", zeroes)
 449         writeMultiple(s, digits, 1)
 450         writeMultiple(s, " ", right)
 451 }
 452
 453 // scan sets z to the integer value corresponding to the longest possible prefix
 454 // read from r representing a signed integer number in a given conversion base.
 455 // It returns z, the actual conversion base used, and an error, if any. In the
 456 // error case, the value of z is undefined but the returned value is nil. The
 457 // syntax follows the syntax of integer literals in Go.
 458 //
 459 // The base argument must be 0 or a value from 2 through MaxBase. If the base
 460 // is 0, the string prefix determines the actual conversion base. A prefix of
 461 // ``0x'' or ``0X'' selects base 16; the ``0'' prefix selects base 8, and a
 462 // ``0b'' or ``0B'' prefix selects base 2. Otherwise the selected base is 10.
 463 //
 464 func (z *Int) scan(r io.RuneScanner, base int) (*Int, int, error) {
 465         // determine sign
 466         ch, _, err := r.ReadRune()
 467         if err != nil {
 468                 return nil, 0, err
 469         }
 470         neg := false
 471         switch ch {
 472         case '-':
 473                 neg = true
 474         case '+': // nothing to do
 475         default:
 476                 r.UnreadRune()
 477         }
 478
 479         // determine mantissa
 480         z.abs, base, err = z.abs.scan(r, base)
 481         if err != nil {
 482                 return nil, base, err
 483         }
 484         z.neg = len(z.abs) > 0 && neg // 0 has no sign
 485
 486         return z, base, nil
 487 }
 488
 489 // Scan is a support routine for fmt.Scanner; it sets z to the value of
 490 // the scanned number. It accepts the formats 'b' (binary), 'o' (octal),
 491 // 'd' (decimal), 'x' (lowercase hexadecimal), and 'X' (uppercase hexadecimal).
 492 func (z *Int) Scan(s fmt.ScanState, ch rune) error {
 493         s.SkipSpace() // skip leading space characters
 494         base := 0
 495         switch ch {
 496         case 'b':
 497                 base = 2
 498         case 'o':
 499                 base = 8
 500         case 'd':
 501                 base = 10
 502         case 'x', 'X':
 503                 base = 16
 504         case 's', 'v':
 505                 // let scan determine the base
 506         default:
 507                 return errors.New("Int.Scan: invalid verb")
 508         }
 509         _, _, err := z.scan(s, base)
 510         return err
 511 }
 512
 513 // Int64 returns the int64 representation of x.
 514 // If x cannot be represented in an int64, the result is undefined.
 515 func (x *Int) Int64() int64 {
 516         v := int64(x.Uint64())
 517         if x.neg {
 518                 v = -v
 519         }
 520         return v
 521 }
 522
 523 // Uint64 returns the uint64 representation of x.
 524 // If x cannot be represented in a uint64, the result is undefined.
 525 func (x *Int) Uint64() uint64 {
 526         if len(x.abs) == 0 {
 527                 return 0
 528         }
 529         v := uint64(x.abs[0])
 530         if _W == 32 && len(x.abs) > 1 {
 531                 v |= uint64(x.abs[1]) << 32
 532         }
 533         return v
 534 }
 535
 536 // SetString sets z to the value of s, interpreted in the given base,
 537 // and returns z and a boolean indicating success. If SetString fails,
 538 // the value of z is undefined but the returned value is nil.
 539 //
 540 // The base argument must be 0 or a value from 2 through MaxBase. If the base
 541 // is 0, the string prefix determines the actual conversion base. A prefix of
 542 // ``0x'' or ``0X'' selects base 16; the ``0'' prefix selects base 8, and a
 543 // ``0b'' or ``0B'' prefix selects base 2. Otherwise the selected base is 10.
 544 //
 545 func (z *Int) SetString(s string, base int) (*Int, bool) {
 546         r := strings.NewReader(s)
 547         _, _, err := z.scan(r, base)
 548         if err != nil {
 549                 return nil, false
 550         }
 551         _, _, err = r.ReadRune()
 552         if err != io.EOF {
 553                 return nil, false
 554         }
 555         return z, true // err == io.EOF => scan consumed all of s
 556 }
 557
 558 // SetBytes interprets buf as the bytes of a big-endian unsigned
 559 // integer, sets z to that value, and returns z.
 560 func (z *Int) SetBytes(buf []byte) *Int {
 561         z.abs = z.abs.setBytes(buf)
 562         z.neg = false
 563         return z
 564 }
 565
 566 // Bytes returns the absolute value of x as a big-endian byte slice.
 567 func (x *Int) Bytes() []byte {
 568         buf := make([]byte, len(x.abs)*_S)
 569         return buf[x.abs.bytes(buf):]
 570 }
 571
 572 // BitLen returns the length of the absolute value of x in bits.
 573 // The bit length of 0 is 0.
 574 func (x *Int) BitLen() int {
 575         return x.abs.bitLen()
 576 }
 577
 578 // Exp sets z = x**y mod |m| (i.e. the sign of m is ignored), and returns z.
 579 // If y <= 0, the result is 1; if m == nil or m == 0, z = x**y.
 580 // See Knuth, volume 2, section 4.6.3.
 581 func (z *Int) Exp(x, y, m *Int) *Int {
 582         if y.neg || len(y.abs) == 0 {
 583                 return z.SetInt64(1)
 584         }
 585         // y > 0
 586
 587         var mWords nat
 588         if m != nil {
 589                 mWords = m.abs // m.abs may be nil for m == 0
 590         }
 591
 592         z.abs = z.abs.expNN(x.abs, y.abs, mWords)
 593         z.neg = len(z.abs) > 0 && x.neg && y.abs[0]&1 == 1 // 0 has no sign
 594         return z
 595 }
 596
 597 // GCD sets z to the greatest common divisor of a and b, which both must
 598 // be > 0, and returns z.
 599 // If x and y are not nil, GCD sets x and y such that z = a*x + b*y.
 600 // If either a or b is <= 0, GCD sets z = x = y = 0.
 601 func (z *Int) GCD(x, y, a, b *Int) *Int {
 602         if a.Sign() <= 0 || b.Sign() <= 0 {
 603                 z.SetInt64(0)
 604                 if x != nil {
 605                         x.SetInt64(0)
 606                 }
 607                 if y != nil {
 608                         y.SetInt64(0)
 609                 }
 610                 return z
 611         }
 612         if x == nil && y == nil {
 613                 return z.binaryGCD(a, b)
 614         }
 615
 616         A := new(Int).Set(a)
 617         B := new(Int).Set(b)
 618
 619         X := new(Int)
 620         Y := new(Int).SetInt64(1)
 621
 622         lastX := new(Int).SetInt64(1)
 623         lastY := new(Int)
 624
 625         q := new(Int)
 626         temp := new(Int)
 627
 628         for len(B.abs) > 0 {
 629                 r := new(Int)
 630                 q, r = q.QuoRem(A, B, r)
 631
 632                 A, B = B, r
 633
 634                 temp.Set(X)
 635                 X.Mul(X, q)
 636                 X.neg = !X.neg
 637                 X.Add(X, lastX)
 638                 lastX.Set(temp)
 639
 640                 temp.Set(Y)
 641                 Y.Mul(Y, q)
 642                 Y.neg = !Y.neg
 643                 Y.Add(Y, lastY)
 644                 lastY.Set(temp)
 645         }
 646
 647         if x != nil {
 648                 *x = *lastX
 649         }
 650
 651         if y != nil {
 652                 *y = *lastY
 653         }
 654
 655         *z = *A
 656         return z
 657 }
 658
 659 // binaryGCD sets z to the greatest common divisor of a and b, which both must
 660 // be > 0, and returns z.
 661 // See Knuth, The Art of Computer Programming, Vol. 2, Section 4.5.2, Algorithm B.
 662 func (z *Int) binaryGCD(a, b *Int) *Int {
 663         u := z
 664         v := new(Int)
 665
 666         // use one Euclidean iteration to ensure that u and v are approx. the same size
 667         switch {
 668         case len(a.abs) > len(b.abs):
 669                 u.Set(b)
 670                 v.Rem(a, b)
 671         case len(a.abs) < len(b.abs):
 672                 u.Set(a)
 673                 v.Rem(b, a)
 674         default:
 675                 u.Set(a)
 676                 v.Set(b)
 677         }
 678
 679         // v might be 0 now
 680         if len(v.abs) == 0 {
 681                 return u
 682         }
 683         // u > 0 && v > 0
 684
 685         // determine largest k such that u = u' << k, v = v' << k
 686         k := u.abs.trailingZeroBits()
 687         if vk := v.abs.trailingZeroBits(); vk < k {
 688                 k = vk
 689         }
 690         u.Rsh(u, k)
 691         v.Rsh(v, k)
 692
 693         // determine t (we know that u > 0)
 694         t := new(Int)
 695         if u.abs[0]&1 != 0 {
 696                 // u is odd
 697                 t.Neg(v)
 698         } else {
 699                 t.Set(u)
 700         }
 701
 702         for len(t.abs) > 0 {
 703                 // reduce t
 704                 t.Rsh(t, t.abs.trailingZeroBits())
 705                 if t.neg {
 706                         v, t = t, v
 707                         v.neg = len(v.abs) > 0 && !v.neg // 0 has no sign
 708                 } else {
 709                         u, t = t, u
 710                 }
 711                 t.Sub(u, v)
 712         }
 713
 714         return z.Lsh(u, k)
 715 }
 716
 717 // ProbablyPrime performs n Miller-Rabin tests to check whether x is prime.
 718 // If it returns true, x is prime with probability 1 - 1/4^n.
 719 // If it returns false, x is not prime.
 720 func (x *Int) ProbablyPrime(n int) bool {
 721         return !x.neg && x.abs.probablyPrime(n)
 722 }
 723
 724 // Rand sets z to a pseudo-random number in [0, n) and returns z.
 725 func (z *Int) Rand(rnd *rand.Rand, n *Int) *Int {
 726         z.neg = false
 727         if n.neg == true || len(n.abs) == 0 {
 728                 z.abs = nil
 729                 return z
 730         }
 731         z.abs = z.abs.random(rnd, n.abs, n.abs.bitLen())
 732         return z
 733 }
 734
 735 // ModInverse sets z to the multiplicative inverse of g in the group ℤ/pℤ (where
 736 // p is a prime) and returns z.
 737 func (z *Int) ModInverse(g, p *Int) *Int {
 738         var d Int
 739         d.GCD(z, nil, g, p)
 740         // x and y are such that g*x + p*y = d. Since p is prime, d = 1. Taking
 741         // that modulo p results in g*x = 1, therefore x is the inverse element.
 742         if z.neg {
 743                 z.Add(z, p)
 744         }
 745         return z
 746 }
 747
 748 // Lsh sets z = x << n and returns z.
 749 func (z *Int) Lsh(x *Int, n uint) *Int {
 750         z.abs = z.abs.shl(x.abs, n)
 751         z.neg = x.neg
 752         return z
 753 }
 754
 755 // Rsh sets z = x >> n and returns z.
 756 func (z *Int) Rsh(x *Int, n uint) *Int {
 757         if x.neg {
 758                 // (-x) >> s == ^(x-1) >> s == ^((x-1) >> s) == -(((x-1) >> s) + 1)
 759                 t := z.abs.sub(x.abs, natOne) // no underflow because |x| > 0
 760                 t = t.shr(t, n)
 761                 z.abs = t.add(t, natOne)
 762                 z.neg = true // z cannot be zero if x is negative
 763                 return z
 764         }
 765
 766         z.abs = z.abs.shr(x.abs, n)
 767         z.neg = false
 768         return z
 769 }
 770
 771 // Bit returns the value of the i'th bit of x. That is, it
 772 // returns (x>>i)&1. The bit index i must be >= 0.
 773 func (x *Int) Bit(i int) uint {
 774         if i == 0 {
 775                 // optimization for common case: odd/even test of x
 776                 if len(x.abs) > 0 {
 777                         return uint(x.abs[0] & 1) // bit 0 is same for -x
 778                 }
 779                 return 0
 780         }
 781         if i < 0 {
 782                 panic("negative bit index")
 783         }
 784         if x.neg {
 785                 t := nat(nil).sub(x.abs, natOne)
 786                 return t.bit(uint(i)) ^ 1
 787         }
 788
 789         return x.abs.bit(uint(i))
 790 }
 791
 792 // SetBit sets z to x, with x's i'th bit set to b (0 or 1).
 793 // That is, if b is 1 SetBit sets z = x | (1 << i);
 794 // if b is 0 SetBit sets z = x &^ (1 << i). If b is not 0 or 1,
 795 // SetBit will panic.
 796 func (z *Int) SetBit(x *Int, i int, b uint) *Int {
 797         if i < 0 {
 798                 panic("negative bit index")
 799         }
 800         if x.neg {
 801                 t := z.abs.sub(x.abs, natOne)
 802                 t = t.setBit(t, uint(i), b^1)
 803                 z.abs = t.add(t, natOne)
 804                 z.neg = len(z.abs) > 0
 805                 return z
 806         }
 807         z.abs = z.abs.setBit(x.abs, uint(i), b)
 808         z.neg = false
 809         return z
 810 }
 811
 812 // And sets z = x & y and returns z.
 813 func (z *Int) And(x, y *Int) *Int {
 814         if x.neg == y.neg {
 815                 if x.neg {
 816                         // (-x) & (-y) == ^(x-1) & ^(y-1) == ^((x-1) | (y-1)) == -(((x-1) | (y-1)) + 1)
 817                         x1 := nat(nil).sub(x.abs, natOne)
 818                         y1 := nat(nil).sub(y.abs, natOne)
 819                         z.abs = z.abs.add(z.abs.or(x1, y1), natOne)
 820                         z.neg = true // z cannot be zero if x and y are negative
 821                         return z
 822                 }
 823
 824                 // x & y == x & y
 825                 z.abs = z.abs.and(x.abs, y.abs)
 826                 z.neg = false
 827                 return z
 828         }
 829
 830         // x.neg != y.neg
 831         if x.neg {
 832                 x, y = y, x // & is symmetric
 833         }
 834
 835         // x & (-y) == x & ^(y-1) == x &^ (y-1)
 836         y1 := nat(nil).sub(y.abs, natOne)
 837         z.abs = z.abs.andNot(x.abs, y1)
 838         z.neg = false
 839         return z
 840 }
 841
 842 // AndNot sets z = x &^ y and returns z.
 843 func (z *Int) AndNot(x, y *Int) *Int {
 844         if x.neg == y.neg {
 845                 if x.neg {
 846                         // (-x) &^ (-y) == ^(x-1) &^ ^(y-1) == ^(x-1) & (y-1) == (y-1) &^ (x-1)
 847                         x1 := nat(nil).sub(x.abs, natOne)
 848                         y1 := nat(nil).sub(y.abs, natOne)
 849                         z.abs = z.abs.andNot(y1, x1)
 850                         z.neg = false
 851                         return z
 852                 }
 853
 854                 // x &^ y == x &^ y
 855                 z.abs = z.abs.andNot(x.abs, y.abs)
 856                 z.neg = false
 857                 return z
 858         }
 859
 860         if x.neg {
 861                 // (-x) &^ y == ^(x-1) &^ y == ^(x-1) & ^y == ^((x-1) | y) == -(((x-1) | y) + 1)
 862                 x1 := nat(nil).sub(x.abs, natOne)
 863                 z.abs = z.abs.add(z.abs.or(x1, y.abs), natOne)
 864                 z.neg = true // z cannot be zero if x is negative and y is positive
 865                 return z
 866         }
 867
 868         // x &^ (-y) == x &^ ^(y-1) == x & (y-1)
 869         y1 := nat(nil).add(y.abs, natOne)
 870         z.abs = z.abs.and(x.abs, y1)
 871         z.neg = false
 872         return z
 873 }
 874
 875 // Or sets z = x | y and returns z.
 876 func (z *Int) Or(x, y *Int) *Int {
 877         if x.neg == y.neg {
 878                 if x.neg {
 879                         // (-x) | (-y) == ^(x-1) | ^(y-1) == ^((x-1) & (y-1)) == -(((x-1) & (y-1)) + 1)
 880                         x1 := nat(nil).sub(x.abs, natOne)
 881                         y1 := nat(nil).sub(y.abs, natOne)
 882                         z.abs = z.abs.add(z.abs.and(x1, y1), natOne)
 883                         z.neg = true // z cannot be zero if x and y are negative
 884                         return z
 885                 }
 886
 887                 // x | y == x | y
 888                 z.abs = z.abs.or(x.abs, y.abs)
 889                 z.neg = false
 890                 return z
 891         }
 892
 893         // x.neg != y.neg
 894         if x.neg {
 895                 x, y = y, x // | is symmetric
 896         }
 897
 898         // x | (-y) == x | ^(y-1) == ^((y-1) &^ x) == -(^((y-1) &^ x) + 1)
 899         y1 := nat(nil).sub(y.abs, natOne)
 900         z.abs = z.abs.add(z.abs.andNot(y1, x.abs), natOne)
 901         z.neg = true // z cannot be zero if one of x or y is negative
 902         return z
 903 }
 904
 905 // Xor sets z = x ^ y and returns z.
 906 func (z *Int) Xor(x, y *Int) *Int {
 907         if x.neg == y.neg {
 908                 if x.neg {
 909                         // (-x) ^ (-y) == ^(x-1) ^ ^(y-1) == (x-1) ^ (y-1)
 910                         x1 := nat(nil).sub(x.abs, natOne)
 911                         y1 := nat(nil).sub(y.abs, natOne)
 912                         z.abs = z.abs.xor(x1, y1)
 913                         z.neg = false
 914                         return z
 915                 }
 916
 917                 // x ^ y == x ^ y
 918                 z.abs = z.abs.xor(x.abs, y.abs)
 919                 z.neg = false
 920                 return z
 921         }
 922
 923         // x.neg != y.neg
 924         if x.neg {
 925                 x, y = y, x // ^ is symmetric
 926         }
 927
 928         // x ^ (-y) == x ^ ^(y-1) == ^(x ^ (y-1)) == -((x ^ (y-1)) + 1)
 929         y1 := nat(nil).sub(y.abs, natOne)
 930         z.abs = z.abs.add(z.abs.xor(x.abs, y1), natOne)
 931         z.neg = true // z cannot be zero if only one of x or y is negative
 932         return z
 933 }
 934
 935 // Not sets z = ^x and returns z.
 936 func (z *Int) Not(x *Int) *Int {
 937         if x.neg {
 938                 // ^(-x) == ^(^(x-1)) == x-1
 939                 z.abs = z.abs.sub(x.abs, natOne)
 940                 z.neg = false
 941                 return z
 942         }
 943
 944         // ^x == -x-1 == -(x+1)
 945         z.abs = z.abs.add(x.abs, natOne)
 946         z.neg = true // z cannot be zero if x is positive
 947         return z
 948 }
 949
 950 // Gob codec version. Permits backward-compatible changes to the encoding.
 951 const intGobVersion byte = 1
 952
 953 // GobEncode implements the gob.GobEncoder interface.
 954 func (x *Int) GobEncode() ([]byte, error) {
 955         if x == nil {
 956                 return nil, nil
 957         }
 958         buf := make([]byte, 1+len(x.abs)*_S) // extra byte for version and sign bit
 959         i := x.abs.bytes(buf) - 1            // i >= 0
 960         b := intGobVersion << 1              // make space for sign bit
 961         if x.neg {
 962                 b |= 1
 963         }
 964         buf[i] = b
 965         return buf[i:], nil
 966 }
 967
 968 // GobDecode implements the gob.GobDecoder interface.
 969 func (z *Int) GobDecode(buf []byte) error {
 970         if len(buf) == 0 {
 971                 // Other side sent a nil or default value.
 972                 *z = Int{}
 973                 return nil
 974         }
 975         b := buf[0]
 976         if b>>1 != intGobVersion {
 977                 return errors.New(fmt.Sprintf("Int.GobDecode: encoding version %d not supported", b>>1))
 978         }
 979         z.neg = b&1 != 0
 980         z.abs = z.abs.setBytes(buf[1:])
 981         return nil
 982 }
 983
 984 // MarshalJSON implements the json.Marshaler interface.
 985 func (x *Int) MarshalJSON() ([]byte, error) {
 986         // TODO(gri): get rid of the []byte/string conversions
 987         return []byte(x.String()), nil
 988 }
 989
 990 // UnmarshalJSON implements the json.Unmarshaler interface.
 991 func (z *Int) UnmarshalJSON(x []byte) error {
 992         // TODO(gri): get rid of the []byte/string conversions
 993         _, ok := z.SetString(string(x), 0)
 994         if !ok {
 995                 return fmt.Errorf("math/big: cannot unmarshal %s into a *big.Int", x)
 996         }
 997         return nil
 998 }