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Source file src/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		"fmt"
    11		"io"
    12		"math/rand"
    13		"strings"
    14	)
    15	
    16	// An Int represents a signed multi-precision integer.
    17	// The zero value for an Int represents the value 0.
    18	type Int struct {
    19		neg bool // sign
    20		abs nat  // absolute value of the integer
    21	}
    22	
    23	var intOne = &Int{false, natOne}
    24	
    25	// Sign returns:
    26	//
    27	//	-1 if x <  0
    28	//	 0 if x == 0
    29	//	+1 if x >  0
    30	//
    31	func (x *Int) Sign() int {
    32		if len(x.abs) == 0 {
    33			return 0
    34		}
    35		if x.neg {
    36			return -1
    37		}
    38		return 1
    39	}
    40	
    41	// SetInt64 sets z to x and returns z.
    42	func (z *Int) SetInt64(x int64) *Int {
    43		neg := false
    44		if x < 0 {
    45			neg = true
    46			x = -x
    47		}
    48		z.abs = z.abs.setUint64(uint64(x))
    49		z.neg = neg
    50		return z
    51	}
    52	
    53	// SetUint64 sets z to x and returns z.
    54	func (z *Int) SetUint64(x uint64) *Int {
    55		z.abs = z.abs.setUint64(x)
    56		z.neg = false
    57		return z
    58	}
    59	
    60	// NewInt allocates and returns a new Int set to x.
    61	func NewInt(x int64) *Int {
    62		return new(Int).SetInt64(x)
    63	}
    64	
    65	// Set sets z to x and returns z.
    66	func (z *Int) Set(x *Int) *Int {
    67		if z != x {
    68			z.abs = z.abs.set(x.abs)
    69			z.neg = x.neg
    70		}
    71		return z
    72	}
    73	
    74	// Bits provides raw (unchecked but fast) access to x by returning its
    75	// absolute value as a little-endian Word slice. The result and x share
    76	// the same underlying array.
    77	// Bits is intended to support implementation of missing low-level Int
    78	// functionality outside this package; it should be avoided otherwise.
    79	func (x *Int) Bits() []Word {
    80		return x.abs
    81	}
    82	
    83	// SetBits provides raw (unchecked but fast) access to z by setting its
    84	// value to abs, interpreted as a little-endian Word slice, and returning
    85	// z. The result and abs share the same underlying array.
    86	// SetBits is intended to support implementation of missing low-level Int
    87	// functionality outside this package; it should be avoided otherwise.
    88	func (z *Int) SetBits(abs []Word) *Int {
    89		z.abs = nat(abs).norm()
    90		z.neg = false
    91		return z
    92	}
    93	
    94	// Abs sets z to |x| (the absolute value of x) and returns z.
    95	func (z *Int) Abs(x *Int) *Int {
    96		z.Set(x)
    97		z.neg = false
    98		return z
    99	}
   100	
   101	// Neg sets z to -x and returns z.
   102	func (z *Int) Neg(x *Int) *Int {
   103		z.Set(x)
   104		z.neg = len(z.abs) > 0 && !z.neg // 0 has no sign
   105		return z
   106	}
   107	
   108	// Add sets z to the sum x+y and returns z.
   109	func (z *Int) Add(x, y *Int) *Int {
   110		neg := x.neg
   111		if x.neg == y.neg {
   112			// x + y == x + y
   113			// (-x) + (-y) == -(x + y)
   114			z.abs = z.abs.add(x.abs, y.abs)
   115		} else {
   116			// x + (-y) == x - y == -(y - x)
   117			// (-x) + y == y - x == -(x - y)
   118			if x.abs.cmp(y.abs) >= 0 {
   119				z.abs = z.abs.sub(x.abs, y.abs)
   120			} else {
   121				neg = !neg
   122				z.abs = z.abs.sub(y.abs, x.abs)
   123			}
   124		}
   125		z.neg = len(z.abs) > 0 && neg // 0 has no sign
   126		return z
   127	}
   128	
   129	// Sub sets z to the difference x-y and returns z.
   130	func (z *Int) Sub(x, y *Int) *Int {
   131		neg := x.neg
   132		if x.neg != y.neg {
   133			// x - (-y) == x + y
   134			// (-x) - y == -(x + y)
   135			z.abs = z.abs.add(x.abs, y.abs)
   136		} else {
   137			// x - y == x - y == -(y - x)
   138			// (-x) - (-y) == y - x == -(x - y)
   139			if x.abs.cmp(y.abs) >= 0 {
   140				z.abs = z.abs.sub(x.abs, y.abs)
   141			} else {
   142				neg = !neg
   143				z.abs = z.abs.sub(y.abs, x.abs)
   144			}
   145		}
   146		z.neg = len(z.abs) > 0 && neg // 0 has no sign
   147		return z
   148	}
   149	
   150	// Mul sets z to the product x*y and returns z.
   151	func (z *Int) Mul(x, y *Int) *Int {
   152		// x * y == x * y
   153		// x * (-y) == -(x * y)
   154		// (-x) * y == -(x * y)
   155		// (-x) * (-y) == x * y
   156		z.abs = z.abs.mul(x.abs, y.abs)
   157		z.neg = len(z.abs) > 0 && x.neg != y.neg // 0 has no sign
   158		return z
   159	}
   160	
   161	// MulRange sets z to the product of all integers
   162	// in the range [a, b] inclusively and returns z.
   163	// If a > b (empty range), the result is 1.
   164	func (z *Int) MulRange(a, b int64) *Int {
   165		switch {
   166		case a > b:
   167			return z.SetInt64(1) // empty range
   168		case a <= 0 && b >= 0:
   169			return z.SetInt64(0) // range includes 0
   170		}
   171		// a <= b && (b < 0 || a > 0)
   172	
   173		neg := false
   174		if a < 0 {
   175			neg = (b-a)&1 == 0
   176			a, b = -b, -a
   177		}
   178	
   179		z.abs = z.abs.mulRange(uint64(a), uint64(b))
   180		z.neg = neg
   181		return z
   182	}
   183	
   184	// Binomial sets z to the binomial coefficient of (n, k) and returns z.
   185	func (z *Int) Binomial(n, k int64) *Int {
   186		// reduce the number of multiplications by reducing k
   187		if n/2 < k && k <= n {
   188			k = n - k // Binomial(n, k) == Binomial(n, n-k)
   189		}
   190		var a, b Int
   191		a.MulRange(n-k+1, n)
   192		b.MulRange(1, k)
   193		return z.Quo(&a, &b)
   194	}
   195	
   196	// Quo sets z to the quotient x/y for y != 0 and returns z.
   197	// If y == 0, a division-by-zero run-time panic occurs.
   198	// Quo implements truncated division (like Go); see QuoRem for more details.
   199	func (z *Int) Quo(x, y *Int) *Int {
   200		z.abs, _ = z.abs.div(nil, x.abs, y.abs)
   201		z.neg = len(z.abs) > 0 && x.neg != y.neg // 0 has no sign
   202		return z
   203	}
   204	
   205	// Rem sets z to the remainder x%y for y != 0 and returns z.
   206	// If y == 0, a division-by-zero run-time panic occurs.
   207	// Rem implements truncated modulus (like Go); see QuoRem for more details.
   208	func (z *Int) Rem(x, y *Int) *Int {
   209		_, z.abs = nat(nil).div(z.abs, x.abs, y.abs)
   210		z.neg = len(z.abs) > 0 && x.neg // 0 has no sign
   211		return z
   212	}
   213	
   214	// QuoRem sets z to the quotient x/y and r to the remainder x%y
   215	// and returns the pair (z, r) for y != 0.
   216	// If y == 0, a division-by-zero run-time panic occurs.
   217	//
   218	// QuoRem implements T-division and modulus (like Go):
   219	//
   220	//	q = x/y      with the result truncated to zero
   221	//	r = x - y*q
   222	//
   223	// (See Daan Leijen, ``Division and Modulus for Computer Scientists''.)
   224	// See DivMod for Euclidean division and modulus (unlike Go).
   225	//
   226	func (z *Int) QuoRem(x, y, r *Int) (*Int, *Int) {
   227		z.abs, r.abs = z.abs.div(r.abs, x.abs, y.abs)
   228		z.neg, r.neg = len(z.abs) > 0 && x.neg != y.neg, len(r.abs) > 0 && x.neg // 0 has no sign
   229		return z, r
   230	}
   231	
   232	// Div sets z to the quotient x/y for y != 0 and returns z.
   233	// If y == 0, a division-by-zero run-time panic occurs.
   234	// Div implements Euclidean division (unlike Go); see DivMod for more details.
   235	func (z *Int) Div(x, y *Int) *Int {
   236		y_neg := y.neg // z may be an alias for y
   237		var r Int
   238		z.QuoRem(x, y, &r)
   239		if r.neg {
   240			if y_neg {
   241				z.Add(z, intOne)
   242			} else {
   243				z.Sub(z, intOne)
   244			}
   245		}
   246		return z
   247	}
   248	
   249	// Mod sets z to the modulus x%y for y != 0 and returns z.
   250	// If y == 0, a division-by-zero run-time panic occurs.
   251	// Mod implements Euclidean modulus (unlike Go); see DivMod for more details.
   252	func (z *Int) Mod(x, y *Int) *Int {
   253		y0 := y // save y
   254		if z == y || alias(z.abs, y.abs) {
   255			y0 = new(Int).Set(y)
   256		}
   257		var q Int
   258		q.QuoRem(x, y, z)
   259		if z.neg {
   260			if y0.neg {
   261				z.Sub(z, y0)
   262			} else {
   263				z.Add(z, y0)
   264			}
   265		}
   266		return z
   267	}
   268	
   269	// DivMod sets z to the quotient x div y and m to the modulus x mod y
   270	// and returns the pair (z, m) for y != 0.
   271	// If y == 0, a division-by-zero run-time panic occurs.
   272	//
   273	// DivMod implements Euclidean division and modulus (unlike Go):
   274	//
   275	//	q = x div y  such that
   276	//	m = x - y*q  with 0 <= m < |q|
   277	//
   278	// (See Raymond T. Boute, ``The Euclidean definition of the functions
   279	// div and mod''. ACM Transactions on Programming Languages and
   280	// Systems (TOPLAS), 14(2):127-144, New York, NY, USA, 4/1992.
   281	// ACM press.)
   282	// See QuoRem for T-division and modulus (like Go).
   283	//
   284	func (z *Int) DivMod(x, y, m *Int) (*Int, *Int) {
   285		y0 := y // save y
   286		if z == y || alias(z.abs, y.abs) {
   287			y0 = new(Int).Set(y)
   288		}
   289		z.QuoRem(x, y, m)
   290		if m.neg {
   291			if y0.neg {
   292				z.Add(z, intOne)
   293				m.Sub(m, y0)
   294			} else {
   295				z.Sub(z, intOne)
   296				m.Add(m, y0)
   297			}
   298		}
   299		return z, m
   300	}
   301	
   302	// Cmp compares x and y and returns:
   303	//
   304	//   -1 if x <  y
   305	//    0 if x == y
   306	//   +1 if x >  y
   307	//
   308	func (x *Int) Cmp(y *Int) (r int) {
   309		// x cmp y == x cmp y
   310		// x cmp (-y) == x
   311		// (-x) cmp y == y
   312		// (-x) cmp (-y) == -(x cmp y)
   313		switch {
   314		case x.neg == y.neg:
   315			r = x.abs.cmp(y.abs)
   316			if x.neg {
   317				r = -r
   318			}
   319		case x.neg:
   320			r = -1
   321		default:
   322			r = 1
   323		}
   324		return
   325	}
   326	
   327	// low32 returns the least significant 32 bits of z.
   328	func low32(z nat) uint32 {
   329		if len(z) == 0 {
   330			return 0
   331		}
   332		return uint32(z[0])
   333	}
   334	
   335	// low64 returns the least significant 64 bits of z.
   336	func low64(z nat) uint64 {
   337		if len(z) == 0 {
   338			return 0
   339		}
   340		v := uint64(z[0])
   341		if _W == 32 && len(z) > 1 {
   342			v |= uint64(z[1]) << 32
   343		}
   344		return v
   345	}
   346	
   347	// Int64 returns the int64 representation of x.
   348	// If x cannot be represented in an int64, the result is undefined.
   349	func (x *Int) Int64() int64 {
   350		v := int64(low64(x.abs))
   351		if x.neg {
   352			v = -v
   353		}
   354		return v
   355	}
   356	
   357	// Uint64 returns the uint64 representation of x.
   358	// If x cannot be represented in a uint64, the result is undefined.
   359	func (x *Int) Uint64() uint64 {
   360		return low64(x.abs)
   361	}
   362	
   363	// SetString sets z to the value of s, interpreted in the given base,
   364	// and returns z and a boolean indicating success. If SetString fails,
   365	// the value of z is undefined but the returned value is nil.
   366	//
   367	// The base argument must be 0 or a value between 2 and MaxBase. If the base
   368	// is 0, the string prefix determines the actual conversion base. A prefix of
   369	// ``0x'' or ``0X'' selects base 16; the ``0'' prefix selects base 8, and a
   370	// ``0b'' or ``0B'' prefix selects base 2. Otherwise the selected base is 10.
   371	//
   372	func (z *Int) SetString(s string, base int) (*Int, bool) {
   373		r := strings.NewReader(s)
   374		_, _, err := z.scan(r, base)
   375		if err != nil {
   376			return nil, false
   377		}
   378		_, err = r.ReadByte()
   379		if err != io.EOF {
   380			return nil, false
   381		}
   382		return z, true // err == io.EOF => scan consumed all of s
   383	}
   384	
   385	// SetBytes interprets buf as the bytes of a big-endian unsigned
   386	// integer, sets z to that value, and returns z.
   387	func (z *Int) SetBytes(buf []byte) *Int {
   388		z.abs = z.abs.setBytes(buf)
   389		z.neg = false
   390		return z
   391	}
   392	
   393	// Bytes returns the absolute value of x as a big-endian byte slice.
   394	func (x *Int) Bytes() []byte {
   395		buf := make([]byte, len(x.abs)*_S)
   396		return buf[x.abs.bytes(buf):]
   397	}
   398	
   399	// BitLen returns the length of the absolute value of x in bits.
   400	// The bit length of 0 is 0.
   401	func (x *Int) BitLen() int {
   402		return x.abs.bitLen()
   403	}
   404	
   405	// Exp sets z = x**y mod |m| (i.e. the sign of m is ignored), and returns z.
   406	// If y <= 0, the result is 1 mod |m|; if m == nil or m == 0, z = x**y.
   407	// See Knuth, volume 2, section 4.6.3.
   408	func (z *Int) Exp(x, y, m *Int) *Int {
   409		var yWords nat
   410		if !y.neg {
   411			yWords = y.abs
   412		}
   413		// y >= 0
   414	
   415		var mWords nat
   416		if m != nil {
   417			mWords = m.abs // m.abs may be nil for m == 0
   418		}
   419	
   420		z.abs = z.abs.expNN(x.abs, yWords, mWords)
   421		z.neg = len(z.abs) > 0 && x.neg && len(yWords) > 0 && yWords[0]&1 == 1 // 0 has no sign
   422		if z.neg && len(mWords) > 0 {
   423			// make modulus result positive
   424			z.abs = z.abs.sub(mWords, z.abs) // z == x**y mod |m| && 0 <= z < |m|
   425			z.neg = false
   426		}
   427	
   428		return z
   429	}
   430	
   431	// GCD sets z to the greatest common divisor of a and b, which both must
   432	// be > 0, and returns z.
   433	// If x and y are not nil, GCD sets x and y such that z = a*x + b*y.
   434	// If either a or b is <= 0, GCD sets z = x = y = 0.
   435	func (z *Int) GCD(x, y, a, b *Int) *Int {
   436		if a.Sign() <= 0 || b.Sign() <= 0 {
   437			z.SetInt64(0)
   438			if x != nil {
   439				x.SetInt64(0)
   440			}
   441			if y != nil {
   442				y.SetInt64(0)
   443			}
   444			return z
   445		}
   446		if x == nil && y == nil {
   447			return z.binaryGCD(a, b)
   448		}
   449	
   450		A := new(Int).Set(a)
   451		B := new(Int).Set(b)
   452	
   453		X := new(Int)
   454		Y := new(Int).SetInt64(1)
   455	
   456		lastX := new(Int).SetInt64(1)
   457		lastY := new(Int)
   458	
   459		q := new(Int)
   460		temp := new(Int)
   461	
   462		for len(B.abs) > 0 {
   463			r := new(Int)
   464			q, r = q.QuoRem(A, B, r)
   465	
   466			A, B = B, r
   467	
   468			temp.Set(X)
   469			X.Mul(X, q)
   470			X.neg = !X.neg
   471			X.Add(X, lastX)
   472			lastX.Set(temp)
   473	
   474			temp.Set(Y)
   475			Y.Mul(Y, q)
   476			Y.neg = !Y.neg
   477			Y.Add(Y, lastY)
   478			lastY.Set(temp)
   479		}
   480	
   481		if x != nil {
   482			*x = *lastX
   483		}
   484	
   485		if y != nil {
   486			*y = *lastY
   487		}
   488	
   489		*z = *A
   490		return z
   491	}
   492	
   493	// binaryGCD sets z to the greatest common divisor of a and b, which both must
   494	// be > 0, and returns z.
   495	// See Knuth, The Art of Computer Programming, Vol. 2, Section 4.5.2, Algorithm B.
   496	func (z *Int) binaryGCD(a, b *Int) *Int {
   497		u := z
   498		v := new(Int)
   499	
   500		// use one Euclidean iteration to ensure that u and v are approx. the same size
   501		switch {
   502		case len(a.abs) > len(b.abs):
   503			// must set v before u since u may be alias for a or b (was issue #11284)
   504			v.Rem(a, b)
   505			u.Set(b)
   506		case len(a.abs) < len(b.abs):
   507			v.Rem(b, a)
   508			u.Set(a)
   509		default:
   510			v.Set(b)
   511			u.Set(a)
   512		}
   513		// a, b must not be used anymore (may be aliases with u)
   514	
   515		// v might be 0 now
   516		if len(v.abs) == 0 {
   517			return u
   518		}
   519		// u > 0 && v > 0
   520	
   521		// determine largest k such that u = u' << k, v = v' << k
   522		k := u.abs.trailingZeroBits()
   523		if vk := v.abs.trailingZeroBits(); vk < k {
   524			k = vk
   525		}
   526		u.Rsh(u, k)
   527		v.Rsh(v, k)
   528	
   529		// determine t (we know that u > 0)
   530		t := new(Int)
   531		if u.abs[0]&1 != 0 {
   532			// u is odd
   533			t.Neg(v)
   534		} else {
   535			t.Set(u)
   536		}
   537	
   538		for len(t.abs) > 0 {
   539			// reduce t
   540			t.Rsh(t, t.abs.trailingZeroBits())
   541			if t.neg {
   542				v, t = t, v
   543				v.neg = len(v.abs) > 0 && !v.neg // 0 has no sign
   544			} else {
   545				u, t = t, u
   546			}
   547			t.Sub(u, v)
   548		}
   549	
   550		return z.Lsh(u, k)
   551	}
   552	
   553	// ProbablyPrime performs n Miller-Rabin tests to check whether x is prime.
   554	// If it returns true, x is prime with probability 1 - 1/4^n.
   555	// If it returns false, x is not prime. n must be > 0.
   556	func (x *Int) ProbablyPrime(n int) bool {
   557		if n <= 0 {
   558			panic("non-positive n for ProbablyPrime")
   559		}
   560		return !x.neg && x.abs.probablyPrime(n)
   561	}
   562	
   563	// Rand sets z to a pseudo-random number in [0, n) and returns z.
   564	func (z *Int) Rand(rnd *rand.Rand, n *Int) *Int {
   565		z.neg = false
   566		if n.neg == true || len(n.abs) == 0 {
   567			z.abs = nil
   568			return z
   569		}
   570		z.abs = z.abs.random(rnd, n.abs, n.abs.bitLen())
   571		return z
   572	}
   573	
   574	// ModInverse sets z to the multiplicative inverse of g in the ring ℤ/nℤ
   575	// and returns z. If g and n are not relatively prime, the result is undefined.
   576	func (z *Int) ModInverse(g, n *Int) *Int {
   577		var d Int
   578		d.GCD(z, nil, g, n)
   579		// x and y are such that g*x + n*y = d. Since g and n are
   580		// relatively prime, d = 1. Taking that modulo n results in
   581		// g*x = 1, therefore x is the inverse element.
   582		if z.neg {
   583			z.Add(z, n)
   584		}
   585		return z
   586	}
   587	
   588	// Jacobi returns the Jacobi symbol (x/y), either +1, -1, or 0.
   589	// The y argument must be an odd integer.
   590	func Jacobi(x, y *Int) int {
   591		if len(y.abs) == 0 || y.abs[0]&1 == 0 {
   592			panic(fmt.Sprintf("big: invalid 2nd argument to Int.Jacobi: need odd integer but got %s", y))
   593		}
   594	
   595		// We use the formulation described in chapter 2, section 2.4,
   596		// "The Yacas Book of Algorithms":
   597		// http://yacas.sourceforge.net/Algo.book.pdf
   598	
   599		var a, b, c Int
   600		a.Set(x)
   601		b.Set(y)
   602		j := 1
   603	
   604		if b.neg {
   605			if a.neg {
   606				j = -1
   607			}
   608			b.neg = false
   609		}
   610	
   611		for {
   612			if b.Cmp(intOne) == 0 {
   613				return j
   614			}
   615			if len(a.abs) == 0 {
   616				return 0
   617			}
   618			a.Mod(&a, &b)
   619			if len(a.abs) == 0 {
   620				return 0
   621			}
   622			// a > 0
   623	
   624			// handle factors of 2 in 'a'
   625			s := a.abs.trailingZeroBits()
   626			if s&1 != 0 {
   627				bmod8 := b.abs[0] & 7
   628				if bmod8 == 3 || bmod8 == 5 {
   629					j = -j
   630				}
   631			}
   632			c.Rsh(&a, s) // a = 2^s*c
   633	
   634			// swap numerator and denominator
   635			if b.abs[0]&3 == 3 && c.abs[0]&3 == 3 {
   636				j = -j
   637			}
   638			a.Set(&b)
   639			b.Set(&c)
   640		}
   641	}
   642	
   643	// ModSqrt sets z to a square root of x mod p if such a square root exists, and
   644	// returns z. The modulus p must be an odd prime. If x is not a square mod p,
   645	// ModSqrt leaves z unchanged and returns nil. This function panics if p is
   646	// not an odd integer.
   647	func (z *Int) ModSqrt(x, p *Int) *Int {
   648		switch Jacobi(x, p) {
   649		case -1:
   650			return nil // x is not a square mod p
   651		case 0:
   652			return z.SetInt64(0) // sqrt(0) mod p = 0
   653		case 1:
   654			break
   655		}
   656		if x.neg || x.Cmp(p) >= 0 { // ensure 0 <= x < p
   657			x = new(Int).Mod(x, p)
   658		}
   659	
   660		// Break p-1 into s*2^e such that s is odd.
   661		var s Int
   662		s.Sub(p, intOne)
   663		e := s.abs.trailingZeroBits()
   664		s.Rsh(&s, e)
   665	
   666		// find some non-square n
   667		var n Int
   668		n.SetInt64(2)
   669		for Jacobi(&n, p) != -1 {
   670			n.Add(&n, intOne)
   671		}
   672	
   673		// Core of the Tonelli-Shanks algorithm. Follows the description in
   674		// section 6 of "Square roots from 1; 24, 51, 10 to Dan Shanks" by Ezra
   675		// Brown:
   676		// https://www.maa.org/sites/default/files/pdf/upload_library/22/Polya/07468342.di020786.02p0470a.pdf
   677		var y, b, g, t Int
   678		y.Add(&s, intOne)
   679		y.Rsh(&y, 1)
   680		y.Exp(x, &y, p)  // y = x^((s+1)/2)
   681		b.Exp(x, &s, p)  // b = x^s
   682		g.Exp(&n, &s, p) // g = n^s
   683		r := e
   684		for {
   685			// find the least m such that ord_p(b) = 2^m
   686			var m uint
   687			t.Set(&b)
   688			for t.Cmp(intOne) != 0 {
   689				t.Mul(&t, &t).Mod(&t, p)
   690				m++
   691			}
   692	
   693			if m == 0 {
   694				return z.Set(&y)
   695			}
   696	
   697			t.SetInt64(0).SetBit(&t, int(r-m-1), 1).Exp(&g, &t, p)
   698			// t = g^(2^(r-m-1)) mod p
   699			g.Mul(&t, &t).Mod(&g, p) // g = g^(2^(r-m)) mod p
   700			y.Mul(&y, &t).Mod(&y, p)
   701			b.Mul(&b, &g).Mod(&b, p)
   702			r = m
   703		}
   704	}
   705	
   706	// Lsh sets z = x << n and returns z.
   707	func (z *Int) Lsh(x *Int, n uint) *Int {
   708		z.abs = z.abs.shl(x.abs, n)
   709		z.neg = x.neg
   710		return z
   711	}
   712	
   713	// Rsh sets z = x >> n and returns z.
   714	func (z *Int) Rsh(x *Int, n uint) *Int {
   715		if x.neg {
   716			// (-x) >> s == ^(x-1) >> s == ^((x-1) >> s) == -(((x-1) >> s) + 1)
   717			t := z.abs.sub(x.abs, natOne) // no underflow because |x| > 0
   718			t = t.shr(t, n)
   719			z.abs = t.add(t, natOne)
   720			z.neg = true // z cannot be zero if x is negative
   721			return z
   722		}
   723	
   724		z.abs = z.abs.shr(x.abs, n)
   725		z.neg = false
   726		return z
   727	}
   728	
   729	// Bit returns the value of the i'th bit of x. That is, it
   730	// returns (x>>i)&1. The bit index i must be >= 0.
   731	func (x *Int) Bit(i int) uint {
   732		if i == 0 {
   733			// optimization for common case: odd/even test of x
   734			if len(x.abs) > 0 {
   735				return uint(x.abs[0] & 1) // bit 0 is same for -x
   736			}
   737			return 0
   738		}
   739		if i < 0 {
   740			panic("negative bit index")
   741		}
   742		if x.neg {
   743			t := nat(nil).sub(x.abs, natOne)
   744			return t.bit(uint(i)) ^ 1
   745		}
   746	
   747		return x.abs.bit(uint(i))
   748	}
   749	
   750	// SetBit sets z to x, with x's i'th bit set to b (0 or 1).
   751	// That is, if b is 1 SetBit sets z = x | (1 << i);
   752	// if b is 0 SetBit sets z = x &^ (1 << i). If b is not 0 or 1,
   753	// SetBit will panic.
   754	func (z *Int) SetBit(x *Int, i int, b uint) *Int {
   755		if i < 0 {
   756			panic("negative bit index")
   757		}
   758		if x.neg {
   759			t := z.abs.sub(x.abs, natOne)
   760			t = t.setBit(t, uint(i), b^1)
   761			z.abs = t.add(t, natOne)
   762			z.neg = len(z.abs) > 0
   763			return z
   764		}
   765		z.abs = z.abs.setBit(x.abs, uint(i), b)
   766		z.neg = false
   767		return z
   768	}
   769	
   770	// And sets z = x & y and returns z.
   771	func (z *Int) And(x, y *Int) *Int {
   772		if x.neg == y.neg {
   773			if x.neg {
   774				// (-x) & (-y) == ^(x-1) & ^(y-1) == ^((x-1) | (y-1)) == -(((x-1) | (y-1)) + 1)
   775				x1 := nat(nil).sub(x.abs, natOne)
   776				y1 := nat(nil).sub(y.abs, natOne)
   777				z.abs = z.abs.add(z.abs.or(x1, y1), natOne)
   778				z.neg = true // z cannot be zero if x and y are negative
   779				return z
   780			}
   781	
   782			// x & y == x & y
   783			z.abs = z.abs.and(x.abs, y.abs)
   784			z.neg = false
   785			return z
   786		}
   787	
   788		// x.neg != y.neg
   789		if x.neg {
   790			x, y = y, x // & is symmetric
   791		}
   792	
   793		// x & (-y) == x & ^(y-1) == x &^ (y-1)
   794		y1 := nat(nil).sub(y.abs, natOne)
   795		z.abs = z.abs.andNot(x.abs, y1)
   796		z.neg = false
   797		return z
   798	}
   799	
   800	// AndNot sets z = x &^ y and returns z.
   801	func (z *Int) AndNot(x, y *Int) *Int {
   802		if x.neg == y.neg {
   803			if x.neg {
   804				// (-x) &^ (-y) == ^(x-1) &^ ^(y-1) == ^(x-1) & (y-1) == (y-1) &^ (x-1)
   805				x1 := nat(nil).sub(x.abs, natOne)
   806				y1 := nat(nil).sub(y.abs, natOne)
   807				z.abs = z.abs.andNot(y1, x1)
   808				z.neg = false
   809				return z
   810			}
   811	
   812			// x &^ y == x &^ y
   813			z.abs = z.abs.andNot(x.abs, y.abs)
   814			z.neg = false
   815			return z
   816		}
   817	
   818		if x.neg {
   819			// (-x) &^ y == ^(x-1) &^ y == ^(x-1) & ^y == ^((x-1) | y) == -(((x-1) | y) + 1)
   820			x1 := nat(nil).sub(x.abs, natOne)
   821			z.abs = z.abs.add(z.abs.or(x1, y.abs), natOne)
   822			z.neg = true // z cannot be zero if x is negative and y is positive
   823			return z
   824		}
   825	
   826		// x &^ (-y) == x &^ ^(y-1) == x & (y-1)
   827		y1 := nat(nil).sub(y.abs, natOne)
   828		z.abs = z.abs.and(x.abs, y1)
   829		z.neg = false
   830		return z
   831	}
   832	
   833	// Or sets z = x | y and returns z.
   834	func (z *Int) Or(x, y *Int) *Int {
   835		if x.neg == y.neg {
   836			if x.neg {
   837				// (-x) | (-y) == ^(x-1) | ^(y-1) == ^((x-1) & (y-1)) == -(((x-1) & (y-1)) + 1)
   838				x1 := nat(nil).sub(x.abs, natOne)
   839				y1 := nat(nil).sub(y.abs, natOne)
   840				z.abs = z.abs.add(z.abs.and(x1, y1), natOne)
   841				z.neg = true // z cannot be zero if x and y are negative
   842				return z
   843			}
   844	
   845			// x | y == x | y
   846			z.abs = z.abs.or(x.abs, y.abs)
   847			z.neg = false
   848			return z
   849		}
   850	
   851		// x.neg != y.neg
   852		if x.neg {
   853			x, y = y, x // | is symmetric
   854		}
   855	
   856		// x | (-y) == x | ^(y-1) == ^((y-1) &^ x) == -(^((y-1) &^ x) + 1)
   857		y1 := nat(nil).sub(y.abs, natOne)
   858		z.abs = z.abs.add(z.abs.andNot(y1, x.abs), natOne)
   859		z.neg = true // z cannot be zero if one of x or y is negative
   860		return z
   861	}
   862	
   863	// Xor sets z = x ^ y and returns z.
   864	func (z *Int) Xor(x, y *Int) *Int {
   865		if x.neg == y.neg {
   866			if x.neg {
   867				// (-x) ^ (-y) == ^(x-1) ^ ^(y-1) == (x-1) ^ (y-1)
   868				x1 := nat(nil).sub(x.abs, natOne)
   869				y1 := nat(nil).sub(y.abs, natOne)
   870				z.abs = z.abs.xor(x1, y1)
   871				z.neg = false
   872				return z
   873			}
   874	
   875			// x ^ y == x ^ y
   876			z.abs = z.abs.xor(x.abs, y.abs)
   877			z.neg = false
   878			return z
   879		}
   880	
   881		// x.neg != y.neg
   882		if x.neg {
   883			x, y = y, x // ^ is symmetric
   884		}
   885	
   886		// x ^ (-y) == x ^ ^(y-1) == ^(x ^ (y-1)) == -((x ^ (y-1)) + 1)
   887		y1 := nat(nil).sub(y.abs, natOne)
   888		z.abs = z.abs.add(z.abs.xor(x.abs, y1), natOne)
   889		z.neg = true // z cannot be zero if only one of x or y is negative
   890		return z
   891	}
   892	
   893	// Not sets z = ^x and returns z.
   894	func (z *Int) Not(x *Int) *Int {
   895		if x.neg {
   896			// ^(-x) == ^(^(x-1)) == x-1
   897			z.abs = z.abs.sub(x.abs, natOne)
   898			z.neg = false
   899			return z
   900		}
   901	
   902		// ^x == -x-1 == -(x+1)
   903		z.abs = z.abs.add(x.abs, natOne)
   904		z.neg = true // z cannot be zero if x is positive
   905		return z
   906	}
   907	
   908	// Gob codec version. Permits backward-compatible changes to the encoding.
   909	const intGobVersion byte = 1
   910	
   911	// GobEncode implements the gob.GobEncoder interface.
   912	func (x *Int) GobEncode() ([]byte, error) {
   913		if x == nil {
   914			return nil, nil
   915		}
   916		buf := make([]byte, 1+len(x.abs)*_S) // extra byte for version and sign bit
   917		i := x.abs.bytes(buf) - 1            // i >= 0
   918		b := intGobVersion << 1              // make space for sign bit
   919		if x.neg {
   920			b |= 1
   921		}
   922		buf[i] = b
   923		return buf[i:], nil
   924	}
   925	
   926	// GobDecode implements the gob.GobDecoder interface.
   927	func (z *Int) GobDecode(buf []byte) error {
   928		if len(buf) == 0 {
   929			// Other side sent a nil or default value.
   930			*z = Int{}
   931			return nil
   932		}
   933		b := buf[0]
   934		if b>>1 != intGobVersion {
   935			return fmt.Errorf("Int.GobDecode: encoding version %d not supported", b>>1)
   936		}
   937		z.neg = b&1 != 0
   938		z.abs = z.abs.setBytes(buf[1:])
   939		return nil
   940	}
   941	
   942	// MarshalJSON implements the json.Marshaler interface.
   943	func (z *Int) MarshalJSON() ([]byte, error) {
   944		// TODO(gri): get rid of the []byte/string conversions
   945		return []byte(z.String()), nil
   946	}
   947	
   948	// UnmarshalJSON implements the json.Unmarshaler interface.
   949	func (z *Int) UnmarshalJSON(text []byte) error {
   950		// TODO(gri): get rid of the []byte/string conversions
   951		if _, ok := z.SetString(string(text), 0); !ok {
   952			return fmt.Errorf("math/big: cannot unmarshal %q into a *big.Int", text)
   953		}
   954		return nil
   955	}
   956	
   957	// MarshalText implements the encoding.TextMarshaler interface.
   958	func (z *Int) MarshalText() (text []byte, err error) {
   959		return []byte(z.String()), nil
   960	}
   961	
   962	// UnmarshalText implements the encoding.TextUnmarshaler interface.
   963	func (z *Int) UnmarshalText(text []byte) error {
   964		if _, ok := z.SetString(string(text), 0); !ok {
   965			return fmt.Errorf("math/big: cannot unmarshal %q into a *big.Int", text)
   966		}
   967		return nil
   968	}
   969	

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