libstdc++: Remove note from the GCC 4.0.1 days

[official-gcc.git] / libgo / go / math / big / int.go

// Copyright 2009 The Go Authors. All rights reserved.
// Use of this source code is governed by a BSD-style
// license that can be found in the LICENSE file.

// This file implements signed multi-precision integers.

package big

import (
        "fmt"
        "io"
        "math/rand"
        "strings"
)

// An Int represents a signed multi-precision integer.
// The zero value for an Int represents the value 0.
//
// Operations always take pointer arguments (*Int) rather
// than Int values, and each unique Int value requires
// its own unique *Int pointer. To "copy" an Int value,
// an existing (or newly allocated) Int must be set to
// a new value using the Int.Set method; shallow copies
// of Ints are not supported and may lead to errors.
type Int struct {
        neg bool // sign
        abs nat  // absolute value of the integer
}

var intOne = &Int{false, natOne}

// Sign returns:
//
//      -1 if x <  0
//       0 if x == 0
//      +1 if x >  0
//
func (x *Int) Sign() int {
        if len(x.abs) == 0 {
                return 0
        }
        if x.neg {
                return -1
        }
        return 1
}

// SetInt64 sets z to x and returns z.
func (z *Int) SetInt64(x int64) *Int {
        neg := false
        if x < 0 {
                neg = true
                x = -x
        }
        z.abs = z.abs.setUint64(uint64(x))
        z.neg = neg
        return z
}

// SetUint64 sets z to x and returns z.
func (z *Int) SetUint64(x uint64) *Int {
        z.abs = z.abs.setUint64(x)
        z.neg = false
        return z
}

// NewInt allocates and returns a new Int set to x.
func NewInt(x int64) *Int {
        return new(Int).SetInt64(x)
}

// Set sets z to x and returns z.
func (z *Int) Set(x *Int) *Int {
        if z != x {
                z.abs = z.abs.set(x.abs)
                z.neg = x.neg
        }
        return z
}

// Bits provides raw (unchecked but fast) access to x by returning its
// absolute value as a little-endian Word slice. The result and x share
// the same underlying array.
// Bits is intended to support implementation of missing low-level Int
// functionality outside this package; it should be avoided otherwise.
func (x *Int) Bits() []Word {
        return x.abs
}

// SetBits provides raw (unchecked but fast) access to z by setting its
// value to abs, interpreted as a little-endian Word slice, and returning
// z. The result and abs share the same underlying array.
// SetBits is intended to support implementation of missing low-level Int
// functionality outside this package; it should be avoided otherwise.
func (z *Int) SetBits(abs []Word) *Int {
        z.abs = nat(abs).norm()
        z.neg = false
        return z
}

// Abs sets z to |x| (the absolute value of x) and returns z.
func (z *Int) Abs(x *Int) *Int {
        z.Set(x)
        z.neg = false
        return z
}

// Neg sets z to -x and returns z.
func (z *Int) Neg(x *Int) *Int {
        z.Set(x)
        z.neg = len(z.abs) > 0 && !z.neg // 0 has no sign
        return z
}

// Add sets z to the sum x+y and returns z.
func (z *Int) Add(x, y *Int) *Int {
        neg := x.neg
        if x.neg == y.neg {
                // x + y == x + y
                // (-x) + (-y) == -(x + y)
                z.abs = z.abs.add(x.abs, y.abs)
        } else {
                // x + (-y) == x - y == -(y - x)
                // (-x) + y == y - x == -(x - y)
                if x.abs.cmp(y.abs) >= 0 {
                        z.abs = z.abs.sub(x.abs, y.abs)
                } else {
                        neg = !neg
                        z.abs = z.abs.sub(y.abs, x.abs)
                }
        }
        z.neg = len(z.abs) > 0 && neg // 0 has no sign
        return z
}

// Sub sets z to the difference x-y and returns z.
func (z *Int) Sub(x, y *Int) *Int {
        neg := x.neg
        if x.neg != y.neg {
                // x - (-y) == x + y
                // (-x) - y == -(x + y)
                z.abs = z.abs.add(x.abs, y.abs)
        } else {
                // x - y == x - y == -(y - x)
                // (-x) - (-y) == y - x == -(x - y)
                if x.abs.cmp(y.abs) >= 0 {
                        z.abs = z.abs.sub(x.abs, y.abs)
                } else {
                        neg = !neg
                        z.abs = z.abs.sub(y.abs, x.abs)
                }
        }
        z.neg = len(z.abs) > 0 && neg // 0 has no sign
        return z
}

// Mul sets z to the product x*y and returns z.
func (z *Int) Mul(x, y *Int) *Int {
        // x * y == x * y
        // x * (-y) == -(x * y)
        // (-x) * y == -(x * y)
        // (-x) * (-y) == x * y
        if x == y {
                z.abs = z.abs.sqr(x.abs)
                z.neg = false
                return z
        }
        z.abs = z.abs.mul(x.abs, y.abs)
        z.neg = len(z.abs) > 0 && x.neg != y.neg // 0 has no sign
        return z
}

// MulRange sets z to the product of all integers
// in the range [a, b] inclusively and returns z.
// If a > b (empty range), the result is 1.
func (z *Int) MulRange(a, b int64) *Int {
        switch {
        case a > b:
                return z.SetInt64(1) // empty range
        case a <= 0 && b >= 0:
                return z.SetInt64(0) // range includes 0
        }
        // a <= b && (b < 0 || a > 0)

        neg := false
        if a < 0 {
                neg = (b-a)&1 == 0
                a, b = -b, -a
        }

        z.abs = z.abs.mulRange(uint64(a), uint64(b))
        z.neg = neg
        return z
}

// Binomial sets z to the binomial coefficient of (n, k) and returns z.
func (z *Int) Binomial(n, k int64) *Int {
        // reduce the number of multiplications by reducing k
        if n/2 < k && k <= n {
                k = n - k // Binomial(n, k) == Binomial(n, n-k)
        }
        var a, b Int
        a.MulRange(n-k+1, n)
        b.MulRange(1, k)
        return z.Quo(&a, &b)
}

// Quo sets z to the quotient x/y for y != 0 and returns z.
// If y == 0, a division-by-zero run-time panic occurs.
// Quo implements truncated division (like Go); see QuoRem for more details.
func (z *Int) Quo(x, y *Int) *Int {
        z.abs, _ = z.abs.div(nil, x.abs, y.abs)
        z.neg = len(z.abs) > 0 && x.neg != y.neg // 0 has no sign
        return z
}

// Rem sets z to the remainder x%y for y != 0 and returns z.
// If y == 0, a division-by-zero run-time panic occurs.
// Rem implements truncated modulus (like Go); see QuoRem for more details.
func (z *Int) Rem(x, y *Int) *Int {
        _, z.abs = nat(nil).div(z.abs, x.abs, y.abs)
        z.neg = len(z.abs) > 0 && x.neg // 0 has no sign
        return z
}

// QuoRem sets z to the quotient x/y and r to the remainder x%y
// and returns the pair (z, r) for y != 0.
// If y == 0, a division-by-zero run-time panic occurs.
//
// QuoRem implements T-division and modulus (like Go):
//
//      q = x/y      with the result truncated to zero
//      r = x - y*q
//
// (See Daan Leijen, ``Division and Modulus for Computer Scientists''.)
// See DivMod for Euclidean division and modulus (unlike Go).
//
func (z *Int) QuoRem(x, y, r *Int) (*Int, *Int) {
        z.abs, r.abs = z.abs.div(r.abs, x.abs, y.abs)
        z.neg, r.neg = len(z.abs) > 0 && x.neg != y.neg, len(r.abs) > 0 && x.neg // 0 has no sign
        return z, r
}

// Div sets z to the quotient x/y for y != 0 and returns z.
// If y == 0, a division-by-zero run-time panic occurs.
// Div implements Euclidean division (unlike Go); see DivMod for more details.
func (z *Int) Div(x, y *Int) *Int {
        y_neg := y.neg // z may be an alias for y
        var r Int
        z.QuoRem(x, y, &r)
        if r.neg {
                if y_neg {
                        z.Add(z, intOne)
                } else {
                        z.Sub(z, intOne)
                }
        }
        return z
}

// Mod sets z to the modulus x%y for y != 0 and returns z.
// If y == 0, a division-by-zero run-time panic occurs.
// Mod implements Euclidean modulus (unlike Go); see DivMod for more details.
func (z *Int) Mod(x, y *Int) *Int {
        y0 := y // save y
        if z == y || alias(z.abs, y.abs) {
                y0 = new(Int).Set(y)
        }
        var q Int
        q.QuoRem(x, y, z)
        if z.neg {
                if y0.neg {
                        z.Sub(z, y0)
                } else {
                        z.Add(z, y0)
                }
        }
        return z
}

// DivMod sets z to the quotient x div y and m to the modulus x mod y
// and returns the pair (z, m) for y != 0.
// If y == 0, a division-by-zero run-time panic occurs.
//
// DivMod implements Euclidean division and modulus (unlike Go):
//
//      q = x div y  such that
//      m = x - y*q  with 0 <= m < |y|
//
// (See Raymond T. Boute, ``The Euclidean definition of the functions
// div and mod''. ACM Transactions on Programming Languages and
// Systems (TOPLAS), 14(2):127-144, New York, NY, USA, 4/1992.
// ACM press.)
// See QuoRem for T-division and modulus (like Go).
//
func (z *Int) DivMod(x, y, m *Int) (*Int, *Int) {
        y0 := y // save y
        if z == y || alias(z.abs, y.abs) {
                y0 = new(Int).Set(y)
        }
        z.QuoRem(x, y, m)
        if m.neg {
                if y0.neg {
                        z.Add(z, intOne)
                        m.Sub(m, y0)
                } else {
                        z.Sub(z, intOne)
                        m.Add(m, y0)
                }
        }
        return z, m
}

// Cmp compares x and y and returns:
//
//   -1 if x <  y
//    0 if x == y
//   +1 if x >  y
//
func (x *Int) Cmp(y *Int) (r int) {
        // x cmp y == x cmp y
        // x cmp (-y) == x
        // (-x) cmp y == y
        // (-x) cmp (-y) == -(x cmp y)
        switch {
        case x == y:
                // nothing to do
        case x.neg == y.neg:
                r = x.abs.cmp(y.abs)
                if x.neg {
                        r = -r
                }
        case x.neg:
                r = -1
        default:
                r = 1
        }
        return
}

// CmpAbs compares the absolute values of x and y and returns:
//
//   -1 if |x| <  |y|
//    0 if |x| == |y|
//   +1 if |x| >  |y|
//
func (x *Int) CmpAbs(y *Int) int {
        return x.abs.cmp(y.abs)
}

// low32 returns the least significant 32 bits of x.
func low32(x nat) uint32 {
        if len(x) == 0 {
                return 0
        }
        return uint32(x[0])
}

// low64 returns the least significant 64 bits of x.
func low64(x nat) uint64 {
        if len(x) == 0 {
                return 0
        }
        v := uint64(x[0])
        if _W == 32 && len(x) > 1 {
                return uint64(x[1])<<32 | v
        }
        return v
}

// Int64 returns the int64 representation of x.
// If x cannot be represented in an int64, the result is undefined.
func (x *Int) Int64() int64 {
        v := int64(low64(x.abs))
        if x.neg {
                v = -v
        }
        return v
}

// Uint64 returns the uint64 representation of x.
// If x cannot be represented in a uint64, the result is undefined.
func (x *Int) Uint64() uint64 {
        return low64(x.abs)
}

// IsInt64 reports whether x can be represented as an int64.
func (x *Int) IsInt64() bool {
        if len(x.abs) <= 64/_W {
                w := int64(low64(x.abs))
                return w >= 0 || x.neg && w == -w
        }
        return false
}

// IsUint64 reports whether x can be represented as a uint64.
func (x *Int) IsUint64() bool {
        return !x.neg && len(x.abs) <= 64/_W
}

// SetString sets z to the value of s, interpreted in the given base,
// and returns z and a boolean indicating success. The entire string
// (not just a prefix) must be valid for success. If SetString fails,
// the value of z is undefined but the returned value is nil.
//
// The base argument must be 0 or a value between 2 and MaxBase.
// For base 0, the number prefix determines the actual base: A prefix of
// ``0b'' or ``0B'' selects base 2, ``0'', ``0o'' or ``0O'' selects base 8,
// and ``0x'' or ``0X'' selects base 16. Otherwise, the selected base is 10
// and no prefix is accepted.
//
// For bases <= 36, lower and upper case letters are considered the same:
// The letters 'a' to 'z' and 'A' to 'Z' represent digit values 10 to 35.
// For bases > 36, the upper case letters 'A' to 'Z' represent the digit
// values 36 to 61.
//
// For base 0, an underscore character ``_'' may appear between a base
// prefix and an adjacent digit, and between successive digits; such
// underscores do not change the value of the number.
// Incorrect placement of underscores is reported as an error if there
// are no other errors. If base != 0, underscores are not recognized
// and act like any other character that is not a valid digit.
//
func (z *Int) SetString(s string, base int) (*Int, bool) {
        return z.setFromScanner(strings.NewReader(s), base)
}

// setFromScanner implements SetString given an io.ByteScanner.
// For documentation see comments of SetString.
func (z *Int) setFromScanner(r io.ByteScanner, base int) (*Int, bool) {
        if _, _, err := z.scan(r, base); err != nil {
                return nil, false
        }
        // entire content must have been consumed
        if _, err := r.ReadByte(); err != io.EOF {
                return nil, false
        }
        return z, true // err == io.EOF => scan consumed all content of r
}

// SetBytes interprets buf as the bytes of a big-endian unsigned
// integer, sets z to that value, and returns z.
func (z *Int) SetBytes(buf []byte) *Int {
        z.abs = z.abs.setBytes(buf)
        z.neg = false
        return z
}

// Bytes returns the absolute value of x as a big-endian byte slice.
//
// To use a fixed length slice, or a preallocated one, use FillBytes.
func (x *Int) Bytes() []byte {
        buf := make([]byte, len(x.abs)*_S)
        return buf[x.abs.bytes(buf):]
}

// FillBytes sets buf to the absolute value of x, storing it as a zero-extended
// big-endian byte slice, and returns buf.
//
// If the absolute value of x doesn't fit in buf, FillBytes will panic.
func (x *Int) FillBytes(buf []byte) []byte {
        // Clear whole buffer. (This gets optimized into a memclr.)
        for i := range buf {
                buf[i] = 0
        }
        x.abs.bytes(buf)
        return buf
}

// BitLen returns the length of the absolute value of x in bits.
// The bit length of 0 is 0.
func (x *Int) BitLen() int {
        return x.abs.bitLen()
}

// TrailingZeroBits returns the number of consecutive least significant zero
// bits of |x|.
func (x *Int) TrailingZeroBits() uint {
        return x.abs.trailingZeroBits()
}

// Exp sets z = x**y mod |m| (i.e. the sign of m is ignored), and returns z.
// If m == nil or m == 0, z = x**y unless y <= 0 then z = 1. If m != 0, y < 0,
// and x and m are not relatively prime, z is unchanged and nil is returned.
//
// Modular exponentiation of inputs of a particular size is not a
// cryptographically constant-time operation.
func (z *Int) Exp(x, y, m *Int) *Int {
        // See Knuth, volume 2, section 4.6.3.
        xWords := x.abs
        if y.neg {
                if m == nil || len(m.abs) == 0 {
                        return z.SetInt64(1)
                }
                // for y < 0: x**y mod m == (x**(-1))**|y| mod m
                inverse := new(Int).ModInverse(x, m)
                if inverse == nil {
                        return nil
                }
                xWords = inverse.abs
        }
        yWords := y.abs

        var mWords nat
        if m != nil {
                mWords = m.abs // m.abs may be nil for m == 0
        }

        z.abs = z.abs.expNN(xWords, yWords, mWords)
        z.neg = len(z.abs) > 0 && x.neg && len(yWords) > 0 && yWords[0]&1 == 1 // 0 has no sign
        if z.neg && len(mWords) > 0 {
                // make modulus result positive
                z.abs = z.abs.sub(mWords, z.abs) // z == x**y mod |m| && 0 <= z < |m|
                z.neg = false
        }

        return z
}

// GCD sets z to the greatest common divisor of a and b and returns z.
// If x or y are not nil, GCD sets their value such that z = a*x + b*y.
//
// a and b may be positive, zero or negative. (Before Go 1.14 both had
// to be > 0.) Regardless of the signs of a and b, z is always >= 0.
//
// If a == b == 0, GCD sets z = x = y = 0.
//
// If a == 0 and b != 0, GCD sets z = |b|, x = 0, y = sign(b) * 1.
//
// If a != 0 and b == 0, GCD sets z = |a|, x = sign(a) * 1, y = 0.
func (z *Int) GCD(x, y, a, b *Int) *Int {
        if len(a.abs) == 0 || len(b.abs) == 0 {
                lenA, lenB, negA, negB := len(a.abs), len(b.abs), a.neg, b.neg
                if lenA == 0 {
                        z.Set(b)
                } else {
                        z.Set(a)
                }
                z.neg = false
                if x != nil {
                        if lenA == 0 {
                                x.SetUint64(0)
                        } else {
                                x.SetUint64(1)
                                x.neg = negA
                        }
                }
                if y != nil {
                        if lenB == 0 {
                                y.SetUint64(0)
                        } else {
                                y.SetUint64(1)
                                y.neg = negB
                        }
                }
                return z
        }

        return z.lehmerGCD(x, y, a, b)
}

// lehmerSimulate attempts to simulate several Euclidean update steps
// using the leading digits of A and B.  It returns u0, u1, v0, v1
// such that A and B can be updated as:
//              A = u0*A + v0*B
//              B = u1*A + v1*B
// Requirements: A >= B and len(B.abs) >= 2
// Since we are calculating with full words to avoid overflow,
// we use 'even' to track the sign of the cosequences.
// For even iterations: u0, v1 >= 0 && u1, v0 <= 0
// For odd  iterations: u0, v1 <= 0 && u1, v0 >= 0
func lehmerSimulate(A, B *Int) (u0, u1, v0, v1 Word, even bool) {
        // initialize the digits
        var a1, a2, u2, v2 Word

        m := len(B.abs) // m >= 2
        n := len(A.abs) // n >= m >= 2

        // extract the top Word of bits from A and B
        h := nlz(A.abs[n-1])
        a1 = A.abs[n-1]<<h | A.abs[n-2]>>(_W-h)
        // B may have implicit zero words in the high bits if the lengths differ
        switch {
        case n == m:
                a2 = B.abs[n-1]<<h | B.abs[n-2]>>(_W-h)
        case n == m+1:
                a2 = B.abs[n-2] >> (_W - h)
        default:
                a2 = 0
        }

        // Since we are calculating with full words to avoid overflow,
        // we use 'even' to track the sign of the cosequences.
        // For even iterations: u0, v1 >= 0 && u1, v0 <= 0
        // For odd  iterations: u0, v1 <= 0 && u1, v0 >= 0
        // The first iteration starts with k=1 (odd).
        even = false
        // variables to track the cosequences
        u0, u1, u2 = 0, 1, 0
        v0, v1, v2 = 0, 0, 1

        // Calculate the quotient and cosequences using Collins' stopping condition.
        // Note that overflow of a Word is not possible when computing the remainder
        // sequence and cosequences since the cosequence size is bounded by the input size.
        // See section 4.2 of Jebelean for details.
        for a2 >= v2 && a1-a2 >= v1+v2 {
                q, r := a1/a2, a1%a2
                a1, a2 = a2, r
                u0, u1, u2 = u1, u2, u1+q*u2
                v0, v1, v2 = v1, v2, v1+q*v2
                even = !even
        }
        return
}

// lehmerUpdate updates the inputs A and B such that:
//              A = u0*A + v0*B
//              B = u1*A + v1*B
// where the signs of u0, u1, v0, v1 are given by even
// For even == true: u0, v1 >= 0 && u1, v0 <= 0
// For even == false: u0, v1 <= 0 && u1, v0 >= 0
// q, r, s, t are temporary variables to avoid allocations in the multiplication
func lehmerUpdate(A, B, q, r, s, t *Int, u0, u1, v0, v1 Word, even bool) {

        t.abs = t.abs.setWord(u0)
        s.abs = s.abs.setWord(v0)
        t.neg = !even
        s.neg = even

        t.Mul(A, t)
        s.Mul(B, s)

        r.abs = r.abs.setWord(u1)
        q.abs = q.abs.setWord(v1)
        r.neg = even
        q.neg = !even

        r.Mul(A, r)
        q.Mul(B, q)

        A.Add(t, s)
        B.Add(r, q)
}

// euclidUpdate performs a single step of the Euclidean GCD algorithm
// if extended is true, it also updates the cosequence Ua, Ub
func euclidUpdate(A, B, Ua, Ub, q, r, s, t *Int, extended bool) {
        q, r = q.QuoRem(A, B, r)

        *A, *B, *r = *B, *r, *A

        if extended {
                // Ua, Ub = Ub, Ua - q*Ub
                t.Set(Ub)
                s.Mul(Ub, q)
                Ub.Sub(Ua, s)
                Ua.Set(t)
        }
}

// lehmerGCD sets z to the greatest common divisor of a and b,
// which both must be != 0, and returns z.
// If x or y are not nil, their values are set such that z = a*x + b*y.
// See Knuth, The Art of Computer Programming, Vol. 2, Section 4.5.2, Algorithm L.
// This implementation uses the improved condition by Collins requiring only one
// quotient and avoiding the possibility of single Word overflow.
// See Jebelean, "Improving the multiprecision Euclidean algorithm",
// Design and Implementation of Symbolic Computation Systems, pp 45-58.
// The cosequences are updated according to Algorithm 10.45 from
// Cohen et al. "Handbook of Elliptic and Hyperelliptic Curve Cryptography" pp 192.
func (z *Int) lehmerGCD(x, y, a, b *Int) *Int {
        var A, B, Ua, Ub *Int

        A = new(Int).Abs(a)
        B = new(Int).Abs(b)

        extended := x != nil || y != nil

        if extended {
                // Ua (Ub) tracks how many times input a has been accumulated into A (B).
                Ua = new(Int).SetInt64(1)
                Ub = new(Int)
        }

        // temp variables for multiprecision update
        q := new(Int)
        r := new(Int)
        s := new(Int)
        t := new(Int)

        // ensure A >= B
        if A.abs.cmp(B.abs) < 0 {
                A, B = B, A
                Ub, Ua = Ua, Ub
        }

        // loop invariant A >= B
        for len(B.abs) > 1 {
                // Attempt to calculate in single-precision using leading words of A and B.
                u0, u1, v0, v1, even := lehmerSimulate(A, B)

                // multiprecision Step
                if v0 != 0 {
                        // Simulate the effect of the single-precision steps using the cosequences.
                        // A = u0*A + v0*B
                        // B = u1*A + v1*B
                        lehmerUpdate(A, B, q, r, s, t, u0, u1, v0, v1, even)

                        if extended {
                                // Ua = u0*Ua + v0*Ub
                                // Ub = u1*Ua + v1*Ub
                                lehmerUpdate(Ua, Ub, q, r, s, t, u0, u1, v0, v1, even)
                        }

                } else {
                        // Single-digit calculations failed to simulate any quotients.
                        // Do a standard Euclidean step.
                        euclidUpdate(A, B, Ua, Ub, q, r, s, t, extended)
                }
        }

        if len(B.abs) > 0 {
                // extended Euclidean algorithm base case if B is a single Word
                if len(A.abs) > 1 {
                        // A is longer than a single Word, so one update is needed.
                        euclidUpdate(A, B, Ua, Ub, q, r, s, t, extended)
                }
                if len(B.abs) > 0 {
                        // A and B are both a single Word.
                        aWord, bWord := A.abs[0], B.abs[0]
                        if extended {
                                var ua, ub, va, vb Word
                                ua, ub = 1, 0
                                va, vb = 0, 1
                                even := true
                                for bWord != 0 {
                                        q, r := aWord/bWord, aWord%bWord
                                        aWord, bWord = bWord, r
                                        ua, ub = ub, ua+q*ub
                                        va, vb = vb, va+q*vb
                                        even = !even
                                }

                                t.abs = t.abs.setWord(ua)
                                s.abs = s.abs.setWord(va)
                                t.neg = !even
                                s.neg = even

                                t.Mul(Ua, t)
                                s.Mul(Ub, s)

                                Ua.Add(t, s)
                        } else {
                                for bWord != 0 {
                                        aWord, bWord = bWord, aWord%bWord
                                }
                        }
                        A.abs[0] = aWord
                }
        }
        negA := a.neg
        if y != nil {
                // avoid aliasing b needed in the division below
                if y == b {
                        B.Set(b)
                } else {
                        B = b
                }
                // y = (z - a*x)/b
                y.Mul(a, Ua) // y can safely alias a
                if negA {
                        y.neg = !y.neg
                }
                y.Sub(A, y)
                y.Div(y, B)
        }

        if x != nil {
                *x = *Ua
                if negA {
                        x.neg = !x.neg
                }
        }

        *z = *A

        return z
}

// Rand sets z to a pseudo-random number in [0, n) and returns z.
//
// As this uses the math/rand package, it must not be used for
// security-sensitive work. Use crypto/rand.Int instead.
func (z *Int) Rand(rnd *rand.Rand, n *Int) *Int {
        z.neg = false
        if n.neg || len(n.abs) == 0 {
                z.abs = nil
                return z
        }
        z.abs = z.abs.random(rnd, n.abs, n.abs.bitLen())
        return z
}

// ModInverse sets z to the multiplicative inverse of g in the ring ℤ/nℤ
// and returns z. If g and n are not relatively prime, g has no multiplicative
// inverse in the ring ℤ/nℤ.  In this case, z is unchanged and the return value
// is nil.
func (z *Int) ModInverse(g, n *Int) *Int {
        // GCD expects parameters a and b to be > 0.
        if n.neg {
                var n2 Int
                n = n2.Neg(n)
        }
        if g.neg {
                var g2 Int
                g = g2.Mod(g, n)
        }
        var d, x Int
        d.GCD(&x, nil, g, n)

        // if and only if d==1, g and n are relatively prime
        if d.Cmp(intOne) != 0 {
                return nil
        }

        // x and y are such that g*x + n*y = 1, therefore x is the inverse element,
        // but it may be negative, so convert to the range 0 <= z < |n|
        if x.neg {
                z.Add(&x, n)
        } else {
                z.Set(&x)
        }
        return z
}

// Jacobi returns the Jacobi symbol (x/y), either +1, -1, or 0.
// The y argument must be an odd integer.
func Jacobi(x, y *Int) int {
        if len(y.abs) == 0 || y.abs[0]&1 == 0 {
                panic(fmt.Sprintf("big: invalid 2nd argument to Int.Jacobi: need odd integer but got %s", y))
        }

        // We use the formulation described in chapter 2, section 2.4,
        // "The Yacas Book of Algorithms":
        // http://yacas.sourceforge.net/Algo.book.pdf

        var a, b, c Int
        a.Set(x)
        b.Set(y)
        j := 1

        if b.neg {
                if a.neg {
                        j = -1
                }
                b.neg = false
        }

        for {
                if b.Cmp(intOne) == 0 {
                        return j
                }
                if len(a.abs) == 0 {
                        return 0
                }
                a.Mod(&a, &b)
                if len(a.abs) == 0 {
                        return 0
                }
                // a > 0

                // handle factors of 2 in 'a'
                s := a.abs.trailingZeroBits()
                if s&1 != 0 {
                        bmod8 := b.abs[0] & 7
                        if bmod8 == 3 || bmod8 == 5 {
                                j = -j
                        }
                }
                c.Rsh(&a, s) // a = 2^s*c

                // swap numerator and denominator
                if b.abs[0]&3 == 3 && c.abs[0]&3 == 3 {
                        j = -j
                }
                a.Set(&b)
                b.Set(&c)
        }
}

// modSqrt3Mod4 uses the identity
//      (a^((p+1)/4))^2  mod p
//   == u^(p+1)          mod p
//   == u^2              mod p
// to calculate the square root of any quadratic residue mod p quickly for 3
// mod 4 primes.
func (z *Int) modSqrt3Mod4Prime(x, p *Int) *Int {
        e := new(Int).Add(p, intOne) // e = p + 1
        e.Rsh(e, 2)                  // e = (p + 1) / 4
        z.Exp(x, e, p)               // z = x^e mod p
        return z
}

// modSqrt5Mod8 uses Atkin's observation that 2 is not a square mod p
//   alpha ==  (2*a)^((p-5)/8)    mod p
//   beta  ==  2*a*alpha^2        mod p  is a square root of -1
//   b     ==  a*alpha*(beta-1)   mod p  is a square root of a
// to calculate the square root of any quadratic residue mod p quickly for 5
// mod 8 primes.
func (z *Int) modSqrt5Mod8Prime(x, p *Int) *Int {
        // p == 5 mod 8 implies p = e*8 + 5
        // e is the quotient and 5 the remainder on division by 8
        e := new(Int).Rsh(p, 3)  // e = (p - 5) / 8
        tx := new(Int).Lsh(x, 1) // tx = 2*x
        alpha := new(Int).Exp(tx, e, p)
        beta := new(Int).Mul(alpha, alpha)
        beta.Mod(beta, p)
        beta.Mul(beta, tx)
        beta.Mod(beta, p)
        beta.Sub(beta, intOne)
        beta.Mul(beta, x)
        beta.Mod(beta, p)
        beta.Mul(beta, alpha)
        z.Mod(beta, p)
        return z
}

// modSqrtTonelliShanks uses the Tonelli-Shanks algorithm to find the square
// root of a quadratic residue modulo any prime.
func (z *Int) modSqrtTonelliShanks(x, p *Int) *Int {
        // Break p-1 into s*2^e such that s is odd.
        var s Int
        s.Sub(p, intOne)
        e := s.abs.trailingZeroBits()
        s.Rsh(&s, e)

        // find some non-square n
        var n Int
        n.SetInt64(2)
        for Jacobi(&n, p) != -1 {
                n.Add(&n, intOne)
        }

        // Core of the Tonelli-Shanks algorithm. Follows the description in
        // section 6 of "Square roots from 1; 24, 51, 10 to Dan Shanks" by Ezra
        // Brown:
        // https://www.maa.org/sites/default/files/pdf/upload_library/22/Polya/07468342.di020786.02p0470a.pdf
        var y, b, g, t Int
        y.Add(&s, intOne)
        y.Rsh(&y, 1)
        y.Exp(x, &y, p)  // y = x^((s+1)/2)
        b.Exp(x, &s, p)  // b = x^s
        g.Exp(&n, &s, p) // g = n^s
        r := e
        for {
                // find the least m such that ord_p(b) = 2^m
                var m uint
                t.Set(&b)
                for t.Cmp(intOne) != 0 {
                        t.Mul(&t, &t).Mod(&t, p)
                        m++
                }

                if m == 0 {
                        return z.Set(&y)
                }

                t.SetInt64(0).SetBit(&t, int(r-m-1), 1).Exp(&g, &t, p)
                // t = g^(2^(r-m-1)) mod p
                g.Mul(&t, &t).Mod(&g, p) // g = g^(2^(r-m)) mod p
                y.Mul(&y, &t).Mod(&y, p)
                b.Mul(&b, &g).Mod(&b, p)
                r = m
        }
}

// ModSqrt sets z to a square root of x mod p if such a square root exists, and
// returns z. The modulus p must be an odd prime. If x is not a square mod p,
// ModSqrt leaves z unchanged and returns nil. This function panics if p is
// not an odd integer.
func (z *Int) ModSqrt(x, p *Int) *Int {
        switch Jacobi(x, p) {
        case -1:
                return nil // x is not a square mod p
        case 0:
                return z.SetInt64(0) // sqrt(0) mod p = 0
        case 1:
                break
        }
        if x.neg || x.Cmp(p) >= 0 { // ensure 0 <= x < p
                x = new(Int).Mod(x, p)
        }

        switch {
        case p.abs[0]%4 == 3:
                // Check whether p is 3 mod 4, and if so, use the faster algorithm.
                return z.modSqrt3Mod4Prime(x, p)
        case p.abs[0]%8 == 5:
                // Check whether p is 5 mod 8, use Atkin's algorithm.
                return z.modSqrt5Mod8Prime(x, p)
        default:
                // Otherwise, use Tonelli-Shanks.
                return z.modSqrtTonelliShanks(x, p)
        }
}

// Lsh sets z = x << n and returns z.
func (z *Int) Lsh(x *Int, n uint) *Int {
        z.abs = z.abs.shl(x.abs, n)
        z.neg = x.neg
        return z
}

// Rsh sets z = x >> n and returns z.
func (z *Int) Rsh(x *Int, n uint) *Int {
        if x.neg {
                // (-x) >> s == ^(x-1) >> s == ^((x-1) >> s) == -(((x-1) >> s) + 1)
                t := z.abs.sub(x.abs, natOne) // no underflow because |x| > 0
                t = t.shr(t, n)
                z.abs = t.add(t, natOne)
                z.neg = true // z cannot be zero if x is negative
                return z
        }

        z.abs = z.abs.shr(x.abs, n)
        z.neg = false
        return z
}

// Bit returns the value of the i'th bit of x. That is, it
// returns (x>>i)&1. The bit index i must be >= 0.
func (x *Int) Bit(i int) uint {
        if i == 0 {
                // optimization for common case: odd/even test of x
                if len(x.abs) > 0 {
                        return uint(x.abs[0] & 1) // bit 0 is same for -x
                }
                return 0
        }
        if i < 0 {
                panic("negative bit index")
        }
        if x.neg {
                t := nat(nil).sub(x.abs, natOne)
                return t.bit(uint(i)) ^ 1
        }

        return x.abs.bit(uint(i))
}

// SetBit sets z to x, with x's i'th bit set to b (0 or 1).
// That is, if b is 1 SetBit sets z = x | (1 << i);
// if b is 0 SetBit sets z = x &^ (1 << i). If b is not 0 or 1,
// SetBit will panic.
func (z *Int) SetBit(x *Int, i int, b uint) *Int {
        if i < 0 {
                panic("negative bit index")
        }
        if x.neg {
                t := z.abs.sub(x.abs, natOne)
                t = t.setBit(t, uint(i), b^1)
                z.abs = t.add(t, natOne)
                z.neg = len(z.abs) > 0
                return z
        }
        z.abs = z.abs.setBit(x.abs, uint(i), b)
        z.neg = false
        return z
}

// And sets z = x & y and returns z.
func (z *Int) And(x, y *Int) *Int {
        if x.neg == y.neg {
                if x.neg {
                        // (-x) & (-y) == ^(x-1) & ^(y-1) == ^((x-1) | (y-1)) == -(((x-1) | (y-1)) + 1)
                        x1 := nat(nil).sub(x.abs, natOne)
                        y1 := nat(nil).sub(y.abs, natOne)
                        z.abs = z.abs.add(z.abs.or(x1, y1), natOne)
                        z.neg = true // z cannot be zero if x and y are negative
                        return z
                }

                // x & y == x & y
                z.abs = z.abs.and(x.abs, y.abs)
                z.neg = false
                return z
        }

        // x.neg != y.neg
        if x.neg {
                x, y = y, x // & is symmetric
        }

        // x & (-y) == x & ^(y-1) == x &^ (y-1)
        y1 := nat(nil).sub(y.abs, natOne)
        z.abs = z.abs.andNot(x.abs, y1)
        z.neg = false
        return z
}

// AndNot sets z = x &^ y and returns z.
func (z *Int) AndNot(x, y *Int) *Int {
        if x.neg == y.neg {
                if x.neg {
                        // (-x) &^ (-y) == ^(x-1) &^ ^(y-1) == ^(x-1) & (y-1) == (y-1) &^ (x-1)
                        x1 := nat(nil).sub(x.abs, natOne)
                        y1 := nat(nil).sub(y.abs, natOne)
                        z.abs = z.abs.andNot(y1, x1)
                        z.neg = false
                        return z
                }

                // x &^ y == x &^ y
                z.abs = z.abs.andNot(x.abs, y.abs)
                z.neg = false
                return z
        }

        if x.neg {
                // (-x) &^ y == ^(x-1) &^ y == ^(x-1) & ^y == ^((x-1) | y) == -(((x-1) | y) + 1)
                x1 := nat(nil).sub(x.abs, natOne)
                z.abs = z.abs.add(z.abs.or(x1, y.abs), natOne)
                z.neg = true // z cannot be zero if x is negative and y is positive
                return z
        }

        // x &^ (-y) == x &^ ^(y-1) == x & (y-1)
        y1 := nat(nil).sub(y.abs, natOne)
        z.abs = z.abs.and(x.abs, y1)
        z.neg = false
        return z
}

// Or sets z = x | y and returns z.
func (z *Int) Or(x, y *Int) *Int {
        if x.neg == y.neg {
                if x.neg {
                        // (-x) | (-y) == ^(x-1) | ^(y-1) == ^((x-1) & (y-1)) == -(((x-1) & (y-1)) + 1)
                        x1 := nat(nil).sub(x.abs, natOne)
                        y1 := nat(nil).sub(y.abs, natOne)
                        z.abs = z.abs.add(z.abs.and(x1, y1), natOne)
                        z.neg = true // z cannot be zero if x and y are negative
                        return z
                }

                // x | y == x | y
                z.abs = z.abs.or(x.abs, y.abs)
                z.neg = false
                return z
        }

        // x.neg != y.neg
        if x.neg {
                x, y = y, x // | is symmetric
        }

        // x | (-y) == x | ^(y-1) == ^((y-1) &^ x) == -(^((y-1) &^ x) + 1)
        y1 := nat(nil).sub(y.abs, natOne)
        z.abs = z.abs.add(z.abs.andNot(y1, x.abs), natOne)
        z.neg = true // z cannot be zero if one of x or y is negative
        return z
}

// Xor sets z = x ^ y and returns z.
func (z *Int) Xor(x, y *Int) *Int {
        if x.neg == y.neg {
                if x.neg {
                        // (-x) ^ (-y) == ^(x-1) ^ ^(y-1) == (x-1) ^ (y-1)
                        x1 := nat(nil).sub(x.abs, natOne)
                        y1 := nat(nil).sub(y.abs, natOne)
                        z.abs = z.abs.xor(x1, y1)
                        z.neg = false
                        return z
                }

                // x ^ y == x ^ y
                z.abs = z.abs.xor(x.abs, y.abs)
                z.neg = false
                return z
        }

        // x.neg != y.neg
        if x.neg {
                x, y = y, x // ^ is symmetric
        }

        // x ^ (-y) == x ^ ^(y-1) == ^(x ^ (y-1)) == -((x ^ (y-1)) + 1)
        y1 := nat(nil).sub(y.abs, natOne)
        z.abs = z.abs.add(z.abs.xor(x.abs, y1), natOne)
        z.neg = true // z cannot be zero if only one of x or y is negative
        return z
}

// Not sets z = ^x and returns z.
func (z *Int) Not(x *Int) *Int {
        if x.neg {
                // ^(-x) == ^(^(x-1)) == x-1
                z.abs = z.abs.sub(x.abs, natOne)
                z.neg = false
                return z
        }

        // ^x == -x-1 == -(x+1)
        z.abs = z.abs.add(x.abs, natOne)
        z.neg = true // z cannot be zero if x is positive
        return z
}

// Sqrt sets z to ⌊√x⌋, the largest integer such that z² ≤ x, and returns z.
// It panics if x is negative.
func (z *Int) Sqrt(x *Int) *Int {
        if x.neg {
                panic("square root of negative number")
        }
        z.neg = false
        z.abs = z.abs.sqrt(x.abs)
        return z
}