993 lines
24 KiB
Go
993 lines
24 KiB
Go
// Copyright 2009 The Go Authors. All rights reserved.
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// Use of this source code is governed by a BSD-style
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// license that can be found in the LICENSE file.
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// This file implements signed multi-precision integers.
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package big
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import (
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"errors"
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"fmt"
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"io"
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"math/rand"
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"strings"
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)
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// An Int represents a signed multi-precision integer.
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// The zero value for an Int represents the value 0.
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type Int struct {
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neg bool // sign
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abs nat // absolute value of the integer
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}
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var intOne = &Int{false, natOne}
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// Sign returns:
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//
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// -1 if x < 0
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// 0 if x == 0
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// +1 if x > 0
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//
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func (x *Int) Sign() int {
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if len(x.abs) == 0 {
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return 0
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}
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if x.neg {
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return -1
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}
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return 1
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}
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// SetInt64 sets z to x and returns z.
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func (z *Int) SetInt64(x int64) *Int {
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neg := false
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if x < 0 {
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neg = true
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x = -x
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}
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z.abs = z.abs.setUint64(uint64(x))
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z.neg = neg
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return z
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}
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// SetUint64 sets z to x and returns z.
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func (z *Int) SetUint64(x uint64) *Int {
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z.abs = z.abs.setUint64(x)
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z.neg = false
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return z
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}
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// NewInt allocates and returns a new Int set to x.
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func NewInt(x int64) *Int {
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return new(Int).SetInt64(x)
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}
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// Set sets z to x and returns z.
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func (z *Int) Set(x *Int) *Int {
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if z != x {
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z.abs = z.abs.set(x.abs)
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z.neg = x.neg
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}
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return z
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}
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// Bits provides raw (unchecked but fast) access to x by returning its
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// absolute value as a little-endian Word slice. The result and x share
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// the same underlying array.
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// Bits is intended to support implementation of missing low-level Int
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// functionality outside this package; it should be avoided otherwise.
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func (x *Int) Bits() []Word {
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return x.abs
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}
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// SetBits provides raw (unchecked but fast) access to z by setting its
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// value to abs, interpreted as a little-endian Word slice, and returning
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// z. The result and abs share the same underlying array.
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// SetBits is intended to support implementation of missing low-level Int
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// functionality outside this package; it should be avoided otherwise.
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func (z *Int) SetBits(abs []Word) *Int {
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z.abs = nat(abs).norm()
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z.neg = false
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return z
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}
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// Abs sets z to |x| (the absolute value of x) and returns z.
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func (z *Int) Abs(x *Int) *Int {
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z.Set(x)
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z.neg = false
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return z
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}
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// Neg sets z to -x and returns z.
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func (z *Int) Neg(x *Int) *Int {
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z.Set(x)
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z.neg = len(z.abs) > 0 && !z.neg // 0 has no sign
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return z
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}
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// Add sets z to the sum x+y and returns z.
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func (z *Int) Add(x, y *Int) *Int {
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neg := x.neg
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if x.neg == y.neg {
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// x + y == x + y
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// (-x) + (-y) == -(x + y)
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z.abs = z.abs.add(x.abs, y.abs)
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} else {
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// x + (-y) == x - y == -(y - x)
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// (-x) + y == y - x == -(x - y)
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if x.abs.cmp(y.abs) >= 0 {
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z.abs = z.abs.sub(x.abs, y.abs)
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} else {
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neg = !neg
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z.abs = z.abs.sub(y.abs, x.abs)
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}
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}
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z.neg = len(z.abs) > 0 && neg // 0 has no sign
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return z
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}
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// Sub sets z to the difference x-y and returns z.
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func (z *Int) Sub(x, y *Int) *Int {
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neg := x.neg
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if x.neg != y.neg {
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// x - (-y) == x + y
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// (-x) - y == -(x + y)
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z.abs = z.abs.add(x.abs, y.abs)
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} else {
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// x - y == x - y == -(y - x)
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// (-x) - (-y) == y - x == -(x - y)
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if x.abs.cmp(y.abs) >= 0 {
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z.abs = z.abs.sub(x.abs, y.abs)
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} else {
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neg = !neg
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z.abs = z.abs.sub(y.abs, x.abs)
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}
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}
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z.neg = len(z.abs) > 0 && neg // 0 has no sign
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return z
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}
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// Mul sets z to the product x*y and returns z.
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func (z *Int) Mul(x, y *Int) *Int {
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// x * y == x * y
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// x * (-y) == -(x * y)
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// (-x) * y == -(x * y)
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// (-x) * (-y) == x * y
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z.abs = z.abs.mul(x.abs, y.abs)
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z.neg = len(z.abs) > 0 && x.neg != y.neg // 0 has no sign
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return z
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}
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// MulRange sets z to the product of all integers
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// in the range [a, b] inclusively and returns z.
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// If a > b (empty range), the result is 1.
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func (z *Int) MulRange(a, b int64) *Int {
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switch {
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case a > b:
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return z.SetInt64(1) // empty range
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case a <= 0 && b >= 0:
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return z.SetInt64(0) // range includes 0
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}
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// a <= b && (b < 0 || a > 0)
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neg := false
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if a < 0 {
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neg = (b-a)&1 == 0
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a, b = -b, -a
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}
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z.abs = z.abs.mulRange(uint64(a), uint64(b))
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z.neg = neg
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return z
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}
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// Binomial sets z to the binomial coefficient of (n, k) and returns z.
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func (z *Int) Binomial(n, k int64) *Int {
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var a, b Int
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a.MulRange(n-k+1, n)
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b.MulRange(1, k)
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return z.Quo(&a, &b)
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}
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// Quo sets z to the quotient x/y for y != 0 and returns z.
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// If y == 0, a division-by-zero run-time panic occurs.
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// Quo implements truncated division (like Go); see QuoRem for more details.
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func (z *Int) Quo(x, y *Int) *Int {
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z.abs, _ = z.abs.div(nil, x.abs, y.abs)
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z.neg = len(z.abs) > 0 && x.neg != y.neg // 0 has no sign
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return z
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}
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// Rem sets z to the remainder x%y for y != 0 and returns z.
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// If y == 0, a division-by-zero run-time panic occurs.
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// Rem implements truncated modulus (like Go); see QuoRem for more details.
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func (z *Int) Rem(x, y *Int) *Int {
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_, z.abs = nat(nil).div(z.abs, x.abs, y.abs)
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z.neg = len(z.abs) > 0 && x.neg // 0 has no sign
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return z
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}
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// QuoRem sets z to the quotient x/y and r to the remainder x%y
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// and returns the pair (z, r) for y != 0.
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// If y == 0, a division-by-zero run-time panic occurs.
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//
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// QuoRem implements T-division and modulus (like Go):
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//
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// q = x/y with the result truncated to zero
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// r = x - y*q
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//
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// (See Daan Leijen, ``Division and Modulus for Computer Scientists''.)
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// See DivMod for Euclidean division and modulus (unlike Go).
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//
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func (z *Int) QuoRem(x, y, r *Int) (*Int, *Int) {
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z.abs, r.abs = z.abs.div(r.abs, x.abs, y.abs)
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z.neg, r.neg = len(z.abs) > 0 && x.neg != y.neg, len(r.abs) > 0 && x.neg // 0 has no sign
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return z, r
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}
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// Div sets z to the quotient x/y for y != 0 and returns z.
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// If y == 0, a division-by-zero run-time panic occurs.
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// Div implements Euclidean division (unlike Go); see DivMod for more details.
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func (z *Int) Div(x, y *Int) *Int {
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y_neg := y.neg // z may be an alias for y
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var r Int
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z.QuoRem(x, y, &r)
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if r.neg {
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if y_neg {
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z.Add(z, intOne)
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} else {
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z.Sub(z, intOne)
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}
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}
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return z
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}
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// Mod sets z to the modulus x%y for y != 0 and returns z.
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// If y == 0, a division-by-zero run-time panic occurs.
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// Mod implements Euclidean modulus (unlike Go); see DivMod for more details.
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func (z *Int) Mod(x, y *Int) *Int {
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y0 := y // save y
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if z == y || alias(z.abs, y.abs) {
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y0 = new(Int).Set(y)
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}
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var q Int
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q.QuoRem(x, y, z)
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if z.neg {
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if y0.neg {
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z.Sub(z, y0)
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} else {
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z.Add(z, y0)
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}
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}
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return z
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}
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// DivMod sets z to the quotient x div y and m to the modulus x mod y
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// and returns the pair (z, m) for y != 0.
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// If y == 0, a division-by-zero run-time panic occurs.
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//
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// DivMod implements Euclidean division and modulus (unlike Go):
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//
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// q = x div y such that
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// m = x - y*q with 0 <= m < |q|
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//
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// (See Raymond T. Boute, ``The Euclidean definition of the functions
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// div and mod''. ACM Transactions on Programming Languages and
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// Systems (TOPLAS), 14(2):127-144, New York, NY, USA, 4/1992.
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// ACM press.)
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// See QuoRem for T-division and modulus (like Go).
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//
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func (z *Int) DivMod(x, y, m *Int) (*Int, *Int) {
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y0 := y // save y
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if z == y || alias(z.abs, y.abs) {
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y0 = new(Int).Set(y)
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}
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z.QuoRem(x, y, m)
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if m.neg {
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if y0.neg {
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z.Add(z, intOne)
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m.Sub(m, y0)
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} else {
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z.Sub(z, intOne)
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m.Add(m, y0)
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}
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}
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return z, m
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}
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// Cmp compares x and y and returns:
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//
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// -1 if x < y
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// 0 if x == y
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// +1 if x > y
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//
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func (x *Int) Cmp(y *Int) (r int) {
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// x cmp y == x cmp y
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// x cmp (-y) == x
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// (-x) cmp y == y
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// (-x) cmp (-y) == -(x cmp y)
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switch {
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case x.neg == y.neg:
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r = x.abs.cmp(y.abs)
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if x.neg {
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r = -r
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}
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case x.neg:
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r = -1
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default:
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r = 1
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}
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return
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}
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func (x *Int) String() string {
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switch {
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case x == nil:
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return "<nil>"
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case x.neg:
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return "-" + x.abs.decimalString()
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}
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return x.abs.decimalString()
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}
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func charset(ch rune) string {
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switch ch {
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case 'b':
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return lowercaseDigits[0:2]
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case 'o':
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return lowercaseDigits[0:8]
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case 'd', 's', 'v':
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return lowercaseDigits[0:10]
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case 'x':
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return lowercaseDigits[0:16]
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case 'X':
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return uppercaseDigits[0:16]
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}
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return "" // unknown format
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}
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// write count copies of text to s
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func writeMultiple(s fmt.State, text string, count int) {
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if len(text) > 0 {
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b := []byte(text)
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for ; count > 0; count-- {
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s.Write(b)
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}
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}
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}
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// Format is a support routine for fmt.Formatter. It accepts
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// the formats 'b' (binary), 'o' (octal), 'd' (decimal), 'x'
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// (lowercase hexadecimal), and 'X' (uppercase hexadecimal).
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// Also supported are the full suite of package fmt's format
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// verbs for integral types, including '+', '-', and ' '
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// for sign control, '#' for leading zero in octal and for
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// hexadecimal, a leading "0x" or "0X" for "%#x" and "%#X"
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// respectively, specification of minimum digits precision,
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// output field width, space or zero padding, and left or
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// right justification.
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//
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func (x *Int) Format(s fmt.State, ch rune) {
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cs := charset(ch)
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// special cases
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switch {
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case cs == "":
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// unknown format
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fmt.Fprintf(s, "%%!%c(big.Int=%s)", ch, x.String())
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return
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case x == nil:
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fmt.Fprint(s, "<nil>")
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return
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}
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// determine sign character
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sign := ""
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switch {
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case x.neg:
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sign = "-"
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case s.Flag('+'): // supersedes ' ' when both specified
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sign = "+"
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case s.Flag(' '):
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sign = " "
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}
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// determine prefix characters for indicating output base
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prefix := ""
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if s.Flag('#') {
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switch ch {
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case 'o': // octal
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prefix = "0"
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case 'x': // hexadecimal
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prefix = "0x"
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case 'X':
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prefix = "0X"
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}
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}
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// determine digits with base set by len(cs) and digit characters from cs
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digits := x.abs.string(cs)
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// number of characters for the three classes of number padding
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var left int // space characters to left of digits for right justification ("%8d")
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var zeroes int // zero characters (actually cs[0]) as left-most digits ("%.8d")
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var right int // space characters to right of digits for left justification ("%-8d")
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// determine number padding from precision: the least number of digits to output
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precision, precisionSet := s.Precision()
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if precisionSet {
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switch {
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case len(digits) < precision:
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zeroes = precision - len(digits) // count of zero padding
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case digits == "0" && precision == 0:
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return // print nothing if zero value (x == 0) and zero precision ("." or ".0")
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}
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}
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// determine field pad from width: the least number of characters to output
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length := len(sign) + len(prefix) + zeroes + len(digits)
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if width, widthSet := s.Width(); widthSet && length < width { // pad as specified
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switch d := width - length; {
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case s.Flag('-'):
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// pad on the right with spaces; supersedes '0' when both specified
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right = d
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case s.Flag('0') && !precisionSet:
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// pad with zeroes unless precision also specified
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zeroes = d
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default:
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// pad on the left with spaces
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left = d
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}
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}
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// print number as [left pad][sign][prefix][zero pad][digits][right pad]
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writeMultiple(s, " ", left)
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writeMultiple(s, sign, 1)
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writeMultiple(s, prefix, 1)
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writeMultiple(s, "0", zeroes)
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writeMultiple(s, digits, 1)
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writeMultiple(s, " ", right)
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}
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// scan sets z to the integer value corresponding to the longest possible prefix
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// read from r representing a signed integer number in a given conversion base.
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// It returns z, the actual conversion base used, and an error, if any. In the
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// error case, the value of z is undefined but the returned value is nil. The
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// syntax follows the syntax of integer literals in Go.
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//
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// The base argument must be 0 or a value from 2 through MaxBase. If the base
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// is 0, the string prefix determines the actual conversion base. A prefix of
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// ``0x'' or ``0X'' selects base 16; the ``0'' prefix selects base 8, and a
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// ``0b'' or ``0B'' prefix selects base 2. Otherwise the selected base is 10.
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//
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func (z *Int) scan(r io.RuneScanner, base int) (*Int, int, error) {
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// determine sign
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ch, _, err := r.ReadRune()
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if err != nil {
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return nil, 0, err
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}
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neg := false
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switch ch {
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case '-':
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neg = true
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case '+': // nothing to do
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default:
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r.UnreadRune()
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}
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// determine mantissa
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z.abs, base, err = z.abs.scan(r, base)
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if err != nil {
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return nil, base, err
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}
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z.neg = len(z.abs) > 0 && neg // 0 has no sign
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return z, base, nil
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}
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// Scan is a support routine for fmt.Scanner; it sets z to the value of
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// the scanned number. It accepts the formats 'b' (binary), 'o' (octal),
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// 'd' (decimal), 'x' (lowercase hexadecimal), and 'X' (uppercase hexadecimal).
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func (z *Int) Scan(s fmt.ScanState, ch rune) error {
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s.SkipSpace() // skip leading space characters
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base := 0
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switch ch {
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case 'b':
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base = 2
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case 'o':
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base = 8
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case 'd':
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base = 10
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case 'x', 'X':
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base = 16
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case 's', 'v':
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// let scan determine the base
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default:
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return errors.New("Int.Scan: invalid verb")
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}
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_, _, err := z.scan(s, base)
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return err
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}
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// Int64 returns the int64 representation of x.
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// If x cannot be represented in an int64, the result is undefined.
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func (x *Int) Int64() int64 {
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v := int64(x.Uint64())
|
||
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 {
|
||
if len(x.abs) == 0 {
|
||
return 0
|
||
}
|
||
v := uint64(x.abs[0])
|
||
if _W == 32 && len(x.abs) > 1 {
|
||
v |= uint64(x.abs[1]) << 32
|
||
}
|
||
return v
|
||
}
|
||
|
||
// SetString sets z to the value of s, interpreted in the given base,
|
||
// and returns z and a boolean indicating 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 from 2 through MaxBase. If the base
|
||
// is 0, the string prefix determines the actual conversion base. A prefix of
|
||
// ``0x'' or ``0X'' selects base 16; the ``0'' prefix selects base 8, and a
|
||
// ``0b'' or ``0B'' prefix selects base 2. Otherwise the selected base is 10.
|
||
//
|
||
func (z *Int) SetString(s string, base int) (*Int, bool) {
|
||
r := strings.NewReader(s)
|
||
_, _, err := z.scan(r, base)
|
||
if err != nil {
|
||
return nil, false
|
||
}
|
||
_, _, err = r.ReadRune()
|
||
if err != io.EOF {
|
||
return nil, false
|
||
}
|
||
return z, true // err == io.EOF => scan consumed all of s
|
||
}
|
||
|
||
// 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 z as a big-endian byte slice.
|
||
func (x *Int) Bytes() []byte {
|
||
buf := make([]byte, len(x.abs)*_S)
|
||
return buf[x.abs.bytes(buf):]
|
||
}
|
||
|
||
// BitLen returns the length of the absolute value of z in bits.
|
||
// The bit length of 0 is 0.
|
||
func (x *Int) BitLen() int {
|
||
return x.abs.bitLen()
|
||
}
|
||
|
||
// Exp sets z = x**y mod |m| (i.e. the sign of m is ignored), and returns z.
|
||
// If y <= 0, the result is 1; if m == nil or m == 0, z = x**y.
|
||
// See Knuth, volume 2, section 4.6.3.
|
||
func (z *Int) Exp(x, y, m *Int) *Int {
|
||
if y.neg || len(y.abs) == 0 {
|
||
return z.SetInt64(1)
|
||
}
|
||
// y > 0
|
||
|
||
var mWords nat
|
||
if m != nil {
|
||
mWords = m.abs // m.abs may be nil for m == 0
|
||
}
|
||
|
||
z.abs = z.abs.expNN(x.abs, y.abs, mWords)
|
||
z.neg = len(z.abs) > 0 && x.neg && y.abs[0]&1 == 1 // 0 has no sign
|
||
return z
|
||
}
|
||
|
||
// GCD sets z to the greatest common divisor of a and b, which both must
|
||
// be > 0, and returns z.
|
||
// If x and y are not nil, GCD sets x and y such that z = a*x + b*y.
|
||
// If either a or b is <= 0, GCD sets z = x = y = 0.
|
||
func (z *Int) GCD(x, y, a, b *Int) *Int {
|
||
if a.Sign() <= 0 || b.Sign() <= 0 {
|
||
z.SetInt64(0)
|
||
if x != nil {
|
||
x.SetInt64(0)
|
||
}
|
||
if y != nil {
|
||
y.SetInt64(0)
|
||
}
|
||
return z
|
||
}
|
||
if x == nil && y == nil {
|
||
return z.binaryGCD(a, b)
|
||
}
|
||
|
||
A := new(Int).Set(a)
|
||
B := new(Int).Set(b)
|
||
|
||
X := new(Int)
|
||
Y := new(Int).SetInt64(1)
|
||
|
||
lastX := new(Int).SetInt64(1)
|
||
lastY := new(Int)
|
||
|
||
q := new(Int)
|
||
temp := new(Int)
|
||
|
||
for len(B.abs) > 0 {
|
||
r := new(Int)
|
||
q, r = q.QuoRem(A, B, r)
|
||
|
||
A, B = B, r
|
||
|
||
temp.Set(X)
|
||
X.Mul(X, q)
|
||
X.neg = !X.neg
|
||
X.Add(X, lastX)
|
||
lastX.Set(temp)
|
||
|
||
temp.Set(Y)
|
||
Y.Mul(Y, q)
|
||
Y.neg = !Y.neg
|
||
Y.Add(Y, lastY)
|
||
lastY.Set(temp)
|
||
}
|
||
|
||
if x != nil {
|
||
*x = *lastX
|
||
}
|
||
|
||
if y != nil {
|
||
*y = *lastY
|
||
}
|
||
|
||
*z = *A
|
||
return z
|
||
}
|
||
|
||
// binaryGCD sets z to the greatest common divisor of a and b, which both must
|
||
// be > 0, and returns z.
|
||
// See Knuth, The Art of Computer Programming, Vol. 2, Section 4.5.2, Algorithm B.
|
||
func (z *Int) binaryGCD(a, b *Int) *Int {
|
||
u := z
|
||
v := new(Int)
|
||
|
||
// use one Euclidean iteration to ensure that u and v are approx. the same size
|
||
switch {
|
||
case len(a.abs) > len(b.abs):
|
||
u.Set(b)
|
||
v.Rem(a, b)
|
||
case len(a.abs) < len(b.abs):
|
||
u.Set(a)
|
||
v.Rem(b, a)
|
||
default:
|
||
u.Set(a)
|
||
v.Set(b)
|
||
}
|
||
|
||
// v might be 0 now
|
||
if len(v.abs) == 0 {
|
||
return u
|
||
}
|
||
// u > 0 && v > 0
|
||
|
||
// determine largest k such that u = u' << k, v = v' << k
|
||
k := u.abs.trailingZeroBits()
|
||
if vk := v.abs.trailingZeroBits(); vk < k {
|
||
k = vk
|
||
}
|
||
u.Rsh(u, k)
|
||
v.Rsh(v, k)
|
||
|
||
// determine t (we know that u > 0)
|
||
t := new(Int)
|
||
if u.abs[0]&1 != 0 {
|
||
// u is odd
|
||
t.Neg(v)
|
||
} else {
|
||
t.Set(u)
|
||
}
|
||
|
||
for len(t.abs) > 0 {
|
||
// reduce t
|
||
t.Rsh(t, t.abs.trailingZeroBits())
|
||
if t.neg {
|
||
v.Neg(t)
|
||
} else {
|
||
u.Set(t)
|
||
}
|
||
t.Sub(u, v)
|
||
}
|
||
|
||
return u.Lsh(u, k)
|
||
}
|
||
|
||
// ProbablyPrime performs n Miller-Rabin tests to check whether x is prime.
|
||
// If it returns true, x is prime with probability 1 - 1/4^n.
|
||
// If it returns false, x is not prime.
|
||
func (x *Int) ProbablyPrime(n int) bool {
|
||
return !x.neg && x.abs.probablyPrime(n)
|
||
}
|
||
|
||
// Rand sets z to a pseudo-random number in [0, n) and returns z.
|
||
func (z *Int) Rand(rnd *rand.Rand, n *Int) *Int {
|
||
z.neg = false
|
||
if n.neg == true || 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 group ℤ/pℤ (where
|
||
// p is a prime) and returns z.
|
||
func (z *Int) ModInverse(g, p *Int) *Int {
|
||
var d Int
|
||
d.GCD(z, nil, g, p)
|
||
// x and y are such that g*x + p*y = d. Since p is prime, d = 1. Taking
|
||
// that modulo p results in g*x = 1, therefore x is the inverse element.
|
||
if z.neg {
|
||
z.Add(z, p)
|
||
}
|
||
return z
|
||
}
|
||
|
||
// 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).add(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
|
||
}
|
||
|
||
// Gob codec version. Permits backward-compatible changes to the encoding.
|
||
const intGobVersion byte = 1
|
||
|
||
// GobEncode implements the gob.GobEncoder interface.
|
||
func (x *Int) GobEncode() ([]byte, error) {
|
||
buf := make([]byte, 1+len(x.abs)*_S) // extra byte for version and sign bit
|
||
i := x.abs.bytes(buf) - 1 // i >= 0
|
||
b := intGobVersion << 1 // make space for sign bit
|
||
if x.neg {
|
||
b |= 1
|
||
}
|
||
buf[i] = b
|
||
return buf[i:], nil
|
||
}
|
||
|
||
// GobDecode implements the gob.GobDecoder interface.
|
||
func (z *Int) GobDecode(buf []byte) error {
|
||
if len(buf) == 0 {
|
||
return errors.New("Int.GobDecode: no data")
|
||
}
|
||
b := buf[0]
|
||
if b>>1 != intGobVersion {
|
||
return errors.New(fmt.Sprintf("Int.GobDecode: encoding version %d not supported", b>>1))
|
||
}
|
||
z.neg = b&1 != 0
|
||
z.abs = z.abs.setBytes(buf[1:])
|
||
return nil
|
||
}
|
||
|
||
// MarshalJSON implements the json.Marshaler interface.
|
||
func (x *Int) MarshalJSON() ([]byte, error) {
|
||
// TODO(gri): get rid of the []byte/string conversions
|
||
return []byte(x.String()), nil
|
||
}
|
||
|
||
// UnmarshalJSON implements the json.Unmarshaler interface.
|
||
func (z *Int) UnmarshalJSON(x []byte) error {
|
||
// TODO(gri): get rid of the []byte/string conversions
|
||
_, ok := z.SetString(string(x), 0)
|
||
if !ok {
|
||
return fmt.Errorf("math/big: cannot unmarshal %s into a *big.Int", x)
|
||
}
|
||
return nil
|
||
}
|