735 lines
15 KiB
Go
735 lines
15 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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"fmt"
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"os"
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"rand"
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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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// 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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z.abs = z.abs.set(x.abs)
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z.neg = x.neg
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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.abs = z.abs.set(x.abs)
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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.abs = z.abs.set(x.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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// 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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// 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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// 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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//
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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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// 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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// 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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//
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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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s := ""
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if x.neg {
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s = "-"
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}
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return s + x.abs.string(10)
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}
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func fmtbase(ch int) int {
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switch ch {
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case 'b':
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return 2
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case 'o':
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return 8
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case 'd':
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return 10
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case 'x':
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return 16
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}
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return 10
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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) and
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// 'x' (hexadecimal).
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//
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func (x *Int) Format(s fmt.State, ch int) {
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if x == nil {
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fmt.Fprint(s, "<nil>")
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return
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}
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if x.neg {
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fmt.Fprint(s, "-")
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}
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fmt.Fprint(s, x.abs.string(fmtbase(ch)))
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}
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// Int64 returns the int64 representation of z.
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// If z cannot be represented in an int64, the result is undefined.
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func (x *Int) Int64() int64 {
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if len(x.abs) == 0 {
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return 0
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}
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v := int64(x.abs[0])
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if _W == 32 && len(x.abs) > 1 {
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v |= int64(x.abs[1]) << 32
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}
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if x.neg {
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v = -v
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}
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return v
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}
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// SetString sets z to the value of s, interpreted in the given base,
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// and returns z and a boolean indicating success. If SetString fails,
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// the value of z is undefined.
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//
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// If the base argument is 0, the string prefix determines the actual
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// conversion base. A prefix of ``0x'' or ``0X'' selects base 16; the
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// ``0'' prefix selects base 8, and a ``0b'' or ``0B'' prefix selects
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// base 2. Otherwise the selected base is 10.
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//
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func (z *Int) SetString(s string, base int) (*Int, bool) {
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if len(s) == 0 || base < 0 || base == 1 || 16 < base {
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return z, false
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}
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neg := s[0] == '-'
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if neg || s[0] == '+' {
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s = s[1:]
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if len(s) == 0 {
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return z, false
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}
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}
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var scanned int
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z.abs, _, scanned = z.abs.scan(s, base)
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if scanned != len(s) {
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return z, false
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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, true
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}
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// SetBytes interprets buf as the bytes of a big-endian unsigned
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// integer, sets z to that value, and returns z.
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func (z *Int) SetBytes(buf []byte) *Int {
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z.abs = z.abs.setBytes(buf)
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z.neg = false
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return z
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}
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// Bytes returns the absolute value of z as a big-endian byte slice.
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func (z *Int) Bytes() []byte {
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buf := make([]byte, len(z.abs)*_S)
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return buf[z.abs.bytes(buf):]
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}
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// BitLen returns the length of the absolute value of z in bits.
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// The bit length of 0 is 0.
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func (z *Int) BitLen() int {
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return z.abs.bitLen()
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}
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// Exp sets z = x**y mod m. If m is nil, z = x**y.
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// See Knuth, volume 2, section 4.6.3.
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func (z *Int) Exp(x, y, m *Int) *Int {
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if y.neg || len(y.abs) == 0 {
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neg := x.neg
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z.SetInt64(1)
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z.neg = neg
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return z
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}
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var mWords nat
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if m != nil {
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mWords = m.abs
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}
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z.abs = z.abs.expNN(x.abs, y.abs, mWords)
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z.neg = len(z.abs) > 0 && x.neg && y.abs[0]&1 == 1 // 0 has no sign
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return z
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}
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// GcdInt sets d to the greatest common divisor of a and b, which must be
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// positive numbers.
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// If x and y are not nil, GcdInt sets x and y such that d = a*x + b*y.
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// If either a or b is not positive, GcdInt sets d = x = y = 0.
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func GcdInt(d, x, y, a, b *Int) {
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if a.neg || b.neg {
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d.SetInt64(0)
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if x != nil {
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x.SetInt64(0)
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}
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if y != nil {
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y.SetInt64(0)
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}
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return
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}
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A := new(Int).Set(a)
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B := new(Int).Set(b)
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X := new(Int)
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Y := new(Int).SetInt64(1)
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lastX := new(Int).SetInt64(1)
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lastY := new(Int)
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q := new(Int)
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temp := new(Int)
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for len(B.abs) > 0 {
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r := new(Int)
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q, r = q.QuoRem(A, B, r)
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A, B = B, r
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temp.Set(X)
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X.Mul(X, q)
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X.neg = !X.neg
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X.Add(X, lastX)
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lastX.Set(temp)
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temp.Set(Y)
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Y.Mul(Y, q)
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Y.neg = !Y.neg
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Y.Add(Y, lastY)
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lastY.Set(temp)
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}
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if x != nil {
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*x = *lastX
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}
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if y != nil {
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*y = *lastY
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}
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*d = *A
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}
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// ProbablyPrime performs n Miller-Rabin tests to check whether z is prime.
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// If it returns true, z is prime with probability 1 - 1/4^n.
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// If it returns false, z is not prime.
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func ProbablyPrime(z *Int, n int) bool {
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return !z.neg && z.abs.probablyPrime(n)
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}
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// Rand sets z to a pseudo-random number in [0, n) and returns z.
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func (z *Int) Rand(rnd *rand.Rand, n *Int) *Int {
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z.neg = false
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if n.neg == true || len(n.abs) == 0 {
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z.abs = nil
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return z
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}
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z.abs = z.abs.random(rnd, n.abs, n.abs.bitLen())
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return z
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}
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// ModInverse sets z to the multiplicative inverse of g in the group ℤ/pℤ (where
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// p is a prime) and returns z.
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func (z *Int) ModInverse(g, p *Int) *Int {
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var d Int
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GcdInt(&d, z, nil, g, p)
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// x and y are such that g*x + p*y = d. Since p is prime, d = 1. Taking
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// that modulo p results in g*x = 1, therefore x is the inverse element.
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if z.neg {
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z.Add(z, p)
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}
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return z
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}
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// Lsh sets z = x << n and returns z.
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func (z *Int) Lsh(x *Int, n uint) *Int {
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z.abs = z.abs.shl(x.abs, n)
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z.neg = x.neg
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return z
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}
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// 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
|
||
}
|
||
|
||
|
||
// 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{}.sub(x.abs, natOne)
|
||
y1 := nat{}.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{}.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{}.sub(x.abs, natOne)
|
||
y1 := nat{}.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{}.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{}.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{}.sub(x.abs, natOne)
|
||
y1 := nat{}.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{}.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{}.sub(x.abs, natOne)
|
||
y1 := nat{}.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{}.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 version byte = 1
|
||
|
||
// GobEncode implements the gob.GobEncoder interface.
|
||
func (z *Int) GobEncode() ([]byte, os.Error) {
|
||
buf := make([]byte, len(z.abs)*_S+1) // extra byte for version and sign bit
|
||
i := z.abs.bytes(buf) - 1 // i >= 0
|
||
b := version << 1 // make space for sign bit
|
||
if z.neg {
|
||
b |= 1
|
||
}
|
||
buf[i] = b
|
||
return buf[i:], nil
|
||
}
|
||
|
||
|
||
// GobDecode implements the gob.GobDecoder interface.
|
||
func (z *Int) GobDecode(buf []byte) os.Error {
|
||
if len(buf) == 0 {
|
||
return os.NewError("Int.GobDecode: no data")
|
||
}
|
||
b := buf[0]
|
||
if b>>1 != version {
|
||
return os.NewError(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
|
||
}
|