830 lines
16 KiB
Go
830 lines
16 KiB
Go
// Copyright (c) 2014 The mathutil 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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// Package mathutil provides utilities supplementing the standard 'math' and
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// 'math/rand' packages.
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//
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// Compatibility issues
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//
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// 2013-12-13: The following functions have been REMOVED
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//
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// func Uint64ToBigInt(n uint64) *big.Int
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// func Uint64FromBigInt(n *big.Int) (uint64, bool)
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//
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// 2013-05-13: The following functions are now DEPRECATED
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//
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// func Uint64ToBigInt(n uint64) *big.Int
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// func Uint64FromBigInt(n *big.Int) (uint64, bool)
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//
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// These functions will be REMOVED with Go release 1.1+1.
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//
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// 2013-01-21: The following functions have been REMOVED
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//
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// func MaxInt() int
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// func MinInt() int
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// func MaxUint() uint
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// func UintPtrBits() int
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//
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// They are now replaced by untyped constants
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//
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// MaxInt
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// MinInt
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// MaxUint
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// UintPtrBits
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//
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// Additionally one more untyped constant was added
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//
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// IntBits
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//
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// This change breaks any existing code depending on the above removed
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// functions. They should have not been published in the first place, that was
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// unfortunate. Instead, defining such architecture and/or implementation
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// specific integer limits and bit widths as untyped constants improves
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// performance and allows for static dead code elimination if it depends on
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// these values. Thanks to minux for pointing it out in the mail list
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// (https://groups.google.com/d/msg/golang-nuts/tlPpLW6aJw8/NT3mpToH-a4J).
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//
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// 2012-12-12: The following functions will be DEPRECATED with Go release
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// 1.0.3+1 and REMOVED with Go release 1.0.3+2, b/c of
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// http://code.google.com/p/go/source/detail?r=954a79ee3ea8
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//
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// func Uint64ToBigInt(n uint64) *big.Int
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// func Uint64FromBigInt(n *big.Int) (uint64, bool)
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package mathutil
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import (
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"math"
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"math/big"
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)
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// Architecture and/or implementation specific integer limits and bit widths.
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const (
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MaxInt = 1<<(IntBits-1) - 1
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MinInt = -MaxInt - 1
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MaxUint = 1<<IntBits - 1
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IntBits = 1 << (^uint(0)>>32&1 + ^uint(0)>>16&1 + ^uint(0)>>8&1 + 3)
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UintPtrBits = 1 << (^uintptr(0)>>32&1 + ^uintptr(0)>>16&1 + ^uintptr(0)>>8&1 + 3)
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)
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var (
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_1 = big.NewInt(1)
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_2 = big.NewInt(2)
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)
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// GCDByte returns the greatest common divisor of a and b. Based on:
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// http://en.wikipedia.org/wiki/Euclidean_algorithm#Implementations
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func GCDByte(a, b byte) byte {
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for b != 0 {
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a, b = b, a%b
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}
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return a
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}
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// GCDUint16 returns the greatest common divisor of a and b.
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func GCDUint16(a, b uint16) uint16 {
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for b != 0 {
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a, b = b, a%b
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}
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return a
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}
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// GCD returns the greatest common divisor of a and b.
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func GCDUint32(a, b uint32) uint32 {
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for b != 0 {
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a, b = b, a%b
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}
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return a
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}
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// GCD64 returns the greatest common divisor of a and b.
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func GCDUint64(a, b uint64) uint64 {
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for b != 0 {
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a, b = b, a%b
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}
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return a
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}
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// ISqrt returns floor(sqrt(n)). Typical run time is few hundreds of ns.
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func ISqrt(n uint32) (x uint32) {
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if n == 0 {
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return
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}
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if n >= math.MaxUint16*math.MaxUint16 {
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return math.MaxUint16
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}
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var px, nx uint32
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for x = n; ; px, x = x, nx {
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nx = (x + n/x) / 2
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if nx == x || nx == px {
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break
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}
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}
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return
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}
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// SqrtUint64 returns floor(sqrt(n)). Typical run time is about 0.5 µs.
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func SqrtUint64(n uint64) (x uint64) {
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if n == 0 {
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return
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}
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if n >= math.MaxUint32*math.MaxUint32 {
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return math.MaxUint32
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}
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var px, nx uint64
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for x = n; ; px, x = x, nx {
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nx = (x + n/x) / 2
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if nx == x || nx == px {
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break
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}
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}
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return
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}
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// SqrtBig returns floor(sqrt(n)). It panics on n < 0.
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func SqrtBig(n *big.Int) (x *big.Int) {
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switch n.Sign() {
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case -1:
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panic(-1)
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case 0:
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return big.NewInt(0)
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}
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var px, nx big.Int
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x = big.NewInt(0)
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x.SetBit(x, n.BitLen()/2+1, 1)
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for {
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nx.Rsh(nx.Add(x, nx.Div(n, x)), 1)
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if nx.Cmp(x) == 0 || nx.Cmp(&px) == 0 {
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break
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}
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px.Set(x)
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x.Set(&nx)
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}
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return
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}
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// Log2Byte returns log base 2 of n. It's the same as index of the highest
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// bit set in n. For n == 0 -1 is returned.
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func Log2Byte(n byte) int {
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return log2[n]
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}
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// Log2Uint16 returns log base 2 of n. It's the same as index of the highest
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// bit set in n. For n == 0 -1 is returned.
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func Log2Uint16(n uint16) int {
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if b := n >> 8; b != 0 {
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return log2[b] + 8
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}
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return log2[n]
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}
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// Log2Uint32 returns log base 2 of n. It's the same as index of the highest
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// bit set in n. For n == 0 -1 is returned.
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func Log2Uint32(n uint32) int {
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if b := n >> 24; b != 0 {
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return log2[b] + 24
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}
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if b := n >> 16; b != 0 {
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return log2[b] + 16
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}
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if b := n >> 8; b != 0 {
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return log2[b] + 8
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}
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return log2[n]
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}
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// Log2Uint64 returns log base 2 of n. It's the same as index of the highest
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// bit set in n. For n == 0 -1 is returned.
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func Log2Uint64(n uint64) int {
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if b := n >> 56; b != 0 {
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return log2[b] + 56
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}
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if b := n >> 48; b != 0 {
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return log2[b] + 48
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}
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if b := n >> 40; b != 0 {
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return log2[b] + 40
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}
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if b := n >> 32; b != 0 {
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return log2[b] + 32
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}
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if b := n >> 24; b != 0 {
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return log2[b] + 24
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}
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if b := n >> 16; b != 0 {
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return log2[b] + 16
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}
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if b := n >> 8; b != 0 {
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return log2[b] + 8
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}
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return log2[n]
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}
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// ModPowByte computes (b^e)%m. It panics for m == 0 || b == e == 0.
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//
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// See also: http://en.wikipedia.org/wiki/Modular_exponentiation#Right-to-left_binary_method
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func ModPowByte(b, e, m byte) byte {
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if b == 0 && e == 0 {
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panic(0)
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}
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if m == 1 {
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return 0
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}
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r := uint16(1)
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for b, m := uint16(b), uint16(m); e > 0; b, e = b*b%m, e>>1 {
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if e&1 == 1 {
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r = r * b % m
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}
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}
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return byte(r)
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}
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// ModPowByte computes (b^e)%m. It panics for m == 0 || b == e == 0.
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func ModPowUint16(b, e, m uint16) uint16 {
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if b == 0 && e == 0 {
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panic(0)
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}
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if m == 1 {
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return 0
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}
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r := uint32(1)
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for b, m := uint32(b), uint32(m); e > 0; b, e = b*b%m, e>>1 {
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if e&1 == 1 {
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r = r * b % m
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}
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}
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return uint16(r)
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}
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// ModPowUint32 computes (b^e)%m. It panics for m == 0 || b == e == 0.
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func ModPowUint32(b, e, m uint32) uint32 {
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if b == 0 && e == 0 {
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panic(0)
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}
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if m == 1 {
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return 0
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}
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r := uint64(1)
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for b, m := uint64(b), uint64(m); e > 0; b, e = b*b%m, e>>1 {
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if e&1 == 1 {
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r = r * b % m
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}
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}
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return uint32(r)
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}
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// ModPowUint64 computes (b^e)%m. It panics for m == 0 || b == e == 0.
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func ModPowUint64(b, e, m uint64) (r uint64) {
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if b == 0 && e == 0 {
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panic(0)
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}
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if m == 1 {
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return 0
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}
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return modPowBigInt(big.NewInt(0).SetUint64(b), big.NewInt(0).SetUint64(e), big.NewInt(0).SetUint64(m)).Uint64()
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}
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func modPowBigInt(b, e, m *big.Int) (r *big.Int) {
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r = big.NewInt(1)
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for i, n := 0, e.BitLen(); i < n; i++ {
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if e.Bit(i) != 0 {
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r.Mod(r.Mul(r, b), m)
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}
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b.Mod(b.Mul(b, b), m)
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}
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return
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}
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// ModPowBigInt computes (b^e)%m. Returns nil for e < 0. It panics for m == 0 || b == e == 0.
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func ModPowBigInt(b, e, m *big.Int) (r *big.Int) {
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if b.Sign() == 0 && e.Sign() == 0 {
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panic(0)
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}
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if m.Cmp(_1) == 0 {
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return big.NewInt(0)
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}
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if e.Sign() < 0 {
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return
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}
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return modPowBigInt(big.NewInt(0).Set(b), big.NewInt(0).Set(e), m)
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}
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var uint64ToBigIntDelta big.Int
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func init() {
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uint64ToBigIntDelta.SetBit(&uint64ToBigIntDelta, 63, 1)
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}
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var uintptrBits int
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func init() {
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x := uint64(math.MaxUint64)
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uintptrBits = BitLenUintptr(uintptr(x))
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}
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// UintptrBits returns the bit width of an uintptr at the executing machine.
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func UintptrBits() int {
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return uintptrBits
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}
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// AddUint128_64 returns the uint128 sum of uint64 a and b.
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func AddUint128_64(a, b uint64) (hi uint64, lo uint64) {
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lo = a + b
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if lo < a {
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hi = 1
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}
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return
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}
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// MulUint128_64 returns the uint128 bit product of uint64 a and b.
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func MulUint128_64(a, b uint64) (hi, lo uint64) {
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/*
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2^(2 W) ahi bhi + 2^W alo bhi + 2^W ahi blo + alo blo
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FEDCBA98 76543210 FEDCBA98 76543210
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---- alo*blo ----
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---- alo*bhi ----
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---- ahi*blo ----
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---- ahi*bhi ----
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*/
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const w = 32
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const m = 1<<w - 1
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ahi, bhi, alo, blo := a>>w, b>>w, a&m, b&m
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lo = alo * blo
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mid1 := alo * bhi
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mid2 := ahi * blo
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c1, lo := AddUint128_64(lo, mid1<<w)
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c2, lo := AddUint128_64(lo, mid2<<w)
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_, hi = AddUint128_64(ahi*bhi, mid1>>w+mid2>>w+uint64(c1+c2))
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return
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}
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// PowerizeBigInt returns (e, p) such that e is the smallest number for which p
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// == b^e is greater or equal n. For n < 0 or b < 2 (0, nil) is returned.
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//
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// NOTE: Run time for large values of n (above about 2^1e6 ~= 1e300000) can be
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// significant and/or unacceptabe. For any smaller values of n the function
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// typically performs in sub second time. For "small" values of n (cca bellow
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// 2^1e3 ~= 1e300) the same can be easily below 10 µs.
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//
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// A special (and trivial) case of b == 2 is handled separately and performs
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// much faster.
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func PowerizeBigInt(b, n *big.Int) (e uint32, p *big.Int) {
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switch {
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case b.Cmp(_2) < 0 || n.Sign() < 0:
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return
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case n.Sign() == 0 || n.Cmp(_1) == 0:
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return 0, big.NewInt(1)
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case b.Cmp(_2) == 0:
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p = big.NewInt(0)
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e = uint32(n.BitLen() - 1)
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p.SetBit(p, int(e), 1)
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if p.Cmp(n) < 0 {
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p.Mul(p, _2)
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e++
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}
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return
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}
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bw := b.BitLen()
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nw := n.BitLen()
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p = big.NewInt(1)
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var bb, r big.Int
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for {
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switch p.Cmp(n) {
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case -1:
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x := uint32((nw - p.BitLen()) / bw)
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if x == 0 {
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x = 1
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}
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e += x
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switch x {
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case 1:
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p.Mul(p, b)
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default:
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r.Set(_1)
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bb.Set(b)
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e := x
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for {
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if e&1 != 0 {
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r.Mul(&r, &bb)
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}
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if e >>= 1; e == 0 {
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break
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}
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bb.Mul(&bb, &bb)
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}
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p.Mul(p, &r)
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}
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case 0, 1:
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return
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}
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}
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}
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// PowerizeUint32BigInt returns (e, p) such that e is the smallest number for
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// which p == b^e is greater or equal n. For n < 0 or b < 2 (0, nil) is
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// returned.
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//
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// More info: see PowerizeBigInt.
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func PowerizeUint32BigInt(b uint32, n *big.Int) (e uint32, p *big.Int) {
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switch {
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case b < 2 || n.Sign() < 0:
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return
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case n.Sign() == 0 || n.Cmp(_1) == 0:
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return 0, big.NewInt(1)
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case b == 2:
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p = big.NewInt(0)
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e = uint32(n.BitLen() - 1)
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p.SetBit(p, int(e), 1)
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if p.Cmp(n) < 0 {
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p.Mul(p, _2)
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e++
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}
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return
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}
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var bb big.Int
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bb.SetInt64(int64(b))
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return PowerizeBigInt(&bb, n)
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}
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/*
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ProbablyPrimeUint32 returns true if n is prime or n is a pseudoprime to base a.
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It implements the Miller-Rabin primality test for one specific value of 'a' and
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k == 1.
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Wrt pseudocode shown at
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http://en.wikipedia.org/wiki/Miller-Rabin_primality_test#Algorithm_and_running_time
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Input: n > 3, an odd integer to be tested for primality;
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Input: k, a parameter that determines the accuracy of the test
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Output: composite if n is composite, otherwise probably prime
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write n − 1 as 2^s·d with d odd by factoring powers of 2 from n − 1
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LOOP: repeat k times:
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pick a random integer a in the range [2, n − 2]
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x ← a^d mod n
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if x = 1 or x = n − 1 then do next LOOP
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for r = 1 .. s − 1
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x ← x^2 mod n
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if x = 1 then return composite
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if x = n − 1 then do next LOOP
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return composite
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return probably prime
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... this function behaves like passing 1 for 'k' and additionaly a
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fixed/non-random 'a'. Otherwise it's the same algorithm.
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See also: http://mathworld.wolfram.com/Rabin-MillerStrongPseudoprimeTest.html
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*/
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func ProbablyPrimeUint32(n, a uint32) bool {
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d, s := n-1, 0
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for ; d&1 == 0; d, s = d>>1, s+1 {
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}
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x := uint64(ModPowUint32(a, d, n))
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if x == 1 || uint32(x) == n-1 {
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return true
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}
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for ; s > 1; s-- {
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if x = x * x % uint64(n); x == 1 {
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return false
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}
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if uint32(x) == n-1 {
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return true
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}
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}
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return false
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}
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// ProbablyPrimeUint64_32 returns true if n is prime or n is a pseudoprime to
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// base a. It implements the Miller-Rabin primality test for one specific value
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// of 'a' and k == 1. See also ProbablyPrimeUint32.
|
||
func ProbablyPrimeUint64_32(n uint64, a uint32) bool {
|
||
d, s := n-1, 0
|
||
for ; d&1 == 0; d, s = d>>1, s+1 {
|
||
}
|
||
x := ModPowUint64(uint64(a), d, n)
|
||
if x == 1 || x == n-1 {
|
||
return true
|
||
}
|
||
|
||
bx, bn := big.NewInt(0).SetUint64(x), big.NewInt(0).SetUint64(n)
|
||
for ; s > 1; s-- {
|
||
if x = bx.Mod(bx.Mul(bx, bx), bn).Uint64(); x == 1 {
|
||
return false
|
||
}
|
||
|
||
if x == n-1 {
|
||
return true
|
||
}
|
||
}
|
||
return false
|
||
}
|
||
|
||
// ProbablyPrimeBigInt_32 returns true if n is prime or n is a pseudoprime to
|
||
// base a. It implements the Miller-Rabin primality test for one specific value
|
||
// of 'a' and k == 1. See also ProbablyPrimeUint32.
|
||
func ProbablyPrimeBigInt_32(n *big.Int, a uint32) bool {
|
||
var d big.Int
|
||
d.Set(n)
|
||
d.Sub(&d, _1) // d <- n-1
|
||
s := 0
|
||
for ; d.Bit(s) == 0; s++ {
|
||
}
|
||
nMinus1 := big.NewInt(0).Set(&d)
|
||
d.Rsh(&d, uint(s))
|
||
|
||
x := ModPowBigInt(big.NewInt(int64(a)), &d, n)
|
||
if x.Cmp(_1) == 0 || x.Cmp(nMinus1) == 0 {
|
||
return true
|
||
}
|
||
|
||
for ; s > 1; s-- {
|
||
if x = x.Mod(x.Mul(x, x), n); x.Cmp(_1) == 0 {
|
||
return false
|
||
}
|
||
|
||
if x.Cmp(nMinus1) == 0 {
|
||
return true
|
||
}
|
||
}
|
||
return false
|
||
}
|
||
|
||
// ProbablyPrimeBigInt returns true if n is prime or n is a pseudoprime to base
|
||
// a. It implements the Miller-Rabin primality test for one specific value of
|
||
// 'a' and k == 1. See also ProbablyPrimeUint32.
|
||
func ProbablyPrimeBigInt(n, a *big.Int) bool {
|
||
var d big.Int
|
||
d.Set(n)
|
||
d.Sub(&d, _1) // d <- n-1
|
||
s := 0
|
||
for ; d.Bit(s) == 0; s++ {
|
||
}
|
||
nMinus1 := big.NewInt(0).Set(&d)
|
||
d.Rsh(&d, uint(s))
|
||
|
||
x := ModPowBigInt(a, &d, n)
|
||
if x.Cmp(_1) == 0 || x.Cmp(nMinus1) == 0 {
|
||
return true
|
||
}
|
||
|
||
for ; s > 1; s-- {
|
||
if x = x.Mod(x.Mul(x, x), n); x.Cmp(_1) == 0 {
|
||
return false
|
||
}
|
||
|
||
if x.Cmp(nMinus1) == 0 {
|
||
return true
|
||
}
|
||
}
|
||
return false
|
||
}
|
||
|
||
// Max returns the larger of a and b.
|
||
func Max(a, b int) int {
|
||
if a > b {
|
||
return a
|
||
}
|
||
|
||
return b
|
||
}
|
||
|
||
// Min returns the smaller of a and b.
|
||
func Min(a, b int) int {
|
||
if a < b {
|
||
return a
|
||
}
|
||
|
||
return b
|
||
}
|
||
|
||
// UMax returns the larger of a and b.
|
||
func UMax(a, b uint) uint {
|
||
if a > b {
|
||
return a
|
||
}
|
||
|
||
return b
|
||
}
|
||
|
||
// UMin returns the smaller of a and b.
|
||
func UMin(a, b uint) uint {
|
||
if a < b {
|
||
return a
|
||
}
|
||
|
||
return b
|
||
}
|
||
|
||
// MaxByte returns the larger of a and b.
|
||
func MaxByte(a, b byte) byte {
|
||
if a > b {
|
||
return a
|
||
}
|
||
|
||
return b
|
||
}
|
||
|
||
// MinByte returns the smaller of a and b.
|
||
func MinByte(a, b byte) byte {
|
||
if a < b {
|
||
return a
|
||
}
|
||
|
||
return b
|
||
}
|
||
|
||
// MaxInt8 returns the larger of a and b.
|
||
func MaxInt8(a, b int8) int8 {
|
||
if a > b {
|
||
return a
|
||
}
|
||
|
||
return b
|
||
}
|
||
|
||
// MinInt8 returns the smaller of a and b.
|
||
func MinInt8(a, b int8) int8 {
|
||
if a < b {
|
||
return a
|
||
}
|
||
|
||
return b
|
||
}
|
||
|
||
// MaxUint16 returns the larger of a and b.
|
||
func MaxUint16(a, b uint16) uint16 {
|
||
if a > b {
|
||
return a
|
||
}
|
||
|
||
return b
|
||
}
|
||
|
||
// MinUint16 returns the smaller of a and b.
|
||
func MinUint16(a, b uint16) uint16 {
|
||
if a < b {
|
||
return a
|
||
}
|
||
|
||
return b
|
||
}
|
||
|
||
// MaxInt16 returns the larger of a and b.
|
||
func MaxInt16(a, b int16) int16 {
|
||
if a > b {
|
||
return a
|
||
}
|
||
|
||
return b
|
||
}
|
||
|
||
// MinInt16 returns the smaller of a and b.
|
||
func MinInt16(a, b int16) int16 {
|
||
if a < b {
|
||
return a
|
||
}
|
||
|
||
return b
|
||
}
|
||
|
||
// MaxUint32 returns the larger of a and b.
|
||
func MaxUint32(a, b uint32) uint32 {
|
||
if a > b {
|
||
return a
|
||
}
|
||
|
||
return b
|
||
}
|
||
|
||
// MinUint32 returns the smaller of a and b.
|
||
func MinUint32(a, b uint32) uint32 {
|
||
if a < b {
|
||
return a
|
||
}
|
||
|
||
return b
|
||
}
|
||
|
||
// MaxInt32 returns the larger of a and b.
|
||
func MaxInt32(a, b int32) int32 {
|
||
if a > b {
|
||
return a
|
||
}
|
||
|
||
return b
|
||
}
|
||
|
||
// MinInt32 returns the smaller of a and b.
|
||
func MinInt32(a, b int32) int32 {
|
||
if a < b {
|
||
return a
|
||
}
|
||
|
||
return b
|
||
}
|
||
|
||
// MaxUint64 returns the larger of a and b.
|
||
func MaxUint64(a, b uint64) uint64 {
|
||
if a > b {
|
||
return a
|
||
}
|
||
|
||
return b
|
||
}
|
||
|
||
// MinUint64 returns the smaller of a and b.
|
||
func MinUint64(a, b uint64) uint64 {
|
||
if a < b {
|
||
return a
|
||
}
|
||
|
||
return b
|
||
}
|
||
|
||
// MaxInt64 returns the larger of a and b.
|
||
func MaxInt64(a, b int64) int64 {
|
||
if a > b {
|
||
return a
|
||
}
|
||
|
||
return b
|
||
}
|
||
|
||
// MinInt64 returns the smaller of a and b.
|
||
func MinInt64(a, b int64) int64 {
|
||
if a < b {
|
||
return a
|
||
}
|
||
|
||
return b
|
||
}
|
||
|
||
// ToBase produces n in base b. For example
|
||
//
|
||
// ToBase(2047, 22) -> [1, 5, 4]
|
||
//
|
||
// 1 * 22^0 1
|
||
// 5 * 22^1 110
|
||
// 4 * 22^2 1936
|
||
// ----
|
||
// 2047
|
||
//
|
||
// ToBase panics for bases < 2.
|
||
func ToBase(n *big.Int, b int) []int {
|
||
var nn big.Int
|
||
nn.Set(n)
|
||
if b < 2 {
|
||
panic("invalid base")
|
||
}
|
||
|
||
k := 1
|
||
switch nn.Sign() {
|
||
case -1:
|
||
nn.Neg(&nn)
|
||
k = -1
|
||
case 0:
|
||
return []int{0}
|
||
}
|
||
|
||
bb := big.NewInt(int64(b))
|
||
var r []int
|
||
rem := big.NewInt(0)
|
||
for nn.Sign() != 0 {
|
||
nn.QuoRem(&nn, bb, rem)
|
||
r = append(r, k*int(rem.Int64()))
|
||
}
|
||
return r
|
||
}
|