1218 lines
30 KiB
Go
1218 lines
30 KiB
Go
// Copyright 2009 The Go Authors. All rights reserved.
|
||
// Use of this source code is governed by a BSD-style
|
||
// license that can be found in the LICENSE file.
|
||
|
||
// This file implements signed multi-precision integers.
|
||
|
||
package big
|
||
|
||
import (
|
||
"fmt"
|
||
"io"
|
||
"math/rand"
|
||
"strings"
|
||
)
|
||
|
||
// An Int represents a signed multi-precision integer.
|
||
// The zero value for an Int represents the value 0.
|
||
//
|
||
// Operations always take pointer arguments (*Int) rather
|
||
// than Int values, and each unique Int value requires
|
||
// its own unique *Int pointer. To "copy" an Int value,
|
||
// an existing (or newly allocated) Int must be set to
|
||
// a new value using the Int.Set method; shallow copies
|
||
// of Ints are not supported and may lead to errors.
|
||
type Int struct {
|
||
neg bool // sign
|
||
abs nat // absolute value of the integer
|
||
}
|
||
|
||
var intOne = &Int{false, natOne}
|
||
|
||
// Sign returns:
|
||
//
|
||
// -1 if x < 0
|
||
// 0 if x == 0
|
||
// +1 if x > 0
|
||
//
|
||
func (x *Int) Sign() int {
|
||
if len(x.abs) == 0 {
|
||
return 0
|
||
}
|
||
if x.neg {
|
||
return -1
|
||
}
|
||
return 1
|
||
}
|
||
|
||
// SetInt64 sets z to x and returns z.
|
||
func (z *Int) SetInt64(x int64) *Int {
|
||
neg := false
|
||
if x < 0 {
|
||
neg = true
|
||
x = -x
|
||
}
|
||
z.abs = z.abs.setUint64(uint64(x))
|
||
z.neg = neg
|
||
return z
|
||
}
|
||
|
||
// SetUint64 sets z to x and returns z.
|
||
func (z *Int) SetUint64(x uint64) *Int {
|
||
z.abs = z.abs.setUint64(x)
|
||
z.neg = false
|
||
return z
|
||
}
|
||
|
||
// NewInt allocates and returns a new Int set to x.
|
||
func NewInt(x int64) *Int {
|
||
return new(Int).SetInt64(x)
|
||
}
|
||
|
||
// Set sets z to x and returns z.
|
||
func (z *Int) Set(x *Int) *Int {
|
||
if z != x {
|
||
z.abs = z.abs.set(x.abs)
|
||
z.neg = x.neg
|
||
}
|
||
return z
|
||
}
|
||
|
||
// Bits provides raw (unchecked but fast) access to x by returning its
|
||
// absolute value as a little-endian Word slice. The result and x share
|
||
// the same underlying array.
|
||
// Bits is intended to support implementation of missing low-level Int
|
||
// functionality outside this package; it should be avoided otherwise.
|
||
func (x *Int) Bits() []Word {
|
||
return x.abs
|
||
}
|
||
|
||
// SetBits provides raw (unchecked but fast) access to z by setting its
|
||
// value to abs, interpreted as a little-endian Word slice, and returning
|
||
// z. The result and abs share the same underlying array.
|
||
// SetBits is intended to support implementation of missing low-level Int
|
||
// functionality outside this package; it should be avoided otherwise.
|
||
func (z *Int) SetBits(abs []Word) *Int {
|
||
z.abs = nat(abs).norm()
|
||
z.neg = false
|
||
return z
|
||
}
|
||
|
||
// Abs sets z to |x| (the absolute value of x) and returns z.
|
||
func (z *Int) Abs(x *Int) *Int {
|
||
z.Set(x)
|
||
z.neg = false
|
||
return z
|
||
}
|
||
|
||
// Neg sets z to -x and returns z.
|
||
func (z *Int) Neg(x *Int) *Int {
|
||
z.Set(x)
|
||
z.neg = len(z.abs) > 0 && !z.neg // 0 has no sign
|
||
return z
|
||
}
|
||
|
||
// Add sets z to the sum x+y and returns z.
|
||
func (z *Int) Add(x, y *Int) *Int {
|
||
neg := x.neg
|
||
if x.neg == y.neg {
|
||
// x + y == x + y
|
||
// (-x) + (-y) == -(x + y)
|
||
z.abs = z.abs.add(x.abs, y.abs)
|
||
} else {
|
||
// x + (-y) == x - y == -(y - x)
|
||
// (-x) + y == y - x == -(x - y)
|
||
if x.abs.cmp(y.abs) >= 0 {
|
||
z.abs = z.abs.sub(x.abs, y.abs)
|
||
} else {
|
||
neg = !neg
|
||
z.abs = z.abs.sub(y.abs, x.abs)
|
||
}
|
||
}
|
||
z.neg = len(z.abs) > 0 && neg // 0 has no sign
|
||
return z
|
||
}
|
||
|
||
// Sub sets z to the difference x-y and returns z.
|
||
func (z *Int) Sub(x, y *Int) *Int {
|
||
neg := x.neg
|
||
if x.neg != y.neg {
|
||
// x - (-y) == x + y
|
||
// (-x) - y == -(x + y)
|
||
z.abs = z.abs.add(x.abs, y.abs)
|
||
} else {
|
||
// x - y == x - y == -(y - x)
|
||
// (-x) - (-y) == y - x == -(x - y)
|
||
if x.abs.cmp(y.abs) >= 0 {
|
||
z.abs = z.abs.sub(x.abs, y.abs)
|
||
} else {
|
||
neg = !neg
|
||
z.abs = z.abs.sub(y.abs, x.abs)
|
||
}
|
||
}
|
||
z.neg = len(z.abs) > 0 && neg // 0 has no sign
|
||
return z
|
||
}
|
||
|
||
// Mul sets z to the product x*y and returns z.
|
||
func (z *Int) Mul(x, y *Int) *Int {
|
||
// x * y == x * y
|
||
// x * (-y) == -(x * y)
|
||
// (-x) * y == -(x * y)
|
||
// (-x) * (-y) == x * y
|
||
if x == y {
|
||
z.abs = z.abs.sqr(x.abs)
|
||
z.neg = false
|
||
return z
|
||
}
|
||
z.abs = z.abs.mul(x.abs, y.abs)
|
||
z.neg = len(z.abs) > 0 && x.neg != y.neg // 0 has no sign
|
||
return z
|
||
}
|
||
|
||
// MulRange sets z to the product of all integers
|
||
// in the range [a, b] inclusively and returns z.
|
||
// If a > b (empty range), the result is 1.
|
||
func (z *Int) MulRange(a, b int64) *Int {
|
||
switch {
|
||
case a > b:
|
||
return z.SetInt64(1) // empty range
|
||
case a <= 0 && b >= 0:
|
||
return z.SetInt64(0) // range includes 0
|
||
}
|
||
// a <= b && (b < 0 || a > 0)
|
||
|
||
neg := false
|
||
if a < 0 {
|
||
neg = (b-a)&1 == 0
|
||
a, b = -b, -a
|
||
}
|
||
|
||
z.abs = z.abs.mulRange(uint64(a), uint64(b))
|
||
z.neg = neg
|
||
return z
|
||
}
|
||
|
||
// Binomial sets z to the binomial coefficient of (n, k) and returns z.
|
||
func (z *Int) Binomial(n, k int64) *Int {
|
||
// reduce the number of multiplications by reducing k
|
||
if n/2 < k && k <= n {
|
||
k = n - k // Binomial(n, k) == Binomial(n, n-k)
|
||
}
|
||
var a, b Int
|
||
a.MulRange(n-k+1, n)
|
||
b.MulRange(1, k)
|
||
return z.Quo(&a, &b)
|
||
}
|
||
|
||
// Quo sets z to the quotient x/y for y != 0 and returns z.
|
||
// If y == 0, a division-by-zero run-time panic occurs.
|
||
// Quo implements truncated division (like Go); see QuoRem for more details.
|
||
func (z *Int) Quo(x, y *Int) *Int {
|
||
z.abs, _ = z.abs.div(nil, x.abs, y.abs)
|
||
z.neg = len(z.abs) > 0 && x.neg != y.neg // 0 has no sign
|
||
return z
|
||
}
|
||
|
||
// Rem sets z to the remainder x%y for y != 0 and returns z.
|
||
// If y == 0, a division-by-zero run-time panic occurs.
|
||
// Rem implements truncated modulus (like Go); see QuoRem for more details.
|
||
func (z *Int) Rem(x, y *Int) *Int {
|
||
_, z.abs = nat(nil).div(z.abs, x.abs, y.abs)
|
||
z.neg = len(z.abs) > 0 && x.neg // 0 has no sign
|
||
return z
|
||
}
|
||
|
||
// QuoRem sets z to the quotient x/y and r to the remainder x%y
|
||
// and returns the pair (z, r) for y != 0.
|
||
// If y == 0, a division-by-zero run-time panic occurs.
|
||
//
|
||
// QuoRem implements T-division and modulus (like Go):
|
||
//
|
||
// q = x/y with the result truncated to zero
|
||
// r = x - y*q
|
||
//
|
||
// (See Daan Leijen, ``Division and Modulus for Computer Scientists''.)
|
||
// See DivMod for Euclidean division and modulus (unlike Go).
|
||
//
|
||
func (z *Int) QuoRem(x, y, r *Int) (*Int, *Int) {
|
||
z.abs, r.abs = z.abs.div(r.abs, x.abs, y.abs)
|
||
z.neg, r.neg = len(z.abs) > 0 && x.neg != y.neg, len(r.abs) > 0 && x.neg // 0 has no sign
|
||
return z, r
|
||
}
|
||
|
||
// Div sets z to the quotient x/y for y != 0 and returns z.
|
||
// If y == 0, a division-by-zero run-time panic occurs.
|
||
// Div implements Euclidean division (unlike Go); see DivMod for more details.
|
||
func (z *Int) Div(x, y *Int) *Int {
|
||
y_neg := y.neg // z may be an alias for y
|
||
var r Int
|
||
z.QuoRem(x, y, &r)
|
||
if r.neg {
|
||
if y_neg {
|
||
z.Add(z, intOne)
|
||
} else {
|
||
z.Sub(z, intOne)
|
||
}
|
||
}
|
||
return z
|
||
}
|
||
|
||
// Mod sets z to the modulus x%y for y != 0 and returns z.
|
||
// If y == 0, a division-by-zero run-time panic occurs.
|
||
// Mod implements Euclidean modulus (unlike Go); see DivMod for more details.
|
||
func (z *Int) Mod(x, y *Int) *Int {
|
||
y0 := y // save y
|
||
if z == y || alias(z.abs, y.abs) {
|
||
y0 = new(Int).Set(y)
|
||
}
|
||
var q Int
|
||
q.QuoRem(x, y, z)
|
||
if z.neg {
|
||
if y0.neg {
|
||
z.Sub(z, y0)
|
||
} else {
|
||
z.Add(z, y0)
|
||
}
|
||
}
|
||
return z
|
||
}
|
||
|
||
// DivMod sets z to the quotient x div y and m to the modulus x mod y
|
||
// and returns the pair (z, m) for y != 0.
|
||
// If y == 0, a division-by-zero run-time panic occurs.
|
||
//
|
||
// DivMod implements Euclidean division and modulus (unlike Go):
|
||
//
|
||
// q = x div y such that
|
||
// m = x - y*q with 0 <= m < |y|
|
||
//
|
||
// (See Raymond T. Boute, ``The Euclidean definition of the functions
|
||
// div and mod''. ACM Transactions on Programming Languages and
|
||
// Systems (TOPLAS), 14(2):127-144, New York, NY, USA, 4/1992.
|
||
// ACM press.)
|
||
// See QuoRem for T-division and modulus (like Go).
|
||
//
|
||
func (z *Int) DivMod(x, y, m *Int) (*Int, *Int) {
|
||
y0 := y // save y
|
||
if z == y || alias(z.abs, y.abs) {
|
||
y0 = new(Int).Set(y)
|
||
}
|
||
z.QuoRem(x, y, m)
|
||
if m.neg {
|
||
if y0.neg {
|
||
z.Add(z, intOne)
|
||
m.Sub(m, y0)
|
||
} else {
|
||
z.Sub(z, intOne)
|
||
m.Add(m, y0)
|
||
}
|
||
}
|
||
return z, m
|
||
}
|
||
|
||
// Cmp compares x and y and returns:
|
||
//
|
||
// -1 if x < y
|
||
// 0 if x == y
|
||
// +1 if x > y
|
||
//
|
||
func (x *Int) Cmp(y *Int) (r int) {
|
||
// x cmp y == x cmp y
|
||
// x cmp (-y) == x
|
||
// (-x) cmp y == y
|
||
// (-x) cmp (-y) == -(x cmp y)
|
||
switch {
|
||
case x == y:
|
||
// nothing to do
|
||
case x.neg == y.neg:
|
||
r = x.abs.cmp(y.abs)
|
||
if x.neg {
|
||
r = -r
|
||
}
|
||
case x.neg:
|
||
r = -1
|
||
default:
|
||
r = 1
|
||
}
|
||
return
|
||
}
|
||
|
||
// CmpAbs compares the absolute values of x and y and returns:
|
||
//
|
||
// -1 if |x| < |y|
|
||
// 0 if |x| == |y|
|
||
// +1 if |x| > |y|
|
||
//
|
||
func (x *Int) CmpAbs(y *Int) int {
|
||
return x.abs.cmp(y.abs)
|
||
}
|
||
|
||
// low32 returns the least significant 32 bits of x.
|
||
func low32(x nat) uint32 {
|
||
if len(x) == 0 {
|
||
return 0
|
||
}
|
||
return uint32(x[0])
|
||
}
|
||
|
||
// low64 returns the least significant 64 bits of x.
|
||
func low64(x nat) uint64 {
|
||
if len(x) == 0 {
|
||
return 0
|
||
}
|
||
v := uint64(x[0])
|
||
if _W == 32 && len(x) > 1 {
|
||
return uint64(x[1])<<32 | v
|
||
}
|
||
return v
|
||
}
|
||
|
||
// Int64 returns the int64 representation of x.
|
||
// If x cannot be represented in an int64, the result is undefined.
|
||
func (x *Int) Int64() int64 {
|
||
v := int64(low64(x.abs))
|
||
if x.neg {
|
||
v = -v
|
||
}
|
||
return v
|
||
}
|
||
|
||
// Uint64 returns the uint64 representation of x.
|
||
// If x cannot be represented in a uint64, the result is undefined.
|
||
func (x *Int) Uint64() uint64 {
|
||
return low64(x.abs)
|
||
}
|
||
|
||
// IsInt64 reports whether x can be represented as an int64.
|
||
func (x *Int) IsInt64() bool {
|
||
if len(x.abs) <= 64/_W {
|
||
w := int64(low64(x.abs))
|
||
return w >= 0 || x.neg && w == -w
|
||
}
|
||
return false
|
||
}
|
||
|
||
// IsUint64 reports whether x can be represented as a uint64.
|
||
func (x *Int) IsUint64() bool {
|
||
return !x.neg && len(x.abs) <= 64/_W
|
||
}
|
||
|
||
// SetString sets z to the value of s, interpreted in the given base,
|
||
// and returns z and a boolean indicating success. The entire string
|
||
// (not just a prefix) must be valid for success. If SetString fails,
|
||
// the value of z is undefined but the returned value is nil.
|
||
//
|
||
// The base argument must be 0 or a value between 2 and MaxBase.
|
||
// For base 0, the number prefix determines the actual base: A prefix of
|
||
// ``0b'' or ``0B'' selects base 2, ``0'', ``0o'' or ``0O'' selects base 8,
|
||
// and ``0x'' or ``0X'' selects base 16. Otherwise, the selected base is 10
|
||
// and no prefix is accepted.
|
||
//
|
||
// For bases <= 36, lower and upper case letters are considered the same:
|
||
// The letters 'a' to 'z' and 'A' to 'Z' represent digit values 10 to 35.
|
||
// For bases > 36, the upper case letters 'A' to 'Z' represent the digit
|
||
// values 36 to 61.
|
||
//
|
||
// For base 0, an underscore character ``_'' may appear between a base
|
||
// prefix and an adjacent digit, and between successive digits; such
|
||
// underscores do not change the value of the number.
|
||
// Incorrect placement of underscores is reported as an error if there
|
||
// are no other errors. If base != 0, underscores are not recognized
|
||
// and act like any other character that is not a valid digit.
|
||
//
|
||
func (z *Int) SetString(s string, base int) (*Int, bool) {
|
||
return z.setFromScanner(strings.NewReader(s), base)
|
||
}
|
||
|
||
// setFromScanner implements SetString given an io.ByteScanner.
|
||
// For documentation see comments of SetString.
|
||
func (z *Int) setFromScanner(r io.ByteScanner, base int) (*Int, bool) {
|
||
if _, _, err := z.scan(r, base); err != nil {
|
||
return nil, false
|
||
}
|
||
// entire content must have been consumed
|
||
if _, err := r.ReadByte(); err != io.EOF {
|
||
return nil, false
|
||
}
|
||
return z, true // err == io.EOF => scan consumed all content of r
|
||
}
|
||
|
||
// SetBytes interprets buf as the bytes of a big-endian unsigned
|
||
// integer, sets z to that value, and returns z.
|
||
func (z *Int) SetBytes(buf []byte) *Int {
|
||
z.abs = z.abs.setBytes(buf)
|
||
z.neg = false
|
||
return z
|
||
}
|
||
|
||
// Bytes returns the absolute value of x as a big-endian byte slice.
|
||
//
|
||
// To use a fixed length slice, or a preallocated one, use FillBytes.
|
||
func (x *Int) Bytes() []byte {
|
||
buf := make([]byte, len(x.abs)*_S)
|
||
return buf[x.abs.bytes(buf):]
|
||
}
|
||
|
||
// FillBytes sets buf to the absolute value of x, storing it as a zero-extended
|
||
// big-endian byte slice, and returns buf.
|
||
//
|
||
// If the absolute value of x doesn't fit in buf, FillBytes will panic.
|
||
func (x *Int) FillBytes(buf []byte) []byte {
|
||
// Clear whole buffer. (This gets optimized into a memclr.)
|
||
for i := range buf {
|
||
buf[i] = 0
|
||
}
|
||
x.abs.bytes(buf)
|
||
return buf
|
||
}
|
||
|
||
// BitLen returns the length of the absolute value of x in bits.
|
||
// The bit length of 0 is 0.
|
||
func (x *Int) BitLen() int {
|
||
return x.abs.bitLen()
|
||
}
|
||
|
||
// TrailingZeroBits returns the number of consecutive least significant zero
|
||
// bits of |x|.
|
||
func (x *Int) TrailingZeroBits() uint {
|
||
return x.abs.trailingZeroBits()
|
||
}
|
||
|
||
// Exp sets z = x**y mod |m| (i.e. the sign of m is ignored), and returns z.
|
||
// If m == nil or m == 0, z = x**y unless y <= 0 then z = 1. If m != 0, y < 0,
|
||
// and x and m are not relatively prime, z is unchanged and nil is returned.
|
||
//
|
||
// Modular exponentiation of inputs of a particular size is not a
|
||
// cryptographically constant-time operation.
|
||
func (z *Int) Exp(x, y, m *Int) *Int {
|
||
// See Knuth, volume 2, section 4.6.3.
|
||
xWords := x.abs
|
||
if y.neg {
|
||
if m == nil || len(m.abs) == 0 {
|
||
return z.SetInt64(1)
|
||
}
|
||
// for y < 0: x**y mod m == (x**(-1))**|y| mod m
|
||
inverse := new(Int).ModInverse(x, m)
|
||
if inverse == nil {
|
||
return nil
|
||
}
|
||
xWords = inverse.abs
|
||
}
|
||
yWords := y.abs
|
||
|
||
var mWords nat
|
||
if m != nil {
|
||
mWords = m.abs // m.abs may be nil for m == 0
|
||
}
|
||
|
||
z.abs = z.abs.expNN(xWords, yWords, mWords)
|
||
z.neg = len(z.abs) > 0 && x.neg && len(yWords) > 0 && yWords[0]&1 == 1 // 0 has no sign
|
||
if z.neg && len(mWords) > 0 {
|
||
// make modulus result positive
|
||
z.abs = z.abs.sub(mWords, z.abs) // z == x**y mod |m| && 0 <= z < |m|
|
||
z.neg = false
|
||
}
|
||
|
||
return z
|
||
}
|
||
|
||
// GCD sets z to the greatest common divisor of a and b and returns z.
|
||
// If x or y are not nil, GCD sets their value such that z = a*x + b*y.
|
||
//
|
||
// a and b may be positive, zero or negative. (Before Go 1.14 both had
|
||
// to be > 0.) Regardless of the signs of a and b, z is always >= 0.
|
||
//
|
||
// If a == b == 0, GCD sets z = x = y = 0.
|
||
//
|
||
// If a == 0 and b != 0, GCD sets z = |b|, x = 0, y = sign(b) * 1.
|
||
//
|
||
// If a != 0 and b == 0, GCD sets z = |a|, x = sign(a) * 1, y = 0.
|
||
func (z *Int) GCD(x, y, a, b *Int) *Int {
|
||
if len(a.abs) == 0 || len(b.abs) == 0 {
|
||
lenA, lenB, negA, negB := len(a.abs), len(b.abs), a.neg, b.neg
|
||
if lenA == 0 {
|
||
z.Set(b)
|
||
} else {
|
||
z.Set(a)
|
||
}
|
||
z.neg = false
|
||
if x != nil {
|
||
if lenA == 0 {
|
||
x.SetUint64(0)
|
||
} else {
|
||
x.SetUint64(1)
|
||
x.neg = negA
|
||
}
|
||
}
|
||
if y != nil {
|
||
if lenB == 0 {
|
||
y.SetUint64(0)
|
||
} else {
|
||
y.SetUint64(1)
|
||
y.neg = negB
|
||
}
|
||
}
|
||
return z
|
||
}
|
||
|
||
return z.lehmerGCD(x, y, a, b)
|
||
}
|
||
|
||
// lehmerSimulate attempts to simulate several Euclidean update steps
|
||
// using the leading digits of A and B. It returns u0, u1, v0, v1
|
||
// such that A and B can be updated as:
|
||
// A = u0*A + v0*B
|
||
// B = u1*A + v1*B
|
||
// Requirements: A >= B and len(B.abs) >= 2
|
||
// Since we are calculating with full words to avoid overflow,
|
||
// we use 'even' to track the sign of the cosequences.
|
||
// For even iterations: u0, v1 >= 0 && u1, v0 <= 0
|
||
// For odd iterations: u0, v1 <= 0 && u1, v0 >= 0
|
||
func lehmerSimulate(A, B *Int) (u0, u1, v0, v1 Word, even bool) {
|
||
// initialize the digits
|
||
var a1, a2, u2, v2 Word
|
||
|
||
m := len(B.abs) // m >= 2
|
||
n := len(A.abs) // n >= m >= 2
|
||
|
||
// extract the top Word of bits from A and B
|
||
h := nlz(A.abs[n-1])
|
||
a1 = A.abs[n-1]<<h | A.abs[n-2]>>(_W-h)
|
||
// B may have implicit zero words in the high bits if the lengths differ
|
||
switch {
|
||
case n == m:
|
||
a2 = B.abs[n-1]<<h | B.abs[n-2]>>(_W-h)
|
||
case n == m+1:
|
||
a2 = B.abs[n-2] >> (_W - h)
|
||
default:
|
||
a2 = 0
|
||
}
|
||
|
||
// Since we are calculating with full words to avoid overflow,
|
||
// we use 'even' to track the sign of the cosequences.
|
||
// For even iterations: u0, v1 >= 0 && u1, v0 <= 0
|
||
// For odd iterations: u0, v1 <= 0 && u1, v0 >= 0
|
||
// The first iteration starts with k=1 (odd).
|
||
even = false
|
||
// variables to track the cosequences
|
||
u0, u1, u2 = 0, 1, 0
|
||
v0, v1, v2 = 0, 0, 1
|
||
|
||
// Calculate the quotient and cosequences using Collins' stopping condition.
|
||
// Note that overflow of a Word is not possible when computing the remainder
|
||
// sequence and cosequences since the cosequence size is bounded by the input size.
|
||
// See section 4.2 of Jebelean for details.
|
||
for a2 >= v2 && a1-a2 >= v1+v2 {
|
||
q, r := a1/a2, a1%a2
|
||
a1, a2 = a2, r
|
||
u0, u1, u2 = u1, u2, u1+q*u2
|
||
v0, v1, v2 = v1, v2, v1+q*v2
|
||
even = !even
|
||
}
|
||
return
|
||
}
|
||
|
||
// lehmerUpdate updates the inputs A and B such that:
|
||
// A = u0*A + v0*B
|
||
// B = u1*A + v1*B
|
||
// where the signs of u0, u1, v0, v1 are given by even
|
||
// For even == true: u0, v1 >= 0 && u1, v0 <= 0
|
||
// For even == false: u0, v1 <= 0 && u1, v0 >= 0
|
||
// q, r, s, t are temporary variables to avoid allocations in the multiplication
|
||
func lehmerUpdate(A, B, q, r, s, t *Int, u0, u1, v0, v1 Word, even bool) {
|
||
|
||
t.abs = t.abs.setWord(u0)
|
||
s.abs = s.abs.setWord(v0)
|
||
t.neg = !even
|
||
s.neg = even
|
||
|
||
t.Mul(A, t)
|
||
s.Mul(B, s)
|
||
|
||
r.abs = r.abs.setWord(u1)
|
||
q.abs = q.abs.setWord(v1)
|
||
r.neg = even
|
||
q.neg = !even
|
||
|
||
r.Mul(A, r)
|
||
q.Mul(B, q)
|
||
|
||
A.Add(t, s)
|
||
B.Add(r, q)
|
||
}
|
||
|
||
// euclidUpdate performs a single step of the Euclidean GCD algorithm
|
||
// if extended is true, it also updates the cosequence Ua, Ub
|
||
func euclidUpdate(A, B, Ua, Ub, q, r, s, t *Int, extended bool) {
|
||
q, r = q.QuoRem(A, B, r)
|
||
|
||
*A, *B, *r = *B, *r, *A
|
||
|
||
if extended {
|
||
// Ua, Ub = Ub, Ua - q*Ub
|
||
t.Set(Ub)
|
||
s.Mul(Ub, q)
|
||
Ub.Sub(Ua, s)
|
||
Ua.Set(t)
|
||
}
|
||
}
|
||
|
||
// lehmerGCD sets z to the greatest common divisor of a and b,
|
||
// which both must be != 0, and returns z.
|
||
// If x or y are not nil, their values are set such that z = a*x + b*y.
|
||
// See Knuth, The Art of Computer Programming, Vol. 2, Section 4.5.2, Algorithm L.
|
||
// This implementation uses the improved condition by Collins requiring only one
|
||
// quotient and avoiding the possibility of single Word overflow.
|
||
// See Jebelean, "Improving the multiprecision Euclidean algorithm",
|
||
// Design and Implementation of Symbolic Computation Systems, pp 45-58.
|
||
// The cosequences are updated according to Algorithm 10.45 from
|
||
// Cohen et al. "Handbook of Elliptic and Hyperelliptic Curve Cryptography" pp 192.
|
||
func (z *Int) lehmerGCD(x, y, a, b *Int) *Int {
|
||
var A, B, Ua, Ub *Int
|
||
|
||
A = new(Int).Abs(a)
|
||
B = new(Int).Abs(b)
|
||
|
||
extended := x != nil || y != nil
|
||
|
||
if extended {
|
||
// Ua (Ub) tracks how many times input a has been accumulated into A (B).
|
||
Ua = new(Int).SetInt64(1)
|
||
Ub = new(Int)
|
||
}
|
||
|
||
// temp variables for multiprecision update
|
||
q := new(Int)
|
||
r := new(Int)
|
||
s := new(Int)
|
||
t := new(Int)
|
||
|
||
// ensure A >= B
|
||
if A.abs.cmp(B.abs) < 0 {
|
||
A, B = B, A
|
||
Ub, Ua = Ua, Ub
|
||
}
|
||
|
||
// loop invariant A >= B
|
||
for len(B.abs) > 1 {
|
||
// Attempt to calculate in single-precision using leading words of A and B.
|
||
u0, u1, v0, v1, even := lehmerSimulate(A, B)
|
||
|
||
// multiprecision Step
|
||
if v0 != 0 {
|
||
// Simulate the effect of the single-precision steps using the cosequences.
|
||
// A = u0*A + v0*B
|
||
// B = u1*A + v1*B
|
||
lehmerUpdate(A, B, q, r, s, t, u0, u1, v0, v1, even)
|
||
|
||
if extended {
|
||
// Ua = u0*Ua + v0*Ub
|
||
// Ub = u1*Ua + v1*Ub
|
||
lehmerUpdate(Ua, Ub, q, r, s, t, u0, u1, v0, v1, even)
|
||
}
|
||
|
||
} else {
|
||
// Single-digit calculations failed to simulate any quotients.
|
||
// Do a standard Euclidean step.
|
||
euclidUpdate(A, B, Ua, Ub, q, r, s, t, extended)
|
||
}
|
||
}
|
||
|
||
if len(B.abs) > 0 {
|
||
// extended Euclidean algorithm base case if B is a single Word
|
||
if len(A.abs) > 1 {
|
||
// A is longer than a single Word, so one update is needed.
|
||
euclidUpdate(A, B, Ua, Ub, q, r, s, t, extended)
|
||
}
|
||
if len(B.abs) > 0 {
|
||
// A and B are both a single Word.
|
||
aWord, bWord := A.abs[0], B.abs[0]
|
||
if extended {
|
||
var ua, ub, va, vb Word
|
||
ua, ub = 1, 0
|
||
va, vb = 0, 1
|
||
even := true
|
||
for bWord != 0 {
|
||
q, r := aWord/bWord, aWord%bWord
|
||
aWord, bWord = bWord, r
|
||
ua, ub = ub, ua+q*ub
|
||
va, vb = vb, va+q*vb
|
||
even = !even
|
||
}
|
||
|
||
t.abs = t.abs.setWord(ua)
|
||
s.abs = s.abs.setWord(va)
|
||
t.neg = !even
|
||
s.neg = even
|
||
|
||
t.Mul(Ua, t)
|
||
s.Mul(Ub, s)
|
||
|
||
Ua.Add(t, s)
|
||
} else {
|
||
for bWord != 0 {
|
||
aWord, bWord = bWord, aWord%bWord
|
||
}
|
||
}
|
||
A.abs[0] = aWord
|
||
}
|
||
}
|
||
negA := a.neg
|
||
if y != nil {
|
||
// avoid aliasing b needed in the division below
|
||
if y == b {
|
||
B.Set(b)
|
||
} else {
|
||
B = b
|
||
}
|
||
// y = (z - a*x)/b
|
||
y.Mul(a, Ua) // y can safely alias a
|
||
if negA {
|
||
y.neg = !y.neg
|
||
}
|
||
y.Sub(A, y)
|
||
y.Div(y, B)
|
||
}
|
||
|
||
if x != nil {
|
||
*x = *Ua
|
||
if negA {
|
||
x.neg = !x.neg
|
||
}
|
||
}
|
||
|
||
*z = *A
|
||
|
||
return z
|
||
}
|
||
|
||
// Rand sets z to a pseudo-random number in [0, n) and returns z.
|
||
//
|
||
// As this uses the math/rand package, it must not be used for
|
||
// security-sensitive work. Use crypto/rand.Int instead.
|
||
func (z *Int) Rand(rnd *rand.Rand, n *Int) *Int {
|
||
z.neg = false
|
||
if n.neg || len(n.abs) == 0 {
|
||
z.abs = nil
|
||
return z
|
||
}
|
||
z.abs = z.abs.random(rnd, n.abs, n.abs.bitLen())
|
||
return z
|
||
}
|
||
|
||
// ModInverse sets z to the multiplicative inverse of g in the ring ℤ/nℤ
|
||
// and returns z. If g and n are not relatively prime, g has no multiplicative
|
||
// inverse in the ring ℤ/nℤ. In this case, z is unchanged and the return value
|
||
// is nil.
|
||
func (z *Int) ModInverse(g, n *Int) *Int {
|
||
// GCD expects parameters a and b to be > 0.
|
||
if n.neg {
|
||
var n2 Int
|
||
n = n2.Neg(n)
|
||
}
|
||
if g.neg {
|
||
var g2 Int
|
||
g = g2.Mod(g, n)
|
||
}
|
||
var d, x Int
|
||
d.GCD(&x, nil, g, n)
|
||
|
||
// if and only if d==1, g and n are relatively prime
|
||
if d.Cmp(intOne) != 0 {
|
||
return nil
|
||
}
|
||
|
||
// x and y are such that g*x + n*y = 1, therefore x is the inverse element,
|
||
// but it may be negative, so convert to the range 0 <= z < |n|
|
||
if x.neg {
|
||
z.Add(&x, n)
|
||
} else {
|
||
z.Set(&x)
|
||
}
|
||
return z
|
||
}
|
||
|
||
// Jacobi returns the Jacobi symbol (x/y), either +1, -1, or 0.
|
||
// The y argument must be an odd integer.
|
||
func Jacobi(x, y *Int) int {
|
||
if len(y.abs) == 0 || y.abs[0]&1 == 0 {
|
||
panic(fmt.Sprintf("big: invalid 2nd argument to Int.Jacobi: need odd integer but got %s", y))
|
||
}
|
||
|
||
// We use the formulation described in chapter 2, section 2.4,
|
||
// "The Yacas Book of Algorithms":
|
||
// http://yacas.sourceforge.net/Algo.book.pdf
|
||
|
||
var a, b, c Int
|
||
a.Set(x)
|
||
b.Set(y)
|
||
j := 1
|
||
|
||
if b.neg {
|
||
if a.neg {
|
||
j = -1
|
||
}
|
||
b.neg = false
|
||
}
|
||
|
||
for {
|
||
if b.Cmp(intOne) == 0 {
|
||
return j
|
||
}
|
||
if len(a.abs) == 0 {
|
||
return 0
|
||
}
|
||
a.Mod(&a, &b)
|
||
if len(a.abs) == 0 {
|
||
return 0
|
||
}
|
||
// a > 0
|
||
|
||
// handle factors of 2 in 'a'
|
||
s := a.abs.trailingZeroBits()
|
||
if s&1 != 0 {
|
||
bmod8 := b.abs[0] & 7
|
||
if bmod8 == 3 || bmod8 == 5 {
|
||
j = -j
|
||
}
|
||
}
|
||
c.Rsh(&a, s) // a = 2^s*c
|
||
|
||
// swap numerator and denominator
|
||
if b.abs[0]&3 == 3 && c.abs[0]&3 == 3 {
|
||
j = -j
|
||
}
|
||
a.Set(&b)
|
||
b.Set(&c)
|
||
}
|
||
}
|
||
|
||
// modSqrt3Mod4 uses the identity
|
||
// (a^((p+1)/4))^2 mod p
|
||
// == u^(p+1) mod p
|
||
// == u^2 mod p
|
||
// to calculate the square root of any quadratic residue mod p quickly for 3
|
||
// mod 4 primes.
|
||
func (z *Int) modSqrt3Mod4Prime(x, p *Int) *Int {
|
||
e := new(Int).Add(p, intOne) // e = p + 1
|
||
e.Rsh(e, 2) // e = (p + 1) / 4
|
||
z.Exp(x, e, p) // z = x^e mod p
|
||
return z
|
||
}
|
||
|
||
// modSqrt5Mod8 uses Atkin's observation that 2 is not a square mod p
|
||
// alpha == (2*a)^((p-5)/8) mod p
|
||
// beta == 2*a*alpha^2 mod p is a square root of -1
|
||
// b == a*alpha*(beta-1) mod p is a square root of a
|
||
// to calculate the square root of any quadratic residue mod p quickly for 5
|
||
// mod 8 primes.
|
||
func (z *Int) modSqrt5Mod8Prime(x, p *Int) *Int {
|
||
// p == 5 mod 8 implies p = e*8 + 5
|
||
// e is the quotient and 5 the remainder on division by 8
|
||
e := new(Int).Rsh(p, 3) // e = (p - 5) / 8
|
||
tx := new(Int).Lsh(x, 1) // tx = 2*x
|
||
alpha := new(Int).Exp(tx, e, p)
|
||
beta := new(Int).Mul(alpha, alpha)
|
||
beta.Mod(beta, p)
|
||
beta.Mul(beta, tx)
|
||
beta.Mod(beta, p)
|
||
beta.Sub(beta, intOne)
|
||
beta.Mul(beta, x)
|
||
beta.Mod(beta, p)
|
||
beta.Mul(beta, alpha)
|
||
z.Mod(beta, p)
|
||
return z
|
||
}
|
||
|
||
// modSqrtTonelliShanks uses the Tonelli-Shanks algorithm to find the square
|
||
// root of a quadratic residue modulo any prime.
|
||
func (z *Int) modSqrtTonelliShanks(x, p *Int) *Int {
|
||
// Break p-1 into s*2^e such that s is odd.
|
||
var s Int
|
||
s.Sub(p, intOne)
|
||
e := s.abs.trailingZeroBits()
|
||
s.Rsh(&s, e)
|
||
|
||
// find some non-square n
|
||
var n Int
|
||
n.SetInt64(2)
|
||
for Jacobi(&n, p) != -1 {
|
||
n.Add(&n, intOne)
|
||
}
|
||
|
||
// Core of the Tonelli-Shanks algorithm. Follows the description in
|
||
// section 6 of "Square roots from 1; 24, 51, 10 to Dan Shanks" by Ezra
|
||
// Brown:
|
||
// https://www.maa.org/sites/default/files/pdf/upload_library/22/Polya/07468342.di020786.02p0470a.pdf
|
||
var y, b, g, t Int
|
||
y.Add(&s, intOne)
|
||
y.Rsh(&y, 1)
|
||
y.Exp(x, &y, p) // y = x^((s+1)/2)
|
||
b.Exp(x, &s, p) // b = x^s
|
||
g.Exp(&n, &s, p) // g = n^s
|
||
r := e
|
||
for {
|
||
// find the least m such that ord_p(b) = 2^m
|
||
var m uint
|
||
t.Set(&b)
|
||
for t.Cmp(intOne) != 0 {
|
||
t.Mul(&t, &t).Mod(&t, p)
|
||
m++
|
||
}
|
||
|
||
if m == 0 {
|
||
return z.Set(&y)
|
||
}
|
||
|
||
t.SetInt64(0).SetBit(&t, int(r-m-1), 1).Exp(&g, &t, p)
|
||
// t = g^(2^(r-m-1)) mod p
|
||
g.Mul(&t, &t).Mod(&g, p) // g = g^(2^(r-m)) mod p
|
||
y.Mul(&y, &t).Mod(&y, p)
|
||
b.Mul(&b, &g).Mod(&b, p)
|
||
r = m
|
||
}
|
||
}
|
||
|
||
// ModSqrt sets z to a square root of x mod p if such a square root exists, and
|
||
// returns z. The modulus p must be an odd prime. If x is not a square mod p,
|
||
// ModSqrt leaves z unchanged and returns nil. This function panics if p is
|
||
// not an odd integer.
|
||
func (z *Int) ModSqrt(x, p *Int) *Int {
|
||
switch Jacobi(x, p) {
|
||
case -1:
|
||
return nil // x is not a square mod p
|
||
case 0:
|
||
return z.SetInt64(0) // sqrt(0) mod p = 0
|
||
case 1:
|
||
break
|
||
}
|
||
if x.neg || x.Cmp(p) >= 0 { // ensure 0 <= x < p
|
||
x = new(Int).Mod(x, p)
|
||
}
|
||
|
||
switch {
|
||
case p.abs[0]%4 == 3:
|
||
// Check whether p is 3 mod 4, and if so, use the faster algorithm.
|
||
return z.modSqrt3Mod4Prime(x, p)
|
||
case p.abs[0]%8 == 5:
|
||
// Check whether p is 5 mod 8, use Atkin's algorithm.
|
||
return z.modSqrt5Mod8Prime(x, p)
|
||
default:
|
||
// Otherwise, use Tonelli-Shanks.
|
||
return z.modSqrtTonelliShanks(x, p)
|
||
}
|
||
}
|
||
|
||
// Lsh sets z = x << n and returns z.
|
||
func (z *Int) Lsh(x *Int, n uint) *Int {
|
||
z.abs = z.abs.shl(x.abs, n)
|
||
z.neg = x.neg
|
||
return z
|
||
}
|
||
|
||
// Rsh sets z = x >> n and returns z.
|
||
func (z *Int) Rsh(x *Int, n uint) *Int {
|
||
if x.neg {
|
||
// (-x) >> s == ^(x-1) >> s == ^((x-1) >> s) == -(((x-1) >> s) + 1)
|
||
t := z.abs.sub(x.abs, natOne) // no underflow because |x| > 0
|
||
t = t.shr(t, n)
|
||
z.abs = t.add(t, natOne)
|
||
z.neg = true // z cannot be zero if x is negative
|
||
return z
|
||
}
|
||
|
||
z.abs = z.abs.shr(x.abs, n)
|
||
z.neg = false
|
||
return z
|
||
}
|
||
|
||
// Bit returns the value of the i'th bit of x. That is, it
|
||
// returns (x>>i)&1. The bit index i must be >= 0.
|
||
func (x *Int) Bit(i int) uint {
|
||
if i == 0 {
|
||
// optimization for common case: odd/even test of x
|
||
if len(x.abs) > 0 {
|
||
return uint(x.abs[0] & 1) // bit 0 is same for -x
|
||
}
|
||
return 0
|
||
}
|
||
if i < 0 {
|
||
panic("negative bit index")
|
||
}
|
||
if x.neg {
|
||
t := nat(nil).sub(x.abs, natOne)
|
||
return t.bit(uint(i)) ^ 1
|
||
}
|
||
|
||
return x.abs.bit(uint(i))
|
||
}
|
||
|
||
// SetBit sets z to x, with x's i'th bit set to b (0 or 1).
|
||
// That is, if b is 1 SetBit sets z = x | (1 << i);
|
||
// if b is 0 SetBit sets z = x &^ (1 << i). If b is not 0 or 1,
|
||
// SetBit will panic.
|
||
func (z *Int) SetBit(x *Int, i int, b uint) *Int {
|
||
if i < 0 {
|
||
panic("negative bit index")
|
||
}
|
||
if x.neg {
|
||
t := z.abs.sub(x.abs, natOne)
|
||
t = t.setBit(t, uint(i), b^1)
|
||
z.abs = t.add(t, natOne)
|
||
z.neg = len(z.abs) > 0
|
||
return z
|
||
}
|
||
z.abs = z.abs.setBit(x.abs, uint(i), b)
|
||
z.neg = false
|
||
return z
|
||
}
|
||
|
||
// And sets z = x & y and returns z.
|
||
func (z *Int) And(x, y *Int) *Int {
|
||
if x.neg == y.neg {
|
||
if x.neg {
|
||
// (-x) & (-y) == ^(x-1) & ^(y-1) == ^((x-1) | (y-1)) == -(((x-1) | (y-1)) + 1)
|
||
x1 := nat(nil).sub(x.abs, natOne)
|
||
y1 := nat(nil).sub(y.abs, natOne)
|
||
z.abs = z.abs.add(z.abs.or(x1, y1), natOne)
|
||
z.neg = true // z cannot be zero if x and y are negative
|
||
return z
|
||
}
|
||
|
||
// x & y == x & y
|
||
z.abs = z.abs.and(x.abs, y.abs)
|
||
z.neg = false
|
||
return z
|
||
}
|
||
|
||
// x.neg != y.neg
|
||
if x.neg {
|
||
x, y = y, x // & is symmetric
|
||
}
|
||
|
||
// x & (-y) == x & ^(y-1) == x &^ (y-1)
|
||
y1 := nat(nil).sub(y.abs, natOne)
|
||
z.abs = z.abs.andNot(x.abs, y1)
|
||
z.neg = false
|
||
return z
|
||
}
|
||
|
||
// AndNot sets z = x &^ y and returns z.
|
||
func (z *Int) AndNot(x, y *Int) *Int {
|
||
if x.neg == y.neg {
|
||
if x.neg {
|
||
// (-x) &^ (-y) == ^(x-1) &^ ^(y-1) == ^(x-1) & (y-1) == (y-1) &^ (x-1)
|
||
x1 := nat(nil).sub(x.abs, natOne)
|
||
y1 := nat(nil).sub(y.abs, natOne)
|
||
z.abs = z.abs.andNot(y1, x1)
|
||
z.neg = false
|
||
return z
|
||
}
|
||
|
||
// x &^ y == x &^ y
|
||
z.abs = z.abs.andNot(x.abs, y.abs)
|
||
z.neg = false
|
||
return z
|
||
}
|
||
|
||
if x.neg {
|
||
// (-x) &^ y == ^(x-1) &^ y == ^(x-1) & ^y == ^((x-1) | y) == -(((x-1) | y) + 1)
|
||
x1 := nat(nil).sub(x.abs, natOne)
|
||
z.abs = z.abs.add(z.abs.or(x1, y.abs), natOne)
|
||
z.neg = true // z cannot be zero if x is negative and y is positive
|
||
return z
|
||
}
|
||
|
||
// x &^ (-y) == x &^ ^(y-1) == x & (y-1)
|
||
y1 := nat(nil).sub(y.abs, natOne)
|
||
z.abs = z.abs.and(x.abs, y1)
|
||
z.neg = false
|
||
return z
|
||
}
|
||
|
||
// Or sets z = x | y and returns z.
|
||
func (z *Int) Or(x, y *Int) *Int {
|
||
if x.neg == y.neg {
|
||
if x.neg {
|
||
// (-x) | (-y) == ^(x-1) | ^(y-1) == ^((x-1) & (y-1)) == -(((x-1) & (y-1)) + 1)
|
||
x1 := nat(nil).sub(x.abs, natOne)
|
||
y1 := nat(nil).sub(y.abs, natOne)
|
||
z.abs = z.abs.add(z.abs.and(x1, y1), natOne)
|
||
z.neg = true // z cannot be zero if x and y are negative
|
||
return z
|
||
}
|
||
|
||
// x | y == x | y
|
||
z.abs = z.abs.or(x.abs, y.abs)
|
||
z.neg = false
|
||
return z
|
||
}
|
||
|
||
// x.neg != y.neg
|
||
if x.neg {
|
||
x, y = y, x // | is symmetric
|
||
}
|
||
|
||
// x | (-y) == x | ^(y-1) == ^((y-1) &^ x) == -(^((y-1) &^ x) + 1)
|
||
y1 := nat(nil).sub(y.abs, natOne)
|
||
z.abs = z.abs.add(z.abs.andNot(y1, x.abs), natOne)
|
||
z.neg = true // z cannot be zero if one of x or y is negative
|
||
return z
|
||
}
|
||
|
||
// Xor sets z = x ^ y and returns z.
|
||
func (z *Int) Xor(x, y *Int) *Int {
|
||
if x.neg == y.neg {
|
||
if x.neg {
|
||
// (-x) ^ (-y) == ^(x-1) ^ ^(y-1) == (x-1) ^ (y-1)
|
||
x1 := nat(nil).sub(x.abs, natOne)
|
||
y1 := nat(nil).sub(y.abs, natOne)
|
||
z.abs = z.abs.xor(x1, y1)
|
||
z.neg = false
|
||
return z
|
||
}
|
||
|
||
// x ^ y == x ^ y
|
||
z.abs = z.abs.xor(x.abs, y.abs)
|
||
z.neg = false
|
||
return z
|
||
}
|
||
|
||
// x.neg != y.neg
|
||
if x.neg {
|
||
x, y = y, x // ^ is symmetric
|
||
}
|
||
|
||
// x ^ (-y) == x ^ ^(y-1) == ^(x ^ (y-1)) == -((x ^ (y-1)) + 1)
|
||
y1 := nat(nil).sub(y.abs, natOne)
|
||
z.abs = z.abs.add(z.abs.xor(x.abs, y1), natOne)
|
||
z.neg = true // z cannot be zero if only one of x or y is negative
|
||
return z
|
||
}
|
||
|
||
// Not sets z = ^x and returns z.
|
||
func (z *Int) Not(x *Int) *Int {
|
||
if x.neg {
|
||
// ^(-x) == ^(^(x-1)) == x-1
|
||
z.abs = z.abs.sub(x.abs, natOne)
|
||
z.neg = false
|
||
return z
|
||
}
|
||
|
||
// ^x == -x-1 == -(x+1)
|
||
z.abs = z.abs.add(x.abs, natOne)
|
||
z.neg = true // z cannot be zero if x is positive
|
||
return z
|
||
}
|
||
|
||
// Sqrt sets z to ⌊√x⌋, the largest integer such that z² ≤ x, and returns z.
|
||
// It panics if x is negative.
|
||
func (z *Int) Sqrt(x *Int) *Int {
|
||
if x.neg {
|
||
panic("square root of negative number")
|
||
}
|
||
z.neg = false
|
||
z.abs = z.abs.sqrt(x.abs)
|
||
return z
|
||
}
|