golang int 代码
文件路径：/src/math/big/int.go
/ Copyright 2009 The Go Authors. All rights reserved.
/ Use of this source code is governed by a BSD-style
/ license that can be found in the LICENSE file.
/ This file implements signed multi-precision integers.
package big
import (
"fmt"
"io"
"math/rand"
"strings"
)
/ An Int represents a signed multi-precision integer.
/ The zero value for an Int represents the value 0.
/
/ Operations always take pointer arguments (*Int) rather
/ than Int values, and each unique Int value requires
/ its own unique *Int pointer. To "copy" an Int value,
/ an existing (or newly allocated) Int must be set to
/ a new value using the Int.Set method; shallow copies
/ of Ints are not supported and may lead to errors.
type Int struct {
neg bool / sign
abs nat / absolute value of the integer
}
var intOne = &Int{false, natOne}
/ Sign returns:
/
/ -1 if x < 0
/ 0 if x == 0
/ +1 if x > 0
func (x *Int) Sign() int {
if len(x.abs) == 0 {
return 0
}
if x.neg {
return -1
}
return 1
}
/ SetInt64 sets z to x and returns z.
func (z *Int) SetInt64(x int64) *Int {
neg := false
if x < 0 {
neg = true
x = -x
}
z.abs = z.abs.setUint64(uint64(x))
z.neg = neg
return z
}
/ SetUint64 sets z to x and returns z.
func (z *Int) SetUint64(x uint64) *Int {
z.abs = z.abs.setUint64(x)
z.neg = false
return z
}
/ NewInt allocates and returns a new Int set to x.
func NewInt(x int64) *Int {
return new(Int).SetInt64(x)
}
/ Set sets z to x and returns z.
func (z *Int) Set(x *Int) *Int {
if z != x {
z.abs = z.abs.set(x.abs)
z.neg = x.neg
}
return z
}
/ Bits provides raw (unchecked but fast) access to x by returning its
/ absolute value as a little-endian Word slice. The result and x share
/ the same underlying array.
/ Bits is intended to support implementation of missing low-level Int
/ functionality outside this package; it should be avoided otherwise.
func (x *Int) Bits() []Word {
return x.abs
}
/ SetBits provides raw (unchecked but fast) access to z by setting its
/ value to abs, interpreted as a little-endian Word slice, and returning
/ z. The result and abs share the same underlying array.
/ SetBits is intended to support implementation of missing low-level Int
/ functionality outside this package; it should be avoided otherwise.
func (z *Int) SetBits(abs []Word) *Int {
z.abs = nat(abs).norm()
z.neg = false
return z
}
/ Abs sets z to |x| (the absolute value of x) and returns z.
func (z *Int) Abs(x *Int) *Int {
z.Set(x)
z.neg = false
return z
}
/ Neg sets z to -x and returns z.
func (z *Int) Neg(x *Int) *Int {
z.Set(x)
z.neg = len(z.abs) > 0 && !z.neg / 0 has no sign
return z
}
/ Add sets z to the sum x+y and returns z.
func (z *Int) Add(x, y *Int) *Int {
neg := x.neg
if x.neg == y.neg {
/ x + y == x + y
/ (-x) + (-y) == -(x + y)
z.abs = z.abs.add(x.abs, y.abs)
} else {
/ x + (-y) == x - y == -(y - x)
/ (-x) + y == y - x == -(x - y)
if x.abs.cmp(y.abs) >= 0 {
z.abs = z.abs.sub(x.abs, y.abs)
} else {
neg = !neg
z.abs = z.abs.sub(y.abs, x.abs)
}
}
z.neg = len(z.abs) > 0 && neg / 0 has no sign
return z
}
/ Sub sets z to the difference x-y and returns z.
func (z *Int) Sub(x, y *Int) *Int {
neg := x.neg
if x.neg != y.neg {
/ x - (-y) == x + y
/ (-x) - y == -(x + y)
z.abs = z.abs.add(x.abs, y.abs)
} else {
/ x - y == x - y == -(y - x)
/ (-x) - (-y) == y - x == -(x - y)
if x.abs.cmp(y.abs) >= 0 {
z.abs = z.abs.sub(x.abs, y.abs)
} else {
neg = !neg
z.abs = z.abs.sub(y.abs, x.abs)
}
}
z.neg = len(z.abs) > 0 && neg / 0 has no sign
return z
}
/ Mul sets z to the product x*y and returns z.
func (z *Int) Mul(x, y *Int) *Int {
/ x * y == x * y
/ x * (-y) == -(x * y)
/ (-x) * y == -(x * y)
/ (-x) * (-y) == x * y
if x == y {
z.abs = z.abs.sqr(x.abs)
z.neg = false
return z
}
z.abs = z.abs.mul(x.abs, y.abs)
z.neg = len(z.abs) > 0 && x.neg != y.neg / 0 has no sign
return z
}
/ MulRange sets z to the product of all integers
/ in the range [a, b] inclusively and returns z.
/ If a > b (empty range), the result is 1.
func (z *Int) MulRange(a, b int64) *Int {
switch {
case a > b:
return z.SetInt64(1) / empty range
case a <= 0 && b >= 0:
return z.SetInt64(0) / range includes 0
}
/ a <= b && (b < 0 || a > 0)
neg := false
if a < 0 {
neg = (b-a)&1 == 0
a, b = -b, -a
}
z.abs = z.abs.mulRange(uint64(a), uint64(b))
z.neg = neg
return z
}
/ Binomial sets z to the binomial coefficient of (n, k) and returns z.
func (z *Int) Binomial(n, k int64) *Int {
/ reduce the number of multiplications by reducing k
if n/2 < k && k <= n {
k = n - k / Binomial(n, k) == Binomial(n, n-k)
}
var a, b Int
a.MulRange(n-k+1, n)
b.MulRange(1, k)
return z.Quo(&a, &b)
}
/ Quo sets z to the quotient x/y for y != 0 and returns z.
/ If y == 0, a division-by-zero run-time panic occurs.
/ Quo implements truncated division (like Go); see QuoRem for more details.
func (z *Int) Quo(x, y *Int) *Int {
z.abs, _ = z.abs.div(nil, x.abs, y.abs)
z.neg = len(z.abs) > 0 && x.neg != y.neg / 0 has no sign
return z
}
/ Rem sets z to the remainder x%y for y != 0 and returns z.
/ If y == 0, a division-by-zero run-time panic occurs.
/ Rem implements truncated modulus (like Go); see QuoRem for more details.
func (z *Int) Rem(x, y *Int) *Int {
_, z.abs = nat(nil).div(z.abs, x.abs, y.abs)
z.neg = len(z.abs) > 0 && x.neg / 0 has no sign
return z
}
/ QuoRem sets z to the quotient x/y and r to the remainder x%y
/ and returns the pair (z, r) for y != 0.
/ If y == 0, a division-by-zero run-time panic occurs.
/
/ QuoRem implements T-division and modulus (like Go):
/
/ q = x/y with the result truncated to zero
/ r = x - y*q
/
/ (See Daan Leijen, “Division and Modulus for Computer Scientists”.)
/ See DivMod for Euclidean division and modulus (unlike Go).
func (z *Int) QuoRem(x, y, r *Int) (*Int, *Int) {
z.abs, r.abs = z.abs.div(r.abs, x.abs, y.abs)
z.neg, r.neg = len(z.abs) > 0 && x.neg != y.neg, len(r.abs) > 0 && x.neg / 0 has no sign
return z, r
}
/ Div sets z to the quotient x/y for y != 0 and returns z.
/ If y == 0, a division-by-zero run-time panic occurs.
/ Div implements Euclidean division (unlike Go); see DivMod for more details.
func (z *Int) Div(x, y *Int) *Int {
y_neg := y.neg / z may be an alias for y
var r Int
z.QuoRem(x, y, &r)
if r.neg {
if y_neg {
z.Add(z, intOne)
} else {
z.Sub(z, intOne)
}
}
return z
}
/ Mod sets z to the modulus x%y for y != 0 and returns z.
/ If y == 0, a division-by-zero run-time panic occurs.
/ Mod implements Euclidean modulus (unlike Go); see DivMod for more details.
func (z *Int) Mod(x, y *Int) *Int {
y0 := y / save y
if z == y || alias(z.abs, y.abs) {
y0 = new(Int).Set(y)
}
var q Int
q.QuoRem(x, y, z)
if z.neg {
if y0.neg {
z.Sub(z, y0)
} else {
z.Add(z, y0)
}
}
return z
}
/ DivMod sets z to the quotient x div y and m to the modulus x mod y
/ and returns the pair (z, m) for y != 0.
/ If y == 0, a division-by-zero run-time panic occurs.
/
/ DivMod implements Euclidean division and modulus (unlike Go):
/
/ q = x div y such that
/ m = x - y*q with 0 <= m < |y|
/
/ (See Raymond T. Boute, “The Euclidean definition of the functions
/ div and mod”. ACM Transactions on Programming Languages and
/ Systems (TOPLAS), 14(2):127-144, New York, NY, USA, 4/1992.
/ ACM press.)
/ See QuoRem for T-division and modulus (like Go).
func (z *Int) DivMod(x, y, m *Int) (*Int, *Int) {
y0 := y / save y
if z == y || alias(z.abs, y.abs) {
y0 = new(Int).Set(y)
}
z.QuoRem(x, y, m)
if m.neg {
if y0.neg {
z.Add(z, intOne)
m.Sub(m, y0)
} else {
z.Sub(z, intOne)
m.Add(m, y0)
}
}
return z, m
}
/ Cmp compares x and y and returns:
/
/ -1 if x < y
/ 0 if x == y
/ +1 if x > y
func (x *Int) Cmp(y *Int) (r int) {
/ x cmp y == x cmp y
/ x cmp (-y) == x
/ (-x) cmp y == y
/ (-x) cmp (-y) == -(x cmp y)
switch {
case x == y:
/ nothing to do
case x.neg == y.neg:
r = x.abs.cmp(y.abs)
if x.neg {
r = -r
}
case x.neg:
r = -1
default:
r = 1
}
return
}
/ CmpAbs compares the absolute values of x and y and returns:
/
/ -1 if |x| < |y|
/ 0 if |x| == |y|
/ +1 if |x| > |y|
func (x *Int) CmpAbs(y *Int) int {
return x.abs.cmp(y.abs)
}
/ low32 returns the least significant 32 bits of x.
func low32(x nat) uint32 {
if len(x) == 0 {
return 0
}
return uint32(x[0])
}
/ low64 returns the least significant 64 bits of x.
func low64(x nat) uint64 {
if len(x) == 0 {
return 0
}
v := uint64(x[0])
if _W == 32 && len(x) > 1 {
return uint64(x[1])<<32 | v
}
return v
}
/ Int64 returns the int64 representation of x.
/ If x cannot be represented in an int64, the result is undefined.
func (x *Int) Int64() int64 {
v := int64(low64(x.abs))
if x.neg {
v = -v
}
return v
}
/ Uint64 returns the uint64 representation of x.
/ If x cannot be represented in a uint64, the result is undefined.
func (x *Int) Uint64() uint64 {
return low64(x.abs)
}
/ IsInt64 reports whether x can be represented as an int64.
func (x *Int) IsInt64() bool {
if len(x.abs) <= 64/_W {
w := int64(low64(x.abs))
return w >= 0 || x.neg && w == -w
}
return false
}
/ IsUint64 reports whether x can be represented as a uint64.
func (x *Int) IsUint64() bool {
return !x.neg && len(x.abs) <= 64/_W
}
/ SetString sets z to the value of s, interpreted in the given base,
/ and returns z and a boolean indicating success. The entire string
/ (not just a prefix) must be valid for success. If SetString fails,
/ the value of z is undefined but the returned value is nil.
/
/ The base argument must be 0 or a value between 2 and MaxBase.
/ For base 0, the number prefix determines the actual base: A prefix of
/ “0b” or “0B” selects base 2, “0”, “0o” or “0O” selects base 8,
/ and “0x” or “0X” selects base 16. Otherwise, the selected base is 10
/ and no prefix is accepted.
/
/ For bases <= 36, lower and upper case letters are considered the same:
/ The letters 'a' to 'z' and 'A' to 'Z' represent digit values 10 to 35.
/ For bases > 36, the upper case letters 'A' to 'Z' represent the digit
/ values 36 to 61.
/
/ For base 0, an underscore character “_” may appear between a base
/ prefix and an adjacent digit, and between successive digits; such
/ underscores do not change the value of the number.
/ Incorrect placement of underscores is reported as an error if there
/ are no other errors. If base != 0, underscores are not recognized
/ and act like any other character that is not a valid digit.
func (z *Int) SetString(s string, base int) (*Int, bool) {
return z.setFromScanner(strings.NewReader(s), base)
}
/ setFromScanner implements SetString given an io.ByteScanner.
/ For documentation see comments of SetString.
func (z *Int) setFromScanner(r io.ByteScanner, base int) (*Int, bool) {
if _, _, err := z.scan(r, base); err != nil {
return nil, false
}
/ entire content must have been consumed
if _, err := r.ReadByte(); err != io.EOF {
return nil, false
}
return z, true / err == io.EOF => scan consumed all content of r
}
/ SetBytes interprets buf as the bytes of a big-endian unsigned
/ integer, sets z to that value, and returns z.
func (z *Int) SetBytes(buf []byte) *Int {
z.abs = z.abs.setBytes(buf)
z.neg = false
return z
}
/ Bytes returns the absolute value of x as a big-endian byte slice.
/
/ To use a fixed length slice, or a preallocated one, use FillBytes.
func (x *Int) Bytes() []byte {
buf := make([]byte, len(x.abs)*_S)
return buf[x.abs.bytes(buf):]
}
/ FillBytes sets buf to the absolute value of x, storing it as a zero-extended
/ big-endian byte slice, and returns buf.
/
/ If the absolute value of x doesn't fit in buf, FillBytes will panic.
func (x *Int) FillBytes(buf []byte) []byte {
/ Clear whole buffer. (This gets optimized into a memclr.)
for i := range buf {
buf[i] = 0
}
x.abs.bytes(buf)
return buf
}
/ BitLen returns the length of the absolute value of x in bits.
/ The bit length of 0 is 0.
func (x *Int) BitLen() int {
return x.abs.bitLen()
}
/ TrailingZeroBits returns the number of consecutive least significant zero
/ bits of |x|.
func (x *Int) TrailingZeroBits() uint {
return x.abs.trailingZeroBits()
}
/ Exp sets z = x**y mod |m| (i.e. the sign of m is ignored), and returns z.
/ If m == nil or m == 0, z = x**y unless y <= 0 then z = 1. If m != 0, y < 0,
/ and x and m are not relatively prime, z is unchanged and nil is returned.
/
/ Modular exponentiation of inputs of a particular size is not a
/ cryptographically constant-time operation.
func (z *Int) Exp(x, y, m *Int) *Int {
/ See Knuth, volume 2, section 4.6.3.
xWords := x.abs
if y.neg {
if m == nil || len(m.abs) == 0 {
return z.SetInt64(1)
}
/ for y < 0: x**y mod m == (x**(-1))**|y| mod m
inverse := new(Int).ModInverse(x, m)
if inverse == nil {
return nil
}
xWords = inverse.abs
}
yWords := y.abs
var mWords nat
if m != nil {
if z == m || alias(z.abs, m.abs) {
m = new(Int).Set(m)
}
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 is not modified before the if check, because z and n might alias.
if n.neg || len(n.abs) == 0 {
z.neg = false
z.abs = nil
return z
}
z.neg = false
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. If n == 0, a division-by-zero run-time panic occurs.
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.String()))
}
/ 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, its behavior is undefined if p is odd but not prime.
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
}