github.com/holiman/uint256 源码阅读
// uint256: Fixed size 256-bit math library
// Copyright 2018-2020 uint256 Authors
// SPDX-License-Identifier: BSD-3-Clause
// Package math provides integer math utilities.
package uint256
import (
"encoding/binary"
"math"
"math/big"
"math/bits"
)
// Int is represented as an array of 4 uint64, in little-endian order,
// so that Int[3] is the most significant, and Int[0] is the least significant
type Int [4]uint64 //Int定义为一个长度为4的uint64数组. 所以一共4*64位无符号.
// 0是最低位, 3是最高位.
// NewInt returns a new initialized Int.
func NewInt(val uint64) *Int {
z := &Int{}
z.SetUint64(val)
return z
}
// SetBytes interprets buf as the bytes of a big-endian unsigned
// integer, sets z to that value, and returns z.
// If buf is larger than 32 bytes, the last 32 bytes is used. This operation
// is semantically equivalent to `FromBig(new(big.Int).SetBytes(buf))`
func (z *Int) SetBytes(buf []byte) *Int {
switch l := len(buf); l {
case 0:
z.Clear()
case 1: //如果buff字符串的长度是1,
z.SetBytes1(buf)
case 2:
z.SetBytes2(buf)
case 3:
z.SetBytes3(buf)
case 4:
z.SetBytes4(buf)
case 5:
z.SetBytes5(buf)
case 6:
z.SetBytes6(buf)
case 7:
z.SetBytes7(buf)
case 8:
z.SetBytes8(buf)
case 9:
z.SetBytes9(buf)
case 10:
z.SetBytes10(buf)
case 11:
z.SetBytes11(buf)
case 12:
z.SetBytes12(buf)
case 13:
z.SetBytes13(buf)
case 14:
z.SetBytes14(buf)
case 15:
z.SetBytes15(buf)
case 16:
z.SetBytes16(buf)
case 17:
z.SetBytes17(buf)
case 18:
z.SetBytes18(buf)
case 19:
z.SetBytes19(buf)
case 20:
z.SetBytes20(buf)
case 21:
z.SetBytes21(buf)
case 22:
z.SetBytes22(buf)
case 23:
z.SetBytes23(buf)
case 24:
z.SetBytes24(buf)
case 25:
z.SetBytes25(buf)
case 26:
z.SetBytes26(buf)
case 27:
z.SetBytes27(buf)
case 28:
z.SetBytes28(buf)
case 29:
z.SetBytes29(buf)
case 30:
z.SetBytes30(buf)
case 31:
z.SetBytes31(buf)
default:
z.SetBytes32(buf[l-32:])
}
return z
}
// Bytes32 returns the value of z as a 32-byte big-endian array.
func (z *Int) Bytes32() [32]byte {
// The PutUint64()s are inlined and we get 4x (load, bswap, store) instructions.
var b [32]byte
binary.BigEndian.PutUint64(b[0:8], z[3])
binary.BigEndian.PutUint64(b[8:16], z[2])
binary.BigEndian.PutUint64(b[16:24], z[1])
binary.BigEndian.PutUint64(b[24:32], z[0])
return b
}
// Bytes20 returns the value of z as a 20-byte big-endian array.
func (z *Int) Bytes20() [20]byte {
var b [20]byte
// The PutUint*()s are inlined and we get 3x (load, bswap, store) instructions.
binary.BigEndian.PutUint32(b[0:4], uint32(z[2]))
binary.BigEndian.PutUint64(b[4:12], z[1])
binary.BigEndian.PutUint64(b[12:20], z[0])
return b
}
// Bytes returns the value of z as a big-endian byte slice.
func (z *Int) Bytes() []byte {
b := z.Bytes32()
return b[32-z.ByteLen():]
}
// WriteToSlice writes the content of z into the given byteslice.
// If dest is larger than 32 bytes, z will fill the first parts, and leave
// the end untouched.
// OBS! If dest is smaller than 32 bytes, only the end parts of z will be used
// for filling the array, making it useful for filling an Address object
func (z *Int) WriteToSlice(dest []byte) {
// ensure 32 bytes
// A too large buffer. Fill last 32 bytes
end := len(dest) - 1
if end > 31 {
end = 31
}
for i := 0; i <= end; i++ {
dest[end-i] = byte(z[i/8] >> uint64(8*(i%8)))
}
}
// WriteToArray32 writes all 32 bytes of z to the destination array, including zero-bytes
func (z *Int) WriteToArray32(dest *[32]byte) {
for i := 0; i < 32; i++ {
dest[31-i] = byte(z[i/8] >> uint64(8*(i%8)))
}
}
// WriteToArray20 writes the last 20 bytes of z to the destination array, including zero-bytes
func (z *Int) WriteToArray20(dest *[20]byte) {
for i := 0; i < 20; i++ {
dest[19-i] = byte(z[i/8] >> uint64(8*(i%8)))
}
}
// Uint64 returns the lower 64-bits of z
func (z *Int) Uint64() uint64 {
return z[0]
}
// Uint64WithOverflow returns the lower 64-bits of z and bool whether overflow occurred
func (z *Int) Uint64WithOverflow() (uint64, bool) {
return z[0], (z[1] | z[2] | z[3]) != 0
}
// Clone creates a new Int identical to z
func (z *Int) Clone() *Int {
return &Int{z[0], z[1], z[2], z[3]}
}
// Add sets z to the sum x+y
func (z *Int) Add(x, y *Int) *Int {
var carry uint64
z[0], carry = bits.Add64(x[0], y[0], 0) //carry 表示是否overflow
z[1], carry = bits.Add64(x[1], y[1], carry)
z[2], carry = bits.Add64(x[2], y[2], carry)
z[3], _ = bits.Add64(x[3], y[3], carry)
return z
}
// AddOverflow sets z to the sum x+y, and returns z and whether overflow occurred
func (z *Int) AddOverflow(x, y *Int) (*Int, bool) {
var carry uint64
z[0], carry = bits.Add64(x[0], y[0], 0)
z[1], carry = bits.Add64(x[1], y[1], carry)
z[2], carry = bits.Add64(x[2], y[2], carry)
z[3], carry = bits.Add64(x[3], y[3], carry)
return z, carry != 0
}
// AddMod sets z to the sum ( x+y ) mod m, and returns z.
// If m == 0, z is set to 0 (OBS: differs from the big.Int)
func (z *Int) AddMod(x, y, m *Int) *Int {
// Fast path for m >= 2^192, with x and y at most slightly bigger than m.
// This is always the case when x and y are already reduced modulo such m.
if (m[3] != 0) && (x[3] <= m[3]) && (y[3] <= m[3]) {
var (
gteC1 uint64
gteC2 uint64
tmpX Int
tmpY Int
res Int
)
// reduce x/y modulo m if they are gte m
tmpX[0], gteC1 = bits.Sub64(x[0], m[0], gteC1)
tmpX[1], gteC1 = bits.Sub64(x[1], m[1], gteC1)
tmpX[2], gteC1 = bits.Sub64(x[2], m[2], gteC1)
tmpX[3], gteC1 = bits.Sub64(x[3], m[3], gteC1)
tmpY[0], gteC2 = bits.Sub64(y[0], m[0], gteC2)
tmpY[1], gteC2 = bits.Sub64(y[1], m[1], gteC2)
tmpY[2], gteC2 = bits.Sub64(y[2], m[2], gteC2)
tmpY[3], gteC2 = bits.Sub64(y[3], m[3], gteC2)
if gteC1 == 0 {
x = &tmpX
}
if gteC2 == 0 {
y = &tmpY
}
var (
c1 uint64
c2 uint64
tmp Int
)
res[0], c1 = bits.Add64(x[0], y[0], c1)
res[1], c1 = bits.Add64(x[1], y[1], c1)
res[2], c1 = bits.Add64(x[2], y[2], c1)
res[3], c1 = bits.Add64(x[3], y[3], c1)
tmp[0], c2 = bits.Sub64(res[0], m[0], c2)
tmp[1], c2 = bits.Sub64(res[1], m[1], c2)
tmp[2], c2 = bits.Sub64(res[2], m[2], c2)
tmp[3], c2 = bits.Sub64(res[3], m[3], c2)
// final sub was unnecessary
if c1 == 0 && c2 != 0 {
copy((*z)[:], res[:])
return z
}
copy((*z)[:], tmp[:])
return z
}
if m.IsZero() {
return z.Clear()
}
if z == m { // z is an alias for m and will be overwritten by AddOverflow before m is read
m = m.Clone()
}
if _, overflow := z.AddOverflow(x, y); overflow {
sum := [5]uint64{z[0], z[1], z[2], z[3], 1}
var quot [5]uint64
rem := udivrem(quot[:], sum[:], m)
return z.Set(&rem)
}
return z.Mod(z, m)
}
// AddUint64 sets z to x + y, where y is a uint64, and returns z
func (z *Int) AddUint64(x *Int, y uint64) *Int {
var carry uint64
z[0], carry = bits.Add64(x[0], y, 0)
z[1], carry = bits.Add64(x[1], 0, carry)
z[2], carry = bits.Add64(x[2], 0, carry)
z[3], _ = bits.Add64(x[3], 0, carry)
return z
}
// PaddedBytes encodes a Int as a 0-padded byte slice. The length
// of the slice is at least n bytes.
// Example, z =1, n = 20 => [0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1]
func (z *Int) PaddedBytes(n int) []byte {
b := make([]byte, n)
for i := 0; i < 32 && i < n; i++ {
b[n-1-i] = byte(z[i/8] >> uint64(8*(i%8)))
}
return b
}
// SubUint64 set z to the difference x - y, where y is a uint64, and returns z
func (z *Int) SubUint64(x *Int, y uint64) *Int {
var carry uint64
z[0], carry = bits.Sub64(x[0], y, carry)
z[1], carry = bits.Sub64(x[1], 0, carry)
z[2], carry = bits.Sub64(x[2], 0, carry)
z[3], _ = bits.Sub64(x[3], 0, carry)
return z
}
// SubOverflow sets z to the difference x-y and returns z and true if the operation underflowed
func (z *Int) SubOverflow(x, y *Int) (*Int, bool) {
var carry uint64
z[0], carry = bits.Sub64(x[0], y[0], 0)
z[1], carry = bits.Sub64(x[1], y[1], carry)
z[2], carry = bits.Sub64(x[2], y[2], carry)
z[3], carry = bits.Sub64(x[3], y[3], carry)
return z, carry != 0
}
// Sub sets z to the difference x-y
func (z *Int) Sub(x, y *Int) *Int {
var carry uint64
z[0], carry = bits.Sub64(x[0], y[0], 0)
z[1], carry = bits.Sub64(x[1], y[1], carry)
z[2], carry = bits.Sub64(x[2], y[2], carry)
z[3], _ = bits.Sub64(x[3], y[3], carry)
return z
}
// umulStep computes (hi * 2^64 + lo) = z + (x * y) + carry.
func umulStep(z, x, y, carry uint64) (hi, lo uint64) {
hi, lo = bits.Mul64(x, y)
lo, carry = bits.Add64(lo, carry, 0)
hi, _ = bits.Add64(hi, 0, carry)
lo, carry = bits.Add64(lo, z, 0)
hi, _ = bits.Add64(hi, 0, carry)
return hi, lo
}
// umulHop computes (hi * 2^64 + lo) = z + (x * y)
func umulHop(z, x, y uint64) (hi, lo uint64) {
hi, lo = bits.Mul64(x, y)
lo, carry := bits.Add64(lo, z, 0)
hi, _ = bits.Add64(hi, 0, carry)
return hi, lo
}
// umul computes full 256 x 256 -> 512 multiplication.
func umul(x, y *Int) [8]uint64 {
var (
res [8]uint64
carry, carry4, carry5, carry6 uint64
res1, res2, res3, res4, res5 uint64
)
carry, res[0] = bits.Mul64(x[0], y[0])
carry, res1 = umulHop(carry, x[1], y[0])
carry, res2 = umulHop(carry, x[2], y[0])
carry4, res3 = umulHop(carry, x[3], y[0])
carry, res[1] = umulHop(res1, x[0], y[1])
carry, res2 = umulStep(res2, x[1], y[1], carry)
carry, res3 = umulStep(res3, x[2], y[1], carry)
carry5, res4 = umulStep(carry4, x[3], y[1], carry)
carry, res[2] = umulHop(res2, x[0], y[2])
carry, res3 = umulStep(res3, x[1], y[2], carry)
carry, res4 = umulStep(res4, x[2], y[2], carry)
carry6, res5 = umulStep(carry5, x[3], y[2], carry)
carry, res[3] = umulHop(res3, x[0], y[3])
carry, res[4] = umulStep(res4, x[1], y[3], carry)
carry, res[5] = umulStep(res5, x[2], y[3], carry)
res[7], res[6] = umulStep(carry6, x[3], y[3], carry)
return res
}
// Mul sets z to the product x*y
func (z *Int) Mul(x, y *Int) *Int {
var (
res Int
carry uint64
res1, res2, res3 uint64
)
carry, res[0] = bits.Mul64(x[0], y[0])
carry, res1 = umulHop(carry, x[1], y[0])
carry, res2 = umulHop(carry, x[2], y[0])
res3 = x[3]*y[0] + carry
carry, res[1] = umulHop(res1, x[0], y[1])
carry, res2 = umulStep(res2, x[1], y[1], carry)
res3 = res3 + x[2]*y[1] + carry
carry, res[2] = umulHop(res2, x[0], y[2])
res3 = res3 + x[1]*y[2] + carry
res[3] = res3 + x[0]*y[3]
return z.Set(&res)
}
// MulOverflow sets z to the product x*y, and returns z and whether overflow occurred
func (z *Int) MulOverflow(x, y *Int) (*Int, bool) {
p := umul(x, y)
copy(z[:], p[:4])
return z, (p[4] | p[5] | p[6] | p[7]) != 0
}
func (z *Int) squared() {
var (
res Int
carry0, carry1, carry2 uint64
res1, res2 uint64
)
carry0, res[0] = bits.Mul64(z[0], z[0])
carry0, res1 = umulHop(carry0, z[0], z[1])
carry0, res2 = umulHop(carry0, z[0], z[2])
carry1, res[1] = umulHop(res1, z[0], z[1])
carry1, res2 = umulStep(res2, z[1], z[1], carry1)
carry2, res[2] = umulHop(res2, z[0], z[2])
res[3] = 2*(z[0]*z[3]+z[1]*z[2]) + carry0 + carry1 + carry2
z.Set(&res)
}
// isBitSet returns true if bit n-th is set, where n = 0 is LSB.
// The n must be <= 255.
func (z *Int) isBitSet(n uint) bool {
return (z[n/64] & (1 << (n % 64))) != 0
}
// addTo computes x += y.
// Requires len(x) >= len(y).
func addTo(x, y []uint64) uint64 {
var carry uint64
for i := 0; i < len(y); i++ {
x[i], carry = bits.Add64(x[i], y[i], carry)
}
return carry
}
// subMulTo computes x -= y * multiplier.
// Requires len(x) >= len(y).
func subMulTo(x, y []uint64, multiplier uint64) uint64 {
var borrow uint64
for i := 0; i < len(y); i++ {
s, carry1 := bits.Sub64(x[i], borrow, 0)
ph, pl := bits.Mul64(y[i], multiplier)
t, carry2 := bits.Sub64(s, pl, 0)
x[i] = t
borrow = ph + carry1 + carry2
}
return borrow
}
// udivremBy1 divides u by single normalized word d and produces both quotient and remainder.
// The quotient is stored in provided quot.
func udivremBy1(quot, u []uint64, d uint64) (rem uint64) {
reciprocal := reciprocal2by1(d)
rem = u[len(u)-1] // Set the top word as remainder.
for j := len(u) - 2; j >= 0; j-- {
quot[j], rem = udivrem2by1(rem, u[j], d, reciprocal)
}
return rem
}
// udivremKnuth implements the division of u by normalized multiple word d from the Knuth's division algorithm.
// The quotient is stored in provided quot - len(u)-len(d) words.
// Updates u to contain the remainder - len(d) words.
func udivremKnuth(quot, u, d []uint64) {
dh := d[len(d)-1]
dl := d[len(d)-2]
reciprocal := reciprocal2by1(dh)
for j := len(u) - len(d) - 1; j >= 0; j-- {
u2 := u[j+len(d)]
u1 := u[j+len(d)-1]
u0 := u[j+len(d)-2]
var qhat, rhat uint64
if u2 >= dh { // Division overflows.
qhat = ^uint64(0)
// TODO: Add "qhat one to big" adjustment (not needed for correctness, but helps avoiding "add back" case).
} else {
qhat, rhat = udivrem2by1(u2, u1, dh, reciprocal)
ph, pl := bits.Mul64(qhat, dl)
if ph > rhat || (ph == rhat && pl > u0) {
qhat--
// TODO: Add "qhat one to big" adjustment (not needed for correctness, but helps avoiding "add back" case).
}
}
// Multiply and subtract.
borrow := subMulTo(u[j:], d, qhat)
u[j+len(d)] = u2 - borrow
if u2 < borrow { // Too much subtracted, add back.
qhat--
u[j+len(d)] += addTo(u[j:], d)
}
quot[j] = qhat // Store quotient digit.
}
}
// udivrem divides u by d and produces both quotient and remainder.
// The quotient is stored in provided quot - len(u)-len(d)+1 words.
// It loosely follows the Knuth's division algorithm (sometimes referenced as "schoolbook" division) using 64-bit words.
// See Knuth, Volume 2, section 4.3.1, Algorithm D.
func udivrem(quot, u []uint64, d *Int) (rem Int) {
var dLen int
for i := len(d) - 1; i >= 0; i-- {
if d[i] != 0 {
dLen = i + 1
break
}
}
shift := uint(bits.LeadingZeros64(d[dLen-1]))
var dnStorage Int
dn := dnStorage[:dLen]
for i := dLen - 1; i > 0; i-- {
dn[i] = (d[i] << shift) | (d[i-1] >> (64 - shift))
}
dn[0] = d[0] << shift
var uLen int
for i := len(u) - 1; i >= 0; i-- {
if u[i] != 0 {
uLen = i + 1
break
}
}
if uLen < dLen {
copy(rem[:], u)
return rem
}
var unStorage [9]uint64
un := unStorage[:uLen+1]
un[uLen] = u[uLen-1] >> (64 - shift)
for i := uLen - 1; i > 0; i-- {
un[i] = (u[i] << shift) | (u[i-1] >> (64 - shift))
}
un[0] = u[0] << shift
// TODO: Skip the highest word of numerator if not significant.
if dLen == 1 {
r := udivremBy1(quot, un, dn[0])
rem.SetUint64(r >> shift)
return rem
}
udivremKnuth(quot, un, dn)
for i := 0; i < dLen-1; i++ {
rem[i] = (un[i] >> shift) | (un[i+1] << (64 - shift))
}
rem[dLen-1] = un[dLen-1] >> shift
return rem
}
// Div sets z to the quotient x/y for returns z.
// If y == 0, z is set to 0
func (z *Int) Div(x, y *Int) *Int {
if y.IsZero() || y.Gt(x) {
return z.Clear()
}
if x.Eq(y) {
return z.SetOne()
}
// Shortcut some cases
if x.IsUint64() {
return z.SetUint64(x.Uint64() / y.Uint64())
}
// At this point, we know
// x/y ; x > y > 0
var quot Int
udivrem(quot[:], x[:], y)
return z.Set(")
}
// Mod sets z to the modulus x%y for y != 0 and returns z.
// If y == 0, z is set to 0 (OBS: differs from the big.Int)
func (z *Int) Mod(x, y *Int) *Int {
if x.IsZero() || y.IsZero() {
return z.Clear()
}
switch x.Cmp(y) {
case -1:
// x < y
copy(z[:], x[:])
return z
case 0:
// x == y
return z.Clear() // They are equal
}
// At this point:
// x != 0
// y != 0
// x > y
// Shortcut trivial case
if x.IsUint64() {
return z.SetUint64(x.Uint64() % y.Uint64())
}
var quot Int
*z = udivrem(quot[:], x[:], y)
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, both z and m are set to 0 (OBS: differs from the big.Int)
func (z *Int) DivMod(x, y, m *Int) (*Int, *Int) {
if y.IsZero() {
return z.Clear(), m.Clear()
}
var quot Int
*m = udivrem(quot[:], x[:], y)
*z = quot
return z, m
}
// SMod interprets x and y as two's complement signed integers,
// sets z to (sign x) * { abs(x) modulus abs(y) }
// If y == 0, z is set to 0 (OBS: differs from the big.Int)
func (z *Int) SMod(x, y *Int) *Int {
ys := y.Sign()
xs := x.Sign()
// abs x
if xs == -1 {
x = new(Int).Neg(x)
}
// abs y
if ys == -1 {
y = new(Int).Neg(y)
}
z.Mod(x, y)
if xs == -1 {
z.Neg(z)
}
return z
}
// MulModWithReciprocal calculates the modulo-m multiplication of x and y
// and returns z, using the reciprocal of m provided as the mu parameter.
// Use uint256.Reciprocal to calculate mu from m.
// If m == 0, z is set to 0 (OBS: differs from the big.Int)
func (z *Int) MulModWithReciprocal(x, y, m *Int, mu *[5]uint64) *Int {
if x.IsZero() || y.IsZero() || m.IsZero() {
return z.Clear()
}
p := umul(x, y)
if m[3] != 0 {
r := reduce4(p, m, *mu)
return z.Set(&r)
}
var (
pl Int
ph Int
)
copy(pl[:], p[:4])
copy(ph[:], p[4:])
// If the multiplication is within 256 bits use Mod().
if ph.IsZero() {
return z.Mod(&pl, m)
}
var quot [8]uint64
rem := udivrem(quot[:], p[:], m)
return z.Set(&rem)
}
// MulMod calculates the modulo-m multiplication of x and y and
// returns z.
// If m == 0, z is set to 0 (OBS: differs from the big.Int)
func (z *Int) MulMod(x, y, m *Int) *Int {
if x.IsZero() || y.IsZero() || m.IsZero() {
return z.Clear()
}
p := umul(x, y)
if m[3] != 0 {
mu := Reciprocal(m)
r := reduce4(p, m, mu)
return z.Set(&r)
}
var (
pl Int
ph Int
)
copy(pl[:], p[:4])
copy(ph[:], p[4:])
// If the multiplication is within 256 bits use Mod().
if ph.IsZero() {
return z.Mod(&pl, m)
}
var quot [8]uint64
rem := udivrem(quot[:], p[:], m)
return z.Set(&rem)
}
// MulDivOverflow calculates (x*y)/d with full precision, returns z and whether overflow occurred in multiply process (result does not fit to 256-bit).
// computes 512-bit multiplication and 512 by 256 division.
func (z *Int) MulDivOverflow(x, y, d *Int) (*Int, bool) {
if x.IsZero() || y.IsZero() || d.IsZero() {
return z.Clear(), false
}
p := umul(x, y)
var quot [8]uint64
udivrem(quot[:], p[:], d)
copy(z[:], quot[:4])
return z, (quot[4] | quot[5] | quot[6] | quot[7]) != 0
}
// Abs interprets x as a two's complement signed number,
// and sets z to the absolute value
//
// Abs(0) = 0
// Abs(1) = 1
// Abs(2**255) = -2**255
// Abs(2**256-1) = -1
func (z *Int) Abs(x *Int) *Int {
if x[3] < 0x8000000000000000 {
return z.Set(x)
}
return z.Sub(new(Int), x)
}
// Neg returns -x mod 2**256.
func (z *Int) Neg(x *Int) *Int {
return z.Sub(new(Int), x)
}
// SDiv interprets n and d as two's complement signed integers,
// does a signed division on the two operands and sets z to the result.
// If d == 0, z is set to 0
func (z *Int) SDiv(n, d *Int) *Int {
if n.Sign() > 0 {
if d.Sign() > 0 {
// pos / pos
z.Div(n, d)
return z
} else {
// pos / neg
z.Div(n, new(Int).Neg(d))
return z.Neg(z)
}
}
if d.Sign() < 0 {
// neg / neg
z.Div(new(Int).Neg(n), new(Int).Neg(d))
return z
}
// neg / pos
z.Div(new(Int).Neg(n), d)
return z.Neg(z)
}
// Sign returns:
//
// -1 if z < 0
// 0 if z == 0
// +1 if z > 0
//
// Where z is interpreted as a two's complement signed number
func (z *Int) Sign() int {
if z.IsZero() {
return 0
}
if z[3] < 0x8000000000000000 {
return 1
}
return -1
}
// BitLen returns the number of bits required to represent z
func (z *Int) BitLen() int {
switch {
case z[3] != 0:
return 192 + bits.Len64(z[3])
case z[2] != 0:
return 128 + bits.Len64(z[2])
case z[1] != 0:
return 64 + bits.Len64(z[1])
default:
return bits.Len64(z[0])
}
}
// ByteLen returns the number of bytes required to represent z
func (z *Int) ByteLen() int {
return (z.BitLen() + 7) / 8
}
func (z *Int) lsh64(x *Int) *Int {
z[3], z[2], z[1], z[0] = x[2], x[1], x[0], 0
return z
}
func (z *Int) lsh128(x *Int) *Int {
z[3], z[2], z[1], z[0] = x[1], x[0], 0, 0
return z
}
func (z *Int) lsh192(x *Int) *Int {
z[3], z[2], z[1], z[0] = x[0], 0, 0, 0
return z
}
func (z *Int) rsh64(x *Int) *Int {
z[3], z[2], z[1], z[0] = 0, x[3], x[2], x[1]
return z
}
func (z *Int) rsh128(x *Int) *Int {
z[3], z[2], z[1], z[0] = 0, 0, x[3], x[2]
return z
}
func (z *Int) rsh192(x *Int) *Int {
z[3], z[2], z[1], z[0] = 0, 0, 0, x[3]
return z
}
func (z *Int) srsh64(x *Int) *Int {
z[3], z[2], z[1], z[0] = math.MaxUint64, x[3], x[2], x[1]
return z
}
func (z *Int) srsh128(x *Int) *Int {
z[3], z[2], z[1], z[0] = math.MaxUint64, math.MaxUint64, x[3], x[2]
return z
}
func (z *Int) srsh192(x *Int) *Int {
z[3], z[2], z[1], z[0] = math.MaxUint64, math.MaxUint64, math.MaxUint64, x[3]
return z
}
// Not sets z = ^x and returns z.
func (z *Int) Not(x *Int) *Int {
z[3], z[2], z[1], z[0] = ^x[3], ^x[2], ^x[1], ^x[0]
return z
}
// Gt returns true if z > x
func (z *Int) Gt(x *Int) bool {
return x.Lt(z)
}
// Slt interprets z and x as signed integers, and returns
// true if z < x
func (z *Int) Slt(x *Int) bool {
zSign := z.Sign()
xSign := x.Sign()
switch {
case zSign >= 0 && xSign < 0:
return false
case zSign < 0 && xSign >= 0:
return true
default:
return z.Lt(x)
}
}
// Sgt interprets z and x as signed integers, and returns
// true if z > x
func (z *Int) Sgt(x *Int) bool {
zSign := z.Sign()
xSign := x.Sign()
switch {
case zSign >= 0 && xSign < 0:
return true
case zSign < 0 && xSign >= 0:
return false
default:
return z.Gt(x)
}
}
// Lt returns true if z < x
func (z *Int) Lt(x *Int) bool {
// z < x <=> z - x < 0 i.e. when subtraction overflows.
_, carry := bits.Sub64(z[0], x[0], 0)
_, carry = bits.Sub64(z[1], x[1], carry)
_, carry = bits.Sub64(z[2], x[2], carry)
_, carry = bits.Sub64(z[3], x[3], carry)
return carry != 0
}
// SetUint64 sets z to the value x //吧一个uint64类型的z 赋值给x.
func (z *Int) SetUint64(x uint64) *Int {
z[3], z[2], z[1], z[0] = 0, 0, 0, x
return z
}
// Eq returns true if z == x
func (z *Int) Eq(x *Int) bool {
return (z[0] == x[0]) && (z[1] == x[1]) && (z[2] == x[2]) && (z[3] == x[3])
}
// Cmp compares z and x and returns:
//
// -1 if z < x
// 0 if z == x
// +1 if z > x
func (z *Int) Cmp(x *Int) (r int) {
// z < x <=> z - x < 0 i.e. when subtraction overflows.
d0, carry := bits.Sub64(z[0], x[0], 0)
d1, carry := bits.Sub64(z[1], x[1], carry)
d2, carry := bits.Sub64(z[2], x[2], carry)
d3, carry := bits.Sub64(z[3], x[3], carry)
if carry == 1 {
return -1
}
if d0|d1|d2|d3 == 0 {
return 0
}
return 1
}
// CmpUint64 compares z and x and returns:
//
// -1 if z < x
// 0 if z == x
// +1 if z > x
func (z *Int) CmpUint64(x uint64) int {
if z[0] > x || (z[1]|z[2]|z[3]) != 0 {
return 1
}
if z[0] == x {
return 0
}
return -1
}
// CmpBig compares z and x and returns:
//
// -1 if z < x
// 0 if z == x
// +1 if z > x
func (z *Int) CmpBig(x *big.Int) (r int) {
// If x is negative, it's surely smaller (z > x)
if x.Sign() == -1 {
return 1
}
y := new(Int)
if y.SetFromBig(x) { // overflow
// z < x
return -1
}
return z.Cmp(y)
}
// LtUint64 returns true if z is smaller than n
func (z *Int) LtUint64(n uint64) bool {
return z[0] < n && (z[1]|z[2]|z[3]) == 0
}
// GtUint64 returns true if z is larger than n
func (z *Int) GtUint64(n uint64) bool {
return z[0] > n || (z[1]|z[2]|z[3]) != 0
}
// IsUint64 reports whether z can be represented as a uint64.
func (z *Int) IsUint64() bool {
return (z[1] | z[2] | z[3]) == 0
}
// IsZero returns true if z == 0
func (z *Int) IsZero() bool {
return (z[0] | z[1] | z[2] | z[3]) == 0
}
// Clear sets z to 0
func (z *Int) Clear() *Int {
z[3], z[2], z[1], z[0] = 0, 0, 0, 0
return z
}
// SetAllOne sets all the bits of z to 1
func (z *Int) SetAllOne() *Int {
z[3], z[2], z[1], z[0] = math.MaxUint64, math.MaxUint64, math.MaxUint64, math.MaxUint64
return z
}
// SetOne sets z to 1
func (z *Int) SetOne() *Int {
z[3], z[2], z[1], z[0] = 0, 0, 0, 1
return z
}
// Lsh sets z = x << n and returns z.
func (z *Int) Lsh(x *Int, n uint) *Int {
// n % 64 == 0
if n&0x3f == 0 {
switch n {
case 0:
return z.Set(x)
case 64:
return z.lsh64(x)
case 128:
return z.lsh128(x)
case 192:
return z.lsh192(x)
default:
return z.Clear()
}
}
var (
a, b uint64
)
// Big swaps first
switch {
case n > 192:
if n > 256 {
return z.Clear()
}
z.lsh192(x)
n -= 192
goto sh192
case n > 128:
z.lsh128(x)
n -= 128
goto sh128
case n > 64:
z.lsh64(x)
n -= 64
goto sh64
default:
z.Set(x)
}
// remaining shifts
a = z[0] >> (64 - n)
z[0] = z[0] << n
sh64:
b = z[1] >> (64 - n)
z[1] = (z[1] << n) | a
sh128:
a = z[2] >> (64 - n)
z[2] = (z[2] << n) | b
sh192:
z[3] = (z[3] << n) | a
return z
}
// Rsh sets z = x >> n and returns z.
func (z *Int) Rsh(x *Int, n uint) *Int {
// n % 64 == 0
if n&0x3f == 0 {
switch n {
case 0:
return z.Set(x)
case 64:
return z.rsh64(x)
case 128:
return z.rsh128(x)
case 192:
return z.rsh192(x)
default:
return z.Clear()
}
}
var (
a, b uint64
)
// Big swaps first
switch {
case n > 192:
if n > 256 {
return z.Clear()
}
z.rsh192(x)
n -= 192
goto sh192
case n > 128:
z.rsh128(x)
n -= 128
goto sh128
case n > 64:
z.rsh64(x)
n -= 64
goto sh64
default:
z.Set(x)
}
// remaining shifts
a = z[3] << (64 - n)
z[3] = z[3] >> n
sh64:
b = z[2] << (64 - n)
z[2] = (z[2] >> n) | a
sh128:
a = z[1] << (64 - n)
z[1] = (z[1] >> n) | b
sh192:
z[0] = (z[0] >> n) | a
return z
}
// SRsh (Signed/Arithmetic right shift)
// considers z to be a signed integer, during right-shift
// and sets z = x >> n and returns z.
func (z *Int) SRsh(x *Int, n uint) *Int {
// If the MSB is 0, SRsh is same as Rsh.
if !x.isBitSet(255) {
return z.Rsh(x, n)
}
if n%64 == 0 {
switch n {
case 0:
return z.Set(x)
case 64:
return z.srsh64(x)
case 128:
return z.srsh128(x)
case 192:
return z.srsh192(x)
default:
return z.SetAllOne()
}
}
var (
a uint64 = math.MaxUint64 << (64 - n%64)
)
// Big swaps first
switch {
case n > 192:
if n > 256 {
return z.SetAllOne()
}
z.srsh192(x)
n -= 192
goto sh192
case n > 128:
z.srsh128(x)
n -= 128
goto sh128
case n > 64:
z.srsh64(x)
n -= 64
goto sh64
default:
z.Set(x)
}
// remaining shifts
z[3], a = (z[3]>>n)|a, z[3]<<(64-n)
sh64:
z[2], a = (z[2]>>n)|a, z[2]<<(64-n)
sh128:
z[1], a = (z[1]>>n)|a, z[1]<<(64-n)
sh192:
z[0] = (z[0] >> n) | a
return z
}
// Set sets z to x and returns z.
func (z *Int) Set(x *Int) *Int {
*z = *x
return z
}
// Or sets z = x | y and returns z.
func (z *Int) Or(x, y *Int) *Int {
z[0] = x[0] | y[0]
z[1] = x[1] | y[1]
z[2] = x[2] | y[2]
z[3] = x[3] | y[3]
return z
}
// And sets z = x & y and returns z.
func (z *Int) And(x, y *Int) *Int {
z[0] = x[0] & y[0]
z[1] = x[1] & y[1]
z[2] = x[2] & y[2]
z[3] = x[3] & y[3]
return z
}
// Xor sets z = x ^ y and returns z.
func (z *Int) Xor(x, y *Int) *Int {
z[0] = x[0] ^ y[0]
z[1] = x[1] ^ y[1]
z[2] = x[2] ^ y[2]
z[3] = x[3] ^ y[3]
return z
}
// Byte sets z to the value of the byte at position n,
// with 'z' considered as a big-endian 32-byte integer
// if 'n' > 32, f is set to 0
// Example: f = '5', n=31 => 5
func (z *Int) Byte(n *Int) *Int {
// in z, z[0] is the least significant
//
if number, overflow := n.Uint64WithOverflow(); !overflow {
if number < 32 {
number := z[4-1-number/8]
offset := (n[0] & 0x7) << 3 // 8*(n.d % 8)
z[0] = (number & (0xff00000000000000 >> offset)) >> (56 - offset)
z[3], z[2], z[1] = 0, 0, 0
return z
}
}
return z.Clear()
}
// Exp sets z = base**exponent mod 2**256, and returns z.
func (z *Int) Exp(base, exponent *Int) *Int {
res := Int{1, 0, 0, 0}
multiplier := *base
expBitLen := exponent.BitLen()
curBit := 0
word := exponent[0]
for ; curBit < expBitLen && curBit < 64; curBit++ {
if word&1 == 1 {
res.Mul(&res, &multiplier)
}
multiplier.squared()
word >>= 1
}
word = exponent[1]
for ; curBit < expBitLen && curBit < 128; curBit++ {
if word&1 == 1 {
res.Mul(&res, &multiplier)
}
multiplier.squared()
word >>= 1
}
word = exponent[2]
for ; curBit < expBitLen && curBit < 192; curBit++ {
if word&1 == 1 {
res.Mul(&res, &multiplier)
}
multiplier.squared()
word >>= 1
}
word = exponent[3]
for ; curBit < expBitLen && curBit < 256; curBit++ {
if word&1 == 1 {
res.Mul(&res, &multiplier)
}
multiplier.squared()
word >>= 1
}
return z.Set(&res)
}
// ExtendSign extends length of two’s complement signed integer,
// sets z to
// - x if byteNum > 31
// - x interpreted as a signed number with sign-bit at (byteNum*8+7), extended to the full 256 bits
//
// and returns z.
func (z *Int) ExtendSign(x, byteNum *Int) *Int {
if byteNum.GtUint64(31) {
return z.Set(x)
}
bit := uint(byteNum.Uint64()*8 + 7)
mask := new(Int).SetOne()
mask.Lsh(mask, bit)
mask.SubUint64(mask, 1)
if x.isBitSet(bit) {
z.Or(x, mask.Not(mask))
} else {
z.And(x, mask)
}
return z
}
// Sqrt sets z to ⌊√x⌋, the largest integer such that z² ≤ x, and returns z.
func (z *Int) Sqrt(x *Int) *Int {
// This implementation of Sqrt is based on big.Int (see math/big/nat.go).
if x.LtUint64(2) {
return z.Set(x)
}
var (
z1 = &Int{1, 0, 0, 0}
z2 = &Int{}
)
// Start with value known to be too large and repeat "z = ⌊(z + ⌊x/z⌋)/2⌋" until it stops getting smaller.
z1 = z1.Lsh(z1, uint(x.BitLen()+1)/2) // must be ≥ √x
for {
z2 = z2.Div(x, z1)
z2 = z2.Add(z2, z1)
{ //z2 = z2.Rsh(z2, 1) -- the code below does a 1-bit rsh faster
a := z2[3] << 63
z2[3] = z2[3] >> 1
b := z2[2] << 63
z2[2] = (z2[2] >> 1) | a
a = z2[1] << 63
z2[1] = (z2[1] >> 1) | b
z2[0] = (z2[0] >> 1) | a
}
// end of inlined bitshift
if z2.Cmp(z1) >= 0 {
// z1 is answer.
return z.Set(z1)
}
z1, z2 = z2, z1
}
}
var (
// lut is a lookuptable of bitlength -> log10, used in Log10().
lut = [257]int8{0, 0, 0, 0, -1, 1, 1, -2, 2, 2, -3, 3, 3, 3, -4, 4, 4, -5, 5, 5, -6, 6, 6, 6, -7, 7, 7, -8, 8, 8, -9, 9, 9, 9, -10, 10, 10, -11, 11, 11, -12, 12, 12, 12, -13, 13, 13, -14, 14, 14, -15, 15, 15, 15, -16, 16, 16, -17, 17, 17, -18, 18, 18, 18, -19, 19, 19, -20, 20, 20, -21, 21, 21, 21, -22, 22, 22, -23, 23, 23, -24, 24, 24, 24, -25, 25, 25, -26, 26, 26, -27, 27, 27, 27, -28, 28, 28, -29, 29, 29, -30, 30, 30, -31, 31, 31, 31, -32, 32, 32, -33, 33, 33, -34, 34, 34, 34, -35, 35, 35, -36, 36, 36, -37, 37, 37, 37, -38, 38, 38, -39, 39, 39, -40, 40, 40, 40, -41, 41, 41, -42, 42, 42, -43, 43, 43, 43, -44, 44, 44, -45, 45, 45, -46, 46, 46, 46, -47, 47, 47, -48, 48, 48, -49, 49, 49, 49, -50, 50, 50, -51, 51, 51, -52, 52, 52, 52, -53, 53, 53, -54, 54, 54, -55, 55, 55, 55, -56, 56, 56, -57, 57, 57, -58, 58, 58, -59, 59, 59, 59, -60, 60, 60, -61, 61, 61, -62, 62, 62, 62, -63, 63, 63, -64, 64, 64, -65, 65, 65, 65, -66, 66, 66, -67, 67, 67, -68, 68, 68, 68, -69, 69, 69, -70, 70, 70, -71, 71, 71, 71, -72, 72, 72, -73, 73, 73, -74, 74, 74, 74, -75, 75, 75, -76, 76, 76, -77}
// pows64 contains 10^0 ... 10^19
pows64 = [20]uint64{
1e0, 1e1, 1e2, 1e3, 1e4, 1e5, 1e6, 1e7, 1e8, 1e9, 1e10, 1e11, 1e12, 1e13, 1e14, 1e15, 1e16, 1e17, 1e18, 1e19,
}
// pows contain 1 ** 20 ... 10 ** 80
pows = [60]Int{
Int{7766279631452241920, 5, 0, 0}, Int{3875820019684212736, 54, 0, 0}, Int{1864712049423024128, 542, 0, 0}, Int{200376420520689664, 5421, 0, 0}, Int{2003764205206896640, 54210, 0, 0}, Int{1590897978359414784, 542101, 0, 0}, Int{15908979783594147840, 5421010, 0, 0}, Int{11515845246265065472, 54210108, 0, 0}, Int{4477988020393345024, 542101086, 0, 0}, Int{7886392056514347008, 5421010862, 0, 0}, Int{5076944270305263616, 54210108624, 0, 0}, Int{13875954555633532928, 542101086242, 0, 0}, Int{9632337040368467968, 5421010862427, 0, 0},
Int{4089650035136921600, 54210108624275, 0, 0}, Int{4003012203950112768, 542101086242752, 0, 0}, Int{3136633892082024448, 5421010862427522, 0, 0}, Int{12919594847110692864, 54210108624275221, 0, 0}, Int{68739955140067328, 542101086242752217, 0, 0}, Int{687399551400673280, 5421010862427522170, 0, 0}, Int{6873995514006732800, 17316620476856118468, 2, 0}, Int{13399722918938673152, 7145508105175220139, 29, 0}, Int{4870020673419870208, 16114848830623546549, 293, 0}, Int{11806718586779598848, 13574535716559052564, 2938, 0},
Int{7386721425538678784, 6618148649623664334, 29387, 0}, Int{80237960548581376, 10841254275107988496, 293873, 0}, Int{802379605485813760, 16178822382532126880, 2938735, 0}, Int{8023796054858137600, 14214271235644855872, 29387358, 0}, Int{6450984253743169536, 13015503840481697412, 293873587, 0}, Int{9169610316303040512, 1027829888850112811, 2938735877, 0}, Int{17909126868192198656, 10278298888501128114, 29387358770, 0}, Int{13070572018536022016, 10549268516463523069, 293873587705, 0}, Int{1578511669393358848, 13258964796087472617, 2938735877055, 0}, Int{15785116693933588480, 3462439444907864858, 29387358770557, 0},
Int{10277214349659471872, 16177650375369096972, 293873587705571, 0}, Int{10538423128046960640, 14202551164014556797, 2938735877055718, 0}, Int{13150510911921848320, 12898303124178706663, 29387358770557187, 0}, Int{2377900603251621888, 18302566799529756941, 293873587705571876, 0}, Int{5332261958806667264, 17004971331911604867, 2938735877055718769, 0}, Int{16429131440647569408, 4029016655730084128, 10940614696847636083, 1}, Int{16717361816799281152, 3396678409881738056, 17172426599928602752, 15}, Int{1152921504606846976, 15520040025107828953, 5703569335900062977, 159}, Int{11529215046068469760, 7626447661401876602, 1695461137871974930, 1593}, Int{4611686018427387904, 2477500319180559562, 16954611378719749304, 15930}, Int{9223372036854775808, 6328259118096044006, 3525417123811528497, 159309},
Int{0, 7942358959831785217, 16807427164405733357, 1593091}, Int{0, 5636613303479645706, 2053574980671369030, 15930919}, Int{0, 1025900813667802212, 2089005733004138687, 159309191}, Int{0, 10259008136678022120, 2443313256331835254, 1593091911}, Int{0, 10356360998232463120, 5986388489608800929, 15930919111}, Int{0, 11329889613776873120, 4523652674959354447, 159309191113}, Int{0, 2618431695511421504, 8343038602174441244, 1593091911132}, Int{0, 7737572881404663424, 9643409726906205977, 15930919111324}, Int{0, 3588752519208427776, 4200376900514301694, 159309191113245}, Int{0, 17440781118374726144, 5110280857723913709, 1593091911132452}, Int{0, 8387114520361296896, 14209320429820033867, 15930919111324522}, Int{0, 10084168908774762496, 12965995782233477362, 159309191113245227}, Int{0, 8607968719199866880, 532749306367912313, 1593091911132452277}, Int{0, 12292710897160462336, 5327493063679123134, 15930919111324522770}, Int{0, 12246644529347313664, 16381442489372128114, 11735238523568814774}, Int{0, 11785980851215826944, 16240472304044868218, 6671920793430838052},
}
)
// Log10 returns the log in base 10, floored to nearest integer.
// **OBS** This method returns '0' for '0', not `-Inf`.
func (z *Int) Log10() uint {
// For some bit-lengths, there's only one possible value. Example:
// three bits can only represent [100 ... 111], or [4 ... 7]
// Ergo, bitlen:3 -> log10 == 0
res := lut[z.BitLen()%257]
if res >= 0 {
return uint(res)
}
// It was negative, which is a signal that we need to do one more check
// do determine which log it is. First remove the negation
res = -res
// We now lookup via the power of tens. Example:
// bitlen 4, [1000 ... 1111], or [8 .. 15]
// In order to figure out if it is '0' or '1', we only need to do one comparison,
// is it larger or smaller than '10'?
// For bitlengths < 20, we can use the uint64-space
if res < 20 {
// Uint64-space
if z.CmpUint64(pows64[res]) < 0 {
return uint(res - 1)
}
return uint(res)
}
// Non-uint64 space
if z.Cmp(&pows[res-20]) < 0 {
return uint(res - 1)
}
return uint(res)
}