github.com/holiman/uint256 源码阅读

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(&quot)
}

// 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)
}