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Patrick
2026-03-06 21:58:16 +00:00
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field/fe.go Normal file
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////////////////////////////////////////////////////////////////////////////////////////////////////////////////////////
//
// Source: https://sources.truenas.cloud/code
// Import: sources.truenas.cloud/code/edwards25519/field
//
////////////////////////////////////////////////////////////////////////////////////////////////////////////////////////
//
// Copyright (c) 2019 The Go Authors. All rights Reserved.
// Use of this source code is goverened by a BSD-style
// license that can be found in the LICENSE file.
//
////////////////////////////////////////////////////////////////////////////////////////////////////////////////////////
package field
import (
"crypto/subtle"
"encoding/binary"
"errors"
"math/bits"
)
////////////////////////////////////////////////////////////////////////////////////////////////////////////////////////
// Element represents an element of the field GF(2^255-19). Note that this
// is not a cryptographically secure group, and should only be used to interact
// with edwards25519.Point coordinates.
//
// This type works similarly to math/big.Int, and all arguments and receivers
// are allowed to alias.
//
// The zero value is a valid zero element.
type Element struct {
// An element t represents the integer
// t.l0 + t.l1*2^51 + t.l2*2^102 + t.l3*2^153 + t.l4*2^204
//
// Between operations, all limbs are expected to be lower than 2^52.
l0 uint64
l1 uint64
l2 uint64
l3 uint64
l4 uint64
}
////////////////////////////////////////////////////////////////////////////////////////////////////////////////////////
const maskLow51Bits uint64 = (1 << 51) - 1
var feZero = &Element{0, 0, 0, 0, 0}
// Zero sets v = 0, and returns v.
func (v *Element) Zero() *Element {
*v = *feZero
return v
}
////////////////////////////////////////////////////////////////////////////////////////////////////////////////////////
var feOne = &Element{1, 0, 0, 0, 0}
// One sets v = 1, and returns v.
func (v *Element) One() *Element {
*v = *feOne
return v
}
////////////////////////////////////////////////////////////////////////////////////////////////////////////////////////
// reduce reduces v modulo 2^255 - 19 and returns it.
func (v *Element) reduce() *Element {
v.carryPropagate()
// After the light reduction we now have a field element representation
// v < 2^255 + 2^13 * 19, but need v < 2^255 - 19.
// If v >= 2^255 - 19, then v + 19 >= 2^255, which would overflow 2^255 - 1,
// generating a carry. That is, c will be 0 if v < 2^255 - 19, and 1 otherwise.
c := (v.l0 + 19) >> 51
c = (v.l1 + c) >> 51
c = (v.l2 + c) >> 51
c = (v.l3 + c) >> 51
c = (v.l4 + c) >> 51
// If v < 2^255 - 19 and c = 0, this will be a no-op. Otherwise, it's
// effectively applying the reduction identity to the carry.
v.l0 += 19 * c
v.l1 += v.l0 >> 51
v.l0 = v.l0 & maskLow51Bits
v.l2 += v.l1 >> 51
v.l1 = v.l1 & maskLow51Bits
v.l3 += v.l2 >> 51
v.l2 = v.l2 & maskLow51Bits
v.l4 += v.l3 >> 51
v.l3 = v.l3 & maskLow51Bits
// no additional carry
v.l4 = v.l4 & maskLow51Bits
return v
}
////////////////////////////////////////////////////////////////////////////////////////////////////////////////////////
// Add sets v = a + b, and returns v.
func (v *Element) Add(a, b *Element) *Element {
v.l0 = a.l0 + b.l0
v.l1 = a.l1 + b.l1
v.l2 = a.l2 + b.l2
v.l3 = a.l3 + b.l3
v.l4 = a.l4 + b.l4
return v.carryPropagate()
}
////////////////////////////////////////////////////////////////////////////////////////////////////////////////////////
// Subtract sets v = a - b, and returns v.
func (v *Element) Subtract(a, b *Element) *Element {
// We first add 2 * p, to guarantee the subtraction won't underflow, and
// then subtract b (which can be up to 2^255 + 2^13 * 19).
v.l0 = (a.l0 + 0xFFFFFFFFFFFDA) - b.l0
v.l1 = (a.l1 + 0xFFFFFFFFFFFFE) - b.l1
v.l2 = (a.l2 + 0xFFFFFFFFFFFFE) - b.l2
v.l3 = (a.l3 + 0xFFFFFFFFFFFFE) - b.l3
v.l4 = (a.l4 + 0xFFFFFFFFFFFFE) - b.l4
return v.carryPropagate()
}
////////////////////////////////////////////////////////////////////////////////////////////////////////////////////////
// Negate sets v = -a, and returns v.
func (v *Element) Negate(a *Element) *Element {
return v.Subtract(feZero, a)
}
// Invert sets v = 1/z mod p, and returns v.
//
// If z == 0, Invert returns v = 0.
func (v *Element) Invert(z *Element) *Element {
// Inversion is implemented as exponentiation with exponent p − 2. It uses the
// same sequence of 255 squarings and 11 multiplications as [Curve25519].
var z2, z9, z11, z2_5_0, z2_10_0, z2_20_0, z2_50_0, z2_100_0, t Element
z2.Square(z) // 2
t.Square(&z2) // 4
t.Square(&t) // 8
z9.Multiply(&t, z) // 9
z11.Multiply(&z9, &z2) // 11
t.Square(&z11) // 22
z2_5_0.Multiply(&t, &z9) // 31 = 2^5 - 2^0
t.Square(&z2_5_0) // 2^6 - 2^1
for i := 0; i < 4; i++ {
t.Square(&t) // 2^10 - 2^5
}
z2_10_0.Multiply(&t, &z2_5_0) // 2^10 - 2^0
t.Square(&z2_10_0) // 2^11 - 2^1
for i := 0; i < 9; i++ {
t.Square(&t) // 2^20 - 2^10
}
z2_20_0.Multiply(&t, &z2_10_0) // 2^20 - 2^0
t.Square(&z2_20_0) // 2^21 - 2^1
for i := 0; i < 19; i++ {
t.Square(&t) // 2^40 - 2^20
}
t.Multiply(&t, &z2_20_0) // 2^40 - 2^0
t.Square(&t) // 2^41 - 2^1
for i := 0; i < 9; i++ {
t.Square(&t) // 2^50 - 2^10
}
z2_50_0.Multiply(&t, &z2_10_0) // 2^50 - 2^0
t.Square(&z2_50_0) // 2^51 - 2^1
for i := 0; i < 49; i++ {
t.Square(&t) // 2^100 - 2^50
}
z2_100_0.Multiply(&t, &z2_50_0) // 2^100 - 2^0
t.Square(&z2_100_0) // 2^101 - 2^1
for i := 0; i < 99; i++ {
t.Square(&t) // 2^200 - 2^100
}
t.Multiply(&t, &z2_100_0) // 2^200 - 2^0
t.Square(&t) // 2^201 - 2^1
for i := 0; i < 49; i++ {
t.Square(&t) // 2^250 - 2^50
}
t.Multiply(&t, &z2_50_0) // 2^250 - 2^0
t.Square(&t) // 2^251 - 2^1
t.Square(&t) // 2^252 - 2^2
t.Square(&t) // 2^253 - 2^3
t.Square(&t) // 2^254 - 2^4
t.Square(&t) // 2^255 - 2^5
return v.Multiply(&t, &z11) // 2^255 - 21
}
////////////////////////////////////////////////////////////////////////////////////////////////////////////////////////
// Set sets v = a, and returns v.
func (v *Element) Set(a *Element) *Element {
*v = *a
return v
}
////////////////////////////////////////////////////////////////////////////////////////////////////////////////////////
// SetBytes sets v to x, where x is a 32-byte little-endian encoding. If x is
// not of the right length, SetBytes returns nil and an error, and the
// receiver is unchanged.
//
// Consistent with RFC 7748, the most significant bit (the high bit of the
// last byte) is ignored, and non-canonical values (2^255-19 through 2^255-1)
// are accepted. Note that this is laxer than specified by RFC 8032, but
// consistent with most Ed25519 implementations.
func (v *Element) SetBytes(x []byte) (*Element, error) {
if len(x) != 32 {
return nil, errors.New("edwards25519: invalid field element input size")
}
// Bits 0:51 (bytes 0:8, bits 0:64, shift 0, mask 51).
v.l0 = binary.LittleEndian.Uint64(x[0:8])
v.l0 &= maskLow51Bits
// Bits 51:102 (bytes 6:14, bits 48:112, shift 3, mask 51).
v.l1 = binary.LittleEndian.Uint64(x[6:14]) >> 3
v.l1 &= maskLow51Bits
// Bits 102:153 (bytes 12:20, bits 96:160, shift 6, mask 51).
v.l2 = binary.LittleEndian.Uint64(x[12:20]) >> 6
v.l2 &= maskLow51Bits
// Bits 153:204 (bytes 19:27, bits 152:216, shift 1, mask 51).
v.l3 = binary.LittleEndian.Uint64(x[19:27]) >> 1
v.l3 &= maskLow51Bits
// Bits 204:255 (bytes 24:32, bits 192:256, shift 12, mask 51).
// Note: not bytes 25:33, shift 4, to avoid overread.
v.l4 = binary.LittleEndian.Uint64(x[24:32]) >> 12
v.l4 &= maskLow51Bits
return v, nil
}
////////////////////////////////////////////////////////////////////////////////////////////////////////////////////////
// Bytes returns the canonical 32-byte little-endian encoding of v.
func (v *Element) Bytes() []byte {
// This function is outlined to make the allocations inline in the caller
// rather than happen on the heap.
var out [32]byte
return v.bytes(&out)
}
////////////////////////////////////////////////////////////////////////////////////////////////////////////////////////
func (v *Element) bytes(out *[32]byte) []byte {
t := *v
t.reduce()
// Pack five 51-bit limbs into four 64-bit words:
//
// 255 204 153 102 51 0
// ├──l4──┼──l3──┼──l2──┼──l1──┼──l0──┤
// ├───u3───┼───u2───┼───u1───┼───u0───┤
// 256 192 128 64 0
u0 := t.l1<<51 | t.l0
u1 := t.l2<<(102-64) | t.l1>>(64-51)
u2 := t.l3<<(153-128) | t.l2>>(128-102)
u3 := t.l4<<(204-192) | t.l3>>(192-153)
binary.LittleEndian.PutUint64(out[0*8:], u0)
binary.LittleEndian.PutUint64(out[1*8:], u1)
binary.LittleEndian.PutUint64(out[2*8:], u2)
binary.LittleEndian.PutUint64(out[3*8:], u3)
return out[:]
}
////////////////////////////////////////////////////////////////////////////////////////////////////////////////////////
// Equal returns 1 if v and u are equal, and 0 otherwise.
func (v *Element) Equal(u *Element) int {
sa, sv := u.Bytes(), v.Bytes()
return subtle.ConstantTimeCompare(sa, sv)
}
////////////////////////////////////////////////////////////////////////////////////////////////////////////////////////
// mask64Bits returns 0xffffffff if cond is 1, and 0 otherwise.
func mask64Bits(cond int) uint64 { return ^(uint64(cond) - 1) }
// Select sets v to a if cond == 1, and to b if cond == 0.
func (v *Element) Select(a, b *Element, cond int) *Element {
m := mask64Bits(cond)
v.l0 = (m & a.l0) | (^m & b.l0)
v.l1 = (m & a.l1) | (^m & b.l1)
v.l2 = (m & a.l2) | (^m & b.l2)
v.l3 = (m & a.l3) | (^m & b.l3)
v.l4 = (m & a.l4) | (^m & b.l4)
return v
}
////////////////////////////////////////////////////////////////////////////////////////////////////////////////////////
// Swap swaps v and u if cond == 1 or leaves them unchanged if cond == 0, and returns v.
func (v *Element) Swap(u *Element, cond int) {
m := mask64Bits(cond)
t := m & (v.l0 ^ u.l0)
v.l0 ^= t
u.l0 ^= t
t = m & (v.l1 ^ u.l1)
v.l1 ^= t
u.l1 ^= t
t = m & (v.l2 ^ u.l2)
v.l2 ^= t
u.l2 ^= t
t = m & (v.l3 ^ u.l3)
v.l3 ^= t
u.l3 ^= t
t = m & (v.l4 ^ u.l4)
v.l4 ^= t
u.l4 ^= t
}
////////////////////////////////////////////////////////////////////////////////////////////////////////////////////////
// IsNegative returns 1 if v is negative, and 0 otherwise.
func (v *Element) IsNegative() int {
return int(v.Bytes()[0] & 1)
}
////////////////////////////////////////////////////////////////////////////////////////////////////////////////////////
// Absolute sets v to |u|, and returns v.
func (v *Element) Absolute(u *Element) *Element {
return v.Select(new(Element).Negate(u), u, u.IsNegative())
}
////////////////////////////////////////////////////////////////////////////////////////////////////////////////////////
// Multiply sets v = x * y, and returns v.
func (v *Element) Multiply(x, y *Element) *Element {
feMul(v, x, y)
return v
}
////////////////////////////////////////////////////////////////////////////////////////////////////////////////////////
// Square sets v = x * x, and returns v.
func (v *Element) Square(x *Element) *Element {
feSquare(v, x)
return v
}
////////////////////////////////////////////////////////////////////////////////////////////////////////////////////////
// Mult32 sets v = x * y, and returns v.
func (v *Element) Mult32(x *Element, y uint32) *Element {
x0lo, x0hi := mul51(x.l0, y)
x1lo, x1hi := mul51(x.l1, y)
x2lo, x2hi := mul51(x.l2, y)
x3lo, x3hi := mul51(x.l3, y)
x4lo, x4hi := mul51(x.l4, y)
v.l0 = x0lo + 19*x4hi // carried over per the reduction identity
v.l1 = x1lo + x0hi
v.l2 = x2lo + x1hi
v.l3 = x3lo + x2hi
v.l4 = x4lo + x3hi
// The hi portions are going to be only 32 bits, plus any previous excess,
// so we can skip the carry propagation.
return v
}
////////////////////////////////////////////////////////////////////////////////////////////////////////////////////////
// mul51 returns lo + hi * 2⁵¹ = a * b.
func mul51(a uint64, b uint32) (lo uint64, hi uint64) {
mh, ml := bits.Mul64(a, uint64(b))
lo = ml & maskLow51Bits
hi = (mh << 13) | (ml >> 51)
return
}
////////////////////////////////////////////////////////////////////////////////////////////////////////////////////////
// Pow22523 set v = x^((p-5)/8), and returns v. (p-5)/8 is 2^252-3.
func (v *Element) Pow22523(x *Element) *Element {
var t0, t1, t2 Element
t0.Square(x) // x^2
t1.Square(&t0) // x^4
t1.Square(&t1) // x^8
t1.Multiply(x, &t1) // x^9
t0.Multiply(&t0, &t1) // x^11
t0.Square(&t0) // x^22
t0.Multiply(&t1, &t0) // x^31
t1.Square(&t0) // x^62
for i := 1; i < 5; i++ { // x^992
t1.Square(&t1)
}
t0.Multiply(&t1, &t0) // x^1023 -> 1023 = 2^10 - 1
t1.Square(&t0) // 2^11 - 2
for i := 1; i < 10; i++ { // 2^20 - 2^10
t1.Square(&t1)
}
t1.Multiply(&t1, &t0) // 2^20 - 1
t2.Square(&t1) // 2^21 - 2
for i := 1; i < 20; i++ { // 2^40 - 2^20
t2.Square(&t2)
}
t1.Multiply(&t2, &t1) // 2^40 - 1
t1.Square(&t1) // 2^41 - 2
for i := 1; i < 10; i++ { // 2^50 - 2^10
t1.Square(&t1)
}
t0.Multiply(&t1, &t0) // 2^50 - 1
t1.Square(&t0) // 2^51 - 2
for i := 1; i < 50; i++ { // 2^100 - 2^50
t1.Square(&t1)
}
t1.Multiply(&t1, &t0) // 2^100 - 1
t2.Square(&t1) // 2^101 - 2
for i := 1; i < 100; i++ { // 2^200 - 2^100
t2.Square(&t2)
}
t1.Multiply(&t2, &t1) // 2^200 - 1
t1.Square(&t1) // 2^201 - 2
for i := 1; i < 50; i++ { // 2^250 - 2^50
t1.Square(&t1)
}
t0.Multiply(&t1, &t0) // 2^250 - 1
t0.Square(&t0) // 2^251 - 2
t0.Square(&t0) // 2^252 - 4
return v.Multiply(&t0, x) // 2^252 - 3 -> x^(2^252-3)
}
////////////////////////////////////////////////////////////////////////////////////////////////////////////////////////
// sqrtM1 is 2^((p-1)/4), which squared is equal to -1 by Euler's Criterion.
var sqrtM1 = &Element{1718705420411056, 234908883556509,
2233514472574048, 2117202627021982, 765476049583133}
// SqrtRatio sets r to the non-negative square root of the ratio of u and v.
//
// If u/v is square, SqrtRatio returns r and 1. If u/v is not square, SqrtRatio
// sets r according to Section 4.3 of draft-irtf-cfrg-ristretto255-decaf448-00,
// and returns r and 0.
func (r *Element) SqrtRatio(u, v *Element) (R *Element, wasSquare int) {
t0 := new(Element)
// r = (u * v3) * (u * v7)^((p-5)/8)
v2 := new(Element).Square(v)
uv3 := new(Element).Multiply(u, t0.Multiply(v2, v))
uv7 := new(Element).Multiply(uv3, t0.Square(v2))
rr := new(Element).Multiply(uv3, t0.Pow22523(uv7))
check := new(Element).Multiply(v, t0.Square(rr)) // check = v * r^2
uNeg := new(Element).Negate(u)
correctSignSqrt := check.Equal(u)
flippedSignSqrt := check.Equal(uNeg)
flippedSignSqrtI := check.Equal(t0.Multiply(uNeg, sqrtM1))
rPrime := new(Element).Multiply(rr, sqrtM1) // r_prime = SQRT_M1 * r
// r = CT_SELECT(r_prime IF flipped_sign_sqrt | flipped_sign_sqrt_i ELSE r)
rr.Select(rPrime, rr, flippedSignSqrt|flippedSignSqrtI)
r.Absolute(rr) // Choose the nonnegative square root.
return r, correctSignSqrt | flippedSignSqrt
}
////////////////////////////////////////////////////////////////////////////////////////////////////////////////////////

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////////////////////////////////////////////////////////////////////////////////////////////////////////////////////////
//
// Source: https://sources.truenas.cloud/code
// Import: sources.truenas.cloud/code/edwards25519/field
//
////////////////////////////////////////////////////////////////////////////////////////////////////////////////////////
//
// Copyright (c) 2019 The Go Authors. All rights Reserved.
// Use of this source code is goverened by a BSD-style
// license that can be found in the LICENSE file.
//
////////////////////////////////////////////////////////////////////////////////////////////////////////////////////////
package field
import (
"testing"
"testing/quick"
)
////////////////////////////////////////////////////////////////////////////////////////////////////////////////////////
func checkAliasingOneArg(f func(v, x *Element) *Element) func(v, x Element) bool {
return func(v, x Element) bool {
x1, v1 := x, x
// Calculate a reference f(x) without aliasing.
if out := f(&v, &x); out != &v && isInBounds(out) {
return false
}
// Test aliasing the argument and the receiver.
if out := f(&v1, &v1); out != &v1 || v1 != v {
return false
}
// Ensure the arguments was not modified.
return x == x1
}
}
////////////////////////////////////////////////////////////////////////////////////////////////////////////////////////
func checkAliasingTwoArgs(f func(v, x, y *Element) *Element) func(v, x, y Element) bool {
return func(v, x, y Element) bool {
x1, y1, v1 := x, y, Element{}
// Calculate a reference f(x, y) without aliasing.
if out := f(&v, &x, &y); out != &v && isInBounds(out) {
return false
}
// Test aliasing the first argument and the receiver.
v1 = x
if out := f(&v1, &v1, &y); out != &v1 || v1 != v {
return false
}
// Test aliasing the second argument and the receiver.
v1 = y
if out := f(&v1, &x, &v1); out != &v1 || v1 != v {
return false
}
// Calculate a reference f(x, x) without aliasing.
if out := f(&v, &x, &x); out != &v {
return false
}
// Test aliasing the first argument and the receiver.
v1 = x
if out := f(&v1, &v1, &x); out != &v1 || v1 != v {
return false
}
// Test aliasing the second argument and the receiver.
v1 = x
if out := f(&v1, &x, &v1); out != &v1 || v1 != v {
return false
}
// Test aliasing both arguments and the receiver.
v1 = x
if out := f(&v1, &v1, &v1); out != &v1 || v1 != v {
return false
}
// Ensure the arguments were not modified.
return x == x1 && y == y1
}
}
////////////////////////////////////////////////////////////////////////////////////////////////////////////////////////
// TestAliasing checks that receivers and arguments can alias each other without
// leading to incorrect results. That is, it ensures that it's safe to write
//
// v.Invert(v)
//
// or
//
// v.Add(v, v)
//
// without any of the inputs getting clobbered by the output being written.
func TestAliasing(t *testing.T) {
type target struct {
name string
oneArgF func(v, x *Element) *Element
twoArgsF func(v, x, y *Element) *Element
}
for _, tt := range []target{
{name: "Absolute", oneArgF: (*Element).Absolute},
{name: "Invert", oneArgF: (*Element).Invert},
{name: "Negate", oneArgF: (*Element).Negate},
{name: "Set", oneArgF: (*Element).Set},
{name: "Square", oneArgF: (*Element).Square},
{name: "Pow22523", oneArgF: (*Element).Pow22523},
{
name: "Mult32",
oneArgF: func(v, x *Element) *Element {
return v.Mult32(x, 0xffffffff)
},
},
{name: "Multiply", twoArgsF: (*Element).Multiply},
{name: "Add", twoArgsF: (*Element).Add},
{name: "Subtract", twoArgsF: (*Element).Subtract},
{
name: "SqrtRatio",
twoArgsF: func(v, x, y *Element) *Element {
r, _ := v.SqrtRatio(x, y)
return r
},
},
{
name: "Select0",
twoArgsF: func(v, x, y *Element) *Element {
return v.Select(x, y, 0)
},
},
{
name: "Select1",
twoArgsF: func(v, x, y *Element) *Element {
return v.Select(x, y, 1)
},
},
} {
var err error
switch {
case tt.oneArgF != nil:
err = quick.Check(checkAliasingOneArg(tt.oneArgF), quickCheckConfig(256))
case tt.twoArgsF != nil:
err = quick.Check(checkAliasingTwoArgs(tt.twoArgsF), quickCheckConfig(256))
}
if err != nil {
t.Errorf("%v: %v", tt.name, err)
}
}
}
////////////////////////////////////////////////////////////////////////////////////////////////////////////////////////

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// Code generated by command: go run fe_amd64_asm.go -out ../fe_amd64.s -stubs ../fe_amd64.go -pkg field. DO NOT EDIT.
//go:build !purego
package field
// feMul sets out = a * b. It works like feMulGeneric.
//
//go:noescape
func feMul(out *Element, a *Element, b *Element)
// feSquare sets out = a * a. It works like feSquareGeneric.
//
//go:noescape
func feSquare(out *Element, a *Element)

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// Code generated by command: go run fe_amd64_asm.go -out ../fe_amd64.s -stubs ../fe_amd64.go -pkg field. DO NOT EDIT.
//go:build !purego
#include "textflag.h"
// func feMul(out *Element, a *Element, b *Element)
TEXT ·feMul(SB), NOSPLIT, $0-24
MOVQ a+8(FP), CX
MOVQ b+16(FP), BX
// r0 = a0×b0
MOVQ (CX), AX
MULQ (BX)
MOVQ AX, DI
MOVQ DX, SI
// r0 += 19×a1×b4
MOVQ 8(CX), DX
LEAQ (DX)(DX*8), AX
LEAQ (DX)(AX*2), AX
MULQ 32(BX)
ADDQ AX, DI
ADCQ DX, SI
// r0 += 19×a2×b3
MOVQ 16(CX), DX
LEAQ (DX)(DX*8), AX
LEAQ (DX)(AX*2), AX
MULQ 24(BX)
ADDQ AX, DI
ADCQ DX, SI
// r0 += 19×a3×b2
MOVQ 24(CX), DX
LEAQ (DX)(DX*8), AX
LEAQ (DX)(AX*2), AX
MULQ 16(BX)
ADDQ AX, DI
ADCQ DX, SI
// r0 += 19×a4×b1
MOVQ 32(CX), DX
LEAQ (DX)(DX*8), AX
LEAQ (DX)(AX*2), AX
MULQ 8(BX)
ADDQ AX, DI
ADCQ DX, SI
// r1 = a0×b1
MOVQ (CX), AX
MULQ 8(BX)
MOVQ AX, R9
MOVQ DX, R8
// r1 += a1×b0
MOVQ 8(CX), AX
MULQ (BX)
ADDQ AX, R9
ADCQ DX, R8
// r1 += 19×a2×b4
MOVQ 16(CX), DX
LEAQ (DX)(DX*8), AX
LEAQ (DX)(AX*2), AX
MULQ 32(BX)
ADDQ AX, R9
ADCQ DX, R8
// r1 += 19×a3×b3
MOVQ 24(CX), DX
LEAQ (DX)(DX*8), AX
LEAQ (DX)(AX*2), AX
MULQ 24(BX)
ADDQ AX, R9
ADCQ DX, R8
// r1 += 19×a4×b2
MOVQ 32(CX), DX
LEAQ (DX)(DX*8), AX
LEAQ (DX)(AX*2), AX
MULQ 16(BX)
ADDQ AX, R9
ADCQ DX, R8
// r2 = a0×b2
MOVQ (CX), AX
MULQ 16(BX)
MOVQ AX, R11
MOVQ DX, R10
// r2 += a1×b1
MOVQ 8(CX), AX
MULQ 8(BX)
ADDQ AX, R11
ADCQ DX, R10
// r2 += a2×b0
MOVQ 16(CX), AX
MULQ (BX)
ADDQ AX, R11
ADCQ DX, R10
// r2 += 19×a3×b4
MOVQ 24(CX), DX
LEAQ (DX)(DX*8), AX
LEAQ (DX)(AX*2), AX
MULQ 32(BX)
ADDQ AX, R11
ADCQ DX, R10
// r2 += 19×a4×b3
MOVQ 32(CX), DX
LEAQ (DX)(DX*8), AX
LEAQ (DX)(AX*2), AX
MULQ 24(BX)
ADDQ AX, R11
ADCQ DX, R10
// r3 = a0×b3
MOVQ (CX), AX
MULQ 24(BX)
MOVQ AX, R13
MOVQ DX, R12
// r3 += a1×b2
MOVQ 8(CX), AX
MULQ 16(BX)
ADDQ AX, R13
ADCQ DX, R12
// r3 += a2×b1
MOVQ 16(CX), AX
MULQ 8(BX)
ADDQ AX, R13
ADCQ DX, R12
// r3 += a3×b0
MOVQ 24(CX), AX
MULQ (BX)
ADDQ AX, R13
ADCQ DX, R12
// r3 += 19×a4×b4
MOVQ 32(CX), DX
LEAQ (DX)(DX*8), AX
LEAQ (DX)(AX*2), AX
MULQ 32(BX)
ADDQ AX, R13
ADCQ DX, R12
// r4 = a0×b4
MOVQ (CX), AX
MULQ 32(BX)
MOVQ AX, R15
MOVQ DX, R14
// r4 += a1×b3
MOVQ 8(CX), AX
MULQ 24(BX)
ADDQ AX, R15
ADCQ DX, R14
// r4 += a2×b2
MOVQ 16(CX), AX
MULQ 16(BX)
ADDQ AX, R15
ADCQ DX, R14
// r4 += a3×b1
MOVQ 24(CX), AX
MULQ 8(BX)
ADDQ AX, R15
ADCQ DX, R14
// r4 += a4×b0
MOVQ 32(CX), AX
MULQ (BX)
ADDQ AX, R15
ADCQ DX, R14
// First reduction chain
MOVQ $0x0007ffffffffffff, AX
SHLQ $0x0d, DI, SI
SHLQ $0x0d, R9, R8
SHLQ $0x0d, R11, R10
SHLQ $0x0d, R13, R12
SHLQ $0x0d, R15, R14
ANDQ AX, DI
IMUL3Q $0x13, R14, R14
ADDQ R14, DI
ANDQ AX, R9
ADDQ SI, R9
ANDQ AX, R11
ADDQ R8, R11
ANDQ AX, R13
ADDQ R10, R13
ANDQ AX, R15
ADDQ R12, R15
// Second reduction chain (carryPropagate)
MOVQ DI, SI
SHRQ $0x33, SI
MOVQ R9, R8
SHRQ $0x33, R8
MOVQ R11, R10
SHRQ $0x33, R10
MOVQ R13, R12
SHRQ $0x33, R12
MOVQ R15, R14
SHRQ $0x33, R14
ANDQ AX, DI
IMUL3Q $0x13, R14, R14
ADDQ R14, DI
ANDQ AX, R9
ADDQ SI, R9
ANDQ AX, R11
ADDQ R8, R11
ANDQ AX, R13
ADDQ R10, R13
ANDQ AX, R15
ADDQ R12, R15
// Store output
MOVQ out+0(FP), AX
MOVQ DI, (AX)
MOVQ R9, 8(AX)
MOVQ R11, 16(AX)
MOVQ R13, 24(AX)
MOVQ R15, 32(AX)
RET
// func feSquare(out *Element, a *Element)
TEXT ·feSquare(SB), NOSPLIT, $0-16
MOVQ a+8(FP), CX
// r0 = l0×l0
MOVQ (CX), AX
MULQ (CX)
MOVQ AX, SI
MOVQ DX, BX
// r0 += 38×l1×l4
MOVQ 8(CX), DX
LEAQ (DX)(DX*8), AX
LEAQ (DX)(AX*2), AX
SHLQ $0x01, AX
MULQ 32(CX)
ADDQ AX, SI
ADCQ DX, BX
// r0 += 38×l2×l3
MOVQ 16(CX), DX
LEAQ (DX)(DX*8), AX
LEAQ (DX)(AX*2), AX
SHLQ $0x01, AX
MULQ 24(CX)
ADDQ AX, SI
ADCQ DX, BX
// r1 = 2×l0×l1
MOVQ (CX), AX
SHLQ $0x01, AX
MULQ 8(CX)
MOVQ AX, R8
MOVQ DX, DI
// r1 += 38×l2×l4
MOVQ 16(CX), DX
LEAQ (DX)(DX*8), AX
LEAQ (DX)(AX*2), AX
SHLQ $0x01, AX
MULQ 32(CX)
ADDQ AX, R8
ADCQ DX, DI
// r1 += 19×l3×l3
MOVQ 24(CX), DX
LEAQ (DX)(DX*8), AX
LEAQ (DX)(AX*2), AX
MULQ 24(CX)
ADDQ AX, R8
ADCQ DX, DI
// r2 = 2×l0×l2
MOVQ (CX), AX
SHLQ $0x01, AX
MULQ 16(CX)
MOVQ AX, R10
MOVQ DX, R9
// r2 += l1×l1
MOVQ 8(CX), AX
MULQ 8(CX)
ADDQ AX, R10
ADCQ DX, R9
// r2 += 38×l3×l4
MOVQ 24(CX), DX
LEAQ (DX)(DX*8), AX
LEAQ (DX)(AX*2), AX
SHLQ $0x01, AX
MULQ 32(CX)
ADDQ AX, R10
ADCQ DX, R9
// r3 = 2×l0×l3
MOVQ (CX), AX
SHLQ $0x01, AX
MULQ 24(CX)
MOVQ AX, R12
MOVQ DX, R11
// r3 += 2×l1×l2
MOVQ 8(CX), AX
SHLQ $0x01, AX
MULQ 16(CX)
ADDQ AX, R12
ADCQ DX, R11
// r3 += 19×l4×l4
MOVQ 32(CX), DX
LEAQ (DX)(DX*8), AX
LEAQ (DX)(AX*2), AX
MULQ 32(CX)
ADDQ AX, R12
ADCQ DX, R11
// r4 = 2×l0×l4
MOVQ (CX), AX
SHLQ $0x01, AX
MULQ 32(CX)
MOVQ AX, R14
MOVQ DX, R13
// r4 += 2×l1×l3
MOVQ 8(CX), AX
SHLQ $0x01, AX
MULQ 24(CX)
ADDQ AX, R14
ADCQ DX, R13
// r4 += l2×l2
MOVQ 16(CX), AX
MULQ 16(CX)
ADDQ AX, R14
ADCQ DX, R13
// First reduction chain
MOVQ $0x0007ffffffffffff, AX
SHLQ $0x0d, SI, BX
SHLQ $0x0d, R8, DI
SHLQ $0x0d, R10, R9
SHLQ $0x0d, R12, R11
SHLQ $0x0d, R14, R13
ANDQ AX, SI
IMUL3Q $0x13, R13, R13
ADDQ R13, SI
ANDQ AX, R8
ADDQ BX, R8
ANDQ AX, R10
ADDQ DI, R10
ANDQ AX, R12
ADDQ R9, R12
ANDQ AX, R14
ADDQ R11, R14
// Second reduction chain (carryPropagate)
MOVQ SI, BX
SHRQ $0x33, BX
MOVQ R8, DI
SHRQ $0x33, DI
MOVQ R10, R9
SHRQ $0x33, R9
MOVQ R12, R11
SHRQ $0x33, R11
MOVQ R14, R13
SHRQ $0x33, R13
ANDQ AX, SI
IMUL3Q $0x13, R13, R13
ADDQ R13, SI
ANDQ AX, R8
ADDQ BX, R8
ANDQ AX, R10
ADDQ DI, R10
ANDQ AX, R12
ADDQ R9, R12
ANDQ AX, R14
ADDQ R11, R14
// Store output
MOVQ out+0(FP), AX
MOVQ SI, (AX)
MOVQ R8, 8(AX)
MOVQ R10, 16(AX)
MOVQ R12, 24(AX)
MOVQ R14, 32(AX)
RET

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////////////////////////////////////////////////////////////////////////////////////////////////////////////////////////
//
// Source: https://sources.truenas.cloud/code
// Import: sources.truenas.cloud/code/edwards25519/field
//
////////////////////////////////////////////////////////////////////////////////////////////////////////////////////////
//
// Copyright (c) 2019 The Go Authors. All rights Reserved.
// Use of this source code is goverened by a BSD-style
// license that can be found in the LICENSE file.
//
////////////////////////////////////////////////////////////////////////////////////////////////////////////////////////
//go:build !amd64 || purego
package field
func feMul(v, x, y *Element) { feMulGeneric(v, x, y) }
func feSquare(v, x *Element) { feSquareGeneric(v, x) }

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////////////////////////////////////////////////////////////////////////////////////////////////////////////////////////
//
// Source: https://sources.truenas.cloud/code
// Import: sources.truenas.cloud/code/edwards25519/field
//
////////////////////////////////////////////////////////////////////////////////////////////////////////////////////////
//
// Copyright (c) 2019 The Go Authors. All rights Reserved.
// Use of this source code is goverened by a BSD-style
// license that can be found in the LICENSE file.
//
////////////////////////////////////////////////////////////////////////////////////////////////////////////////////////
package field
////////////////////////////////////////////////////////////////////////////////////////////////////////////////////////
import "testing"
////////////////////////////////////////////////////////////////////////////////////////////////////////////////////////
func BenchmarkAdd(b *testing.B) {
x := new(Element).One()
y := new(Element).Add(x, x)
b.ResetTimer()
for i := 0; i < b.N; i++ {
x.Add(x, y)
}
}
////////////////////////////////////////////////////////////////////////////////////////////////////////////////////////
func BenchmarkMultiply(b *testing.B) {
x := new(Element).One()
y := new(Element).Add(x, x)
b.ResetTimer()
for i := 0; i < b.N; i++ {
x.Multiply(x, y)
}
}
////////////////////////////////////////////////////////////////////////////////////////////////////////////////////////
func BenchmarkSquare(b *testing.B) {
x := new(Element).Add(feOne, feOne)
b.ResetTimer()
for i := 0; i < b.N; i++ {
x.Square(x)
}
}
////////////////////////////////////////////////////////////////////////////////////////////////////////////////////////
func BenchmarkInvert(b *testing.B) {
x := new(Element).Add(feOne, feOne)
b.ResetTimer()
for i := 0; i < b.N; i++ {
x.Invert(x)
}
}
////////////////////////////////////////////////////////////////////////////////////////////////////////////////////////
func BenchmarkMult32(b *testing.B) {
x := new(Element).One()
b.ResetTimer()
for i := 0; i < b.N; i++ {
x.Mult32(x, 0xaa42aa42)
}
}
////////////////////////////////////////////////////////////////////////////////////////////////////////////////////////
func BenchmarkBytes(b *testing.B) {
x := new(Element).One()
b.ResetTimer()
for i := 0; i < b.N; i++ {
x.Bytes()
}
}
////////////////////////////////////////////////////////////////////////////////////////////////////////////////////////

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////////////////////////////////////////////////////////////////////////////////////////////////////////////////////////
//
// Source: https://sources.truenas.cloud/code
// Import: sources.truenas.cloud/code/edwards25519/field
//
////////////////////////////////////////////////////////////////////////////////////////////////////////////////////////
//
// Copyright (c) 2019 The Go Authors. All rights Reserved.
// Use of this source code is goverened by a BSD-style
// license that can be found in the LICENSE file.
//
////////////////////////////////////////////////////////////////////////////////////////////////////////////////////////
package field
import "errors"
////////////////////////////////////////////////////////////////////////////////////////////////////////////////////////
// This file contains additional functionality that is not included in the
// upstream crypto/ed25519/edwards25519/field package.
// SetWideBytes sets v to x, where x is a 64-byte little-endian encoding, which
// is reduced modulo the field order. If x is not of the right length,
// SetWideBytes returns nil and an error, and the receiver is unchanged.
//
// SetWideBytes is not necessary to select a uniformly distributed value, and is
// only provided for compatibility: SetBytes can be used instead as the chance
// of bias is less than 2⁻²⁵⁰.
func (v *Element) SetWideBytes(x []byte) (*Element, error) {
if len(x) != 64 {
return nil, errors.New("edwards25519: invalid SetWideBytes input size")
}
// Split the 64 bytes into two elements, and extract the most significant
// bit of each, which is ignored by SetBytes.
lo, _ := new(Element).SetBytes(x[:32])
loMSB := uint64(x[31] >> 7)
hi, _ := new(Element).SetBytes(x[32:])
hiMSB := uint64(x[63] >> 7)
// The output we want is
//
// v = lo + loMSB * 2²⁵⁵ + hi * 2²⁵⁶ + hiMSB * 2⁵¹¹
//
// which applying the reduction identity comes out to
//
// v = lo + loMSB * 19 + hi * 2 * 19 + hiMSB * 2 * 19²
//
// l0 will be the sum of a 52 bits value (lo.l0), plus a 5 bits value
// (loMSB * 19), a 6 bits value (hi.l0 * 2 * 19), and a 10 bits value
// (hiMSB * 2 * 19²), so it fits in a uint64.
v.l0 = lo.l0 + loMSB*19 + hi.l0*2*19 + hiMSB*2*19*19
v.l1 = lo.l1 + hi.l1*2*19
v.l2 = lo.l2 + hi.l2*2*19
v.l3 = lo.l3 + hi.l3*2*19
v.l4 = lo.l4 + hi.l4*2*19
return v.carryPropagate(), nil
}
////////////////////////////////////////////////////////////////////////////////////////////////////////////////////////

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////////////////////////////////////////////////////////////////////////////////////////////////////////////////////////
//
// Source: https://sources.truenas.cloud/code
// Import: sources.truenas.cloud/code/edwards25519/field
//
////////////////////////////////////////////////////////////////////////////////////////////////////////////////////////
//
// Copyright (c) 2019 The Go Authors. All rights Reserved.
// Use of this source code is goverened by a BSD-style
// license that can be found in the LICENSE file.
//
////////////////////////////////////////////////////////////////////////////////////////////////////////////////////////
package field
import (
"math/big"
"testing"
"testing/quick"
)
////////////////////////////////////////////////////////////////////////////////////////////////////////////////////////
var bigP = new(big.Int).Sub(new(big.Int).Lsh(big.NewInt(1), 255), big.NewInt(19))
func TestSetWideBytes(t *testing.T) {
f1 := func(in [64]byte, fe Element) bool {
fe1 := new(Element).Set(&fe)
if out, err := fe.SetWideBytes([]byte{42}); err == nil || out != nil ||
fe.Equal(fe1) != 1 {
return false
}
if out, err := fe.SetWideBytes(in[:]); err != nil || out != &fe {
return false
}
b := new(big.Int).SetBytes(swapEndianness(in[:]))
fe1.fromBig(b.Mod(b, bigP))
return fe.Equal(fe1) == 1 && isInBounds(&fe) && isInBounds(fe1)
}
if err := quick.Check(f1, nil); err != nil {
t.Error(err)
}
}
////////////////////////////////////////////////////////////////////////////////////////////////////////////////////////

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////////////////////////////////////////////////////////////////////////////////////////////////////////////////////////
//
// Source: https://sources.truenas.cloud/code
// Import: sources.truenas.cloud/code/edwards25519/field
//
////////////////////////////////////////////////////////////////////////////////////////////////////////////////////////
//
// Copyright (c) 2019 The Go Authors. All rights Reserved.
// Use of this source code is goverened by a BSD-style
// license that can be found in the LICENSE file.
//
////////////////////////////////////////////////////////////////////////////////////////////////////////////////////////
package field
import "math/bits"
////////////////////////////////////////////////////////////////////////////////////////////////////////////////////////
// uint128 holds a 128-bit number as two 64-bit limbs, for use with the
// bits.Mul64 and bits.Add64 intrinsics.
type uint128 struct {
lo, hi uint64
}
////////////////////////////////////////////////////////////////////////////////////////////////////////////////////////
// mul returns a * b.
func mul(a, b uint64) uint128 {
hi, lo := bits.Mul64(a, b)
return uint128{lo, hi}
}
////////////////////////////////////////////////////////////////////////////////////////////////////////////////////////
// addMul returns v + a * b.
func addMul(v uint128, a, b uint64) uint128 {
hi, lo := bits.Mul64(a, b)
lo, c := bits.Add64(lo, v.lo, 0)
hi, _ = bits.Add64(hi, v.hi, c)
return uint128{lo, hi}
}
////////////////////////////////////////////////////////////////////////////////////////////////////////////////////////
// mul19 returns v * 19.
func mul19(v uint64) uint64 {
// Using this approach seems to yield better optimizations than *19.
return v + (v+v<<3)<<1
}
////////////////////////////////////////////////////////////////////////////////////////////////////////////////////////
// addMul19 returns v + 19 * a * b, where a and b are at most 52 bits.
func addMul19(v uint128, a, b uint64) uint128 {
hi, lo := bits.Mul64(mul19(a), b)
lo, c := bits.Add64(lo, v.lo, 0)
hi, _ = bits.Add64(hi, v.hi, c)
return uint128{lo, hi}
}
////////////////////////////////////////////////////////////////////////////////////////////////////////////////////////
// addMul38 returns v + 38 * a * b, where a and b are at most 52 bits.
func addMul38(v uint128, a, b uint64) uint128 {
hi, lo := bits.Mul64(mul19(a), b*2)
lo, c := bits.Add64(lo, v.lo, 0)
hi, _ = bits.Add64(hi, v.hi, c)
return uint128{lo, hi}
}
////////////////////////////////////////////////////////////////////////////////////////////////////////////////////////
// shiftRightBy51 returns a >> 51. a is assumed to be at most 115 bits.
func shiftRightBy51(a uint128) uint64 {
return (a.hi << (64 - 51)) | (a.lo >> 51)
}
////////////////////////////////////////////////////////////////////////////////////////////////////////////////////////
func feMulGeneric(v, a, b *Element) {
a0 := a.l0
a1 := a.l1
a2 := a.l2
a3 := a.l3
a4 := a.l4
b0 := b.l0
b1 := b.l1
b2 := b.l2
b3 := b.l3
b4 := b.l4
// Limb multiplication works like pen-and-paper columnar multiplication, but
// with 51-bit limbs instead of digits.
//
// a4 a3 a2 a1 a0 x
// b4 b3 b2 b1 b0 =
// ------------------------
// a4b0 a3b0 a2b0 a1b0 a0b0 +
// a4b1 a3b1 a2b1 a1b1 a0b1 +
// a4b2 a3b2 a2b2 a1b2 a0b2 +
// a4b3 a3b3 a2b3 a1b3 a0b3 +
// a4b4 a3b4 a2b4 a1b4 a0b4 =
// ----------------------------------------------
// r8 r7 r6 r5 r4 r3 r2 r1 r0
//
// We can then use the reduction identity (a * 2²⁵⁵ + b = a * 19 + b) to
// reduce the limbs that would overflow 255 bits. r5 * 2²⁵⁵ becomes 19 * r5,
// r6 * 2³⁰⁶ becomes 19 * r6 * 2⁵¹, etc.
//
// Reduction can be carried out simultaneously to multiplication. For
// example, we do not compute r5: whenever the result of a multiplication
// belongs to r5, like a1b4, we multiply it by 19 and add the result to r0.
//
// a4b0 a3b0 a2b0 a1b0 a0b0 +
// a3b1 a2b1 a1b1 a0b1 19×a4b1 +
// a2b2 a1b2 a0b2 19×a4b2 19×a3b2 +
// a1b3 a0b3 19×a4b3 19×a3b3 19×a2b3 +
// a0b4 19×a4b4 19×a3b4 19×a2b4 19×a1b4 =
// --------------------------------------
// r4 r3 r2 r1 r0
//
// Finally we add up the columns into wide, overlapping limbs.
// r0 = a0×b0 + 19×(a1×b4 + a2×b3 + a3×b2 + a4×b1)
r0 := mul(a0, b0)
r0 = addMul19(r0, a1, b4)
r0 = addMul19(r0, a2, b3)
r0 = addMul19(r0, a3, b2)
r0 = addMul19(r0, a4, b1)
// r1 = a0×b1 + a1×b0 + 19×(a2×b4 + a3×b3 + a4×b2)
r1 := mul(a0, b1)
r1 = addMul(r1, a1, b0)
r1 = addMul19(r1, a2, b4)
r1 = addMul19(r1, a3, b3)
r1 = addMul19(r1, a4, b2)
// r2 = a0×b2 + a1×b1 + a2×b0 + 19×(a3×b4 + a4×b3)
r2 := mul(a0, b2)
r2 = addMul(r2, a1, b1)
r2 = addMul(r2, a2, b0)
r2 = addMul19(r2, a3, b4)
r2 = addMul19(r2, a4, b3)
// r3 = a0×b3 + a1×b2 + a2×b1 + a3×b0 + 19×a4×b4
r3 := mul(a0, b3)
r3 = addMul(r3, a1, b2)
r3 = addMul(r3, a2, b1)
r3 = addMul(r3, a3, b0)
r3 = addMul19(r3, a4, b4)
// r4 = a0×b4 + a1×b3 + a2×b2 + a3×b1 + a4×b0
r4 := mul(a0, b4)
r4 = addMul(r4, a1, b3)
r4 = addMul(r4, a2, b2)
r4 = addMul(r4, a3, b1)
r4 = addMul(r4, a4, b0)
// After the multiplication, we need to reduce (carry) the five coefficients
// to obtain a result with limbs that are at most slightly larger than 2⁵¹,
// to respect the Element invariant.
//
// Overall, the reduction works the same as carryPropagate, except with
// wider inputs: we take the carry for each coefficient by shifting it right
// by 51, and add it to the limb above it. The top carry is multiplied by 19
// according to the reduction identity and added to the lowest limb.
//
// The largest coefficient (r0) will be at most 111 bits, which guarantees
// that all carries are at most 111 - 51 = 60 bits, which fits in a uint64.
//
// r0 = a0×b0 + 19×(a1×b4 + a2×b3 + a3×b2 + a4×b1)
// r0 < 2⁵²×2⁵² + 19×(2⁵²×2⁵² + 2⁵²×2⁵² + 2⁵²×2⁵² + 2⁵²×2⁵²)
// r0 < (1 + 19 × 4) × 2⁵² × 2⁵²
// r0 < 2⁷ × 2⁵² × 2⁵²
// r0 < 2¹¹¹
//
// Moreover, the top coefficient (r4) is at most 107 bits, so c4 is at most
// 56 bits, and c4 * 19 is at most 61 bits, which again fits in a uint64 and
// allows us to easily apply the reduction identity.
//
// r4 = a0×b4 + a1×b3 + a2×b2 + a3×b1 + a4×b0
// r4 < 5 × 2⁵² × 2⁵²
// r4 < 2¹⁰⁷
//
c0 := shiftRightBy51(r0)
c1 := shiftRightBy51(r1)
c2 := shiftRightBy51(r2)
c3 := shiftRightBy51(r3)
c4 := shiftRightBy51(r4)
rr0 := r0.lo&maskLow51Bits + mul19(c4)
rr1 := r1.lo&maskLow51Bits + c0
rr2 := r2.lo&maskLow51Bits + c1
rr3 := r3.lo&maskLow51Bits + c2
rr4 := r4.lo&maskLow51Bits + c3
// Now all coefficients fit into 64-bit registers but are still too large to
// be passed around as an Element. We therefore do one last carry chain,
// where the carries will be small enough to fit in the wiggle room above 2⁵¹.
v.l0 = rr0&maskLow51Bits + mul19(rr4>>51)
v.l1 = rr1&maskLow51Bits + rr0>>51
v.l2 = rr2&maskLow51Bits + rr1>>51
v.l3 = rr3&maskLow51Bits + rr2>>51
v.l4 = rr4&maskLow51Bits + rr3>>51
}
////////////////////////////////////////////////////////////////////////////////////////////////////////////////////////
func feSquareGeneric(v, a *Element) {
l0 := a.l0
l1 := a.l1
l2 := a.l2
l3 := a.l3
l4 := a.l4
// Squaring works precisely like multiplication above, but thanks to its
// symmetry we get to group a few terms together.
//
// l4 l3 l2 l1 l0 x
// l4 l3 l2 l1 l0 =
// ------------------------
// l4l0 l3l0 l2l0 l1l0 l0l0 +
// l4l1 l3l1 l2l1 l1l1 l0l1 +
// l4l2 l3l2 l2l2 l1l2 l0l2 +
// l4l3 l3l3 l2l3 l1l3 l0l3 +
// l4l4 l3l4 l2l4 l1l4 l0l4 =
// ----------------------------------------------
// r8 r7 r6 r5 r4 r3 r2 r1 r0
//
// l4l0 l3l0 l2l0 l1l0 l0l0 +
// l3l1 l2l1 l1l1 l0l1 19×l4l1 +
// l2l2 l1l2 l0l2 19×l4l2 19×l3l2 +
// l1l3 l0l3 19×l4l3 19×l3l3 19×l2l3 +
// l0l4 19×l4l4 19×l3l4 19×l2l4 19×l1l4 =
// --------------------------------------
// r4 r3 r2 r1 r0
// r0 = l0×l0 + 19×(l1×l4 + l2×l3 + l3×l2 + l4×l1) = l0×l0 + 19×2×(l1×l4 + l2×l3)
r0 := mul(l0, l0)
r0 = addMul38(r0, l1, l4)
r0 = addMul38(r0, l2, l3)
// r1 = l0×l1 + l1×l0 + 19×(l2×l4 + l3×l3 + l4×l2) = 2×l0×l1 + 19×2×l2×l4 + 19×l3×l3
r1 := mul(l0*2, l1)
r1 = addMul38(r1, l2, l4)
r1 = addMul19(r1, l3, l3)
// r2 = l0×l2 + l1×l1 + l2×l0 + 19×(l3×l4 + l4×l3) = 2×l0×l2 + l1×l1 + 19×2×l3×l4
r2 := mul(l0*2, l2)
r2 = addMul(r2, l1, l1)
r2 = addMul38(r2, l3, l4)
// r3 = l0×l3 + l1×l2 + l2×l1 + l3×l0 + 19×l4×l4 = 2×l0×l3 + 2×l1×l2 + 19×l4×l4
r3 := mul(l0*2, l3)
r3 = addMul(r3, l1*2, l2)
r3 = addMul19(r3, l4, l4)
// r4 = l0×l4 + l1×l3 + l2×l2 + l3×l1 + l4×l0 = 2×l0×l4 + 2×l1×l3 + l2×l2
r4 := mul(l0*2, l4)
r4 = addMul(r4, l1*2, l3)
r4 = addMul(r4, l2, l2)
c0 := shiftRightBy51(r0)
c1 := shiftRightBy51(r1)
c2 := shiftRightBy51(r2)
c3 := shiftRightBy51(r3)
c4 := shiftRightBy51(r4)
rr0 := r0.lo&maskLow51Bits + mul19(c4)
rr1 := r1.lo&maskLow51Bits + c0
rr2 := r2.lo&maskLow51Bits + c1
rr3 := r3.lo&maskLow51Bits + c2
rr4 := r4.lo&maskLow51Bits + c3
v.l0 = rr0&maskLow51Bits + mul19(rr4>>51)
v.l1 = rr1&maskLow51Bits + rr0>>51
v.l2 = rr2&maskLow51Bits + rr1>>51
v.l3 = rr3&maskLow51Bits + rr2>>51
v.l4 = rr4&maskLow51Bits + rr3>>51
}
////////////////////////////////////////////////////////////////////////////////////////////////////////////////////////
// carryPropagate brings the limbs below 52 bits by applying the reduction
// identity (a * 2²⁵⁵ + b = a * 19 + b) to the l4 carry.
func (v *Element) carryPropagate() *Element {
// (l4>>51) is at most 64 - 51 = 13 bits, so (l4>>51)*19 is at most 18 bits, and
// the final l0 will be at most 52 bits. Similarly for the rest.
l0 := v.l0
v.l0 = v.l0&maskLow51Bits + mul19(v.l4>>51)
v.l4 = v.l4&maskLow51Bits + v.l3>>51
v.l3 = v.l3&maskLow51Bits + v.l2>>51
v.l2 = v.l2&maskLow51Bits + v.l1>>51
v.l1 = v.l1&maskLow51Bits + l0>>51
return v
}
////////////////////////////////////////////////////////////////////////////////////////////////////////////////////////

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////////////////////////////////////////////////////////////////////////////////////////////////////////////////////////
//
// Source: https://sources.truenas.cloud/code
// Import: sources.truenas.cloud/code/edwards25519/field
//
////////////////////////////////////////////////////////////////////////////////////////////////////////////////////////
//
// Copyright (c) 2019 The Go Authors. All rights Reserved.
// Use of this source code is goverened by a BSD-style
// license that can be found in the LICENSE file.
//
////////////////////////////////////////////////////////////////////////////////////////////////////////////////////////
package field
////////////////////////////////////////////////////////////////////////////////////////////////////////////////////////
import (
"bytes"
"crypto/rand"
"encoding/hex"
"io"
"math/big"
"math/bits"
mathrand "math/rand"
"reflect"
"testing"
"testing/quick"
)
////////////////////////////////////////////////////////////////////////////////////////////////////////////////////////
func (v Element) String() string {
return hex.EncodeToString(v.Bytes())
}
////////////////////////////////////////////////////////////////////////////////////////////////////////////////////////
// quickCheckConfig returns a quick.Config that scales the max count by the
// given factor if the -short flag is not set.
func quickCheckConfig(slowScale int) *quick.Config {
cfg := new(quick.Config)
if !testing.Short() {
cfg.MaxCountScale = float64(slowScale)
}
return cfg
}
////////////////////////////////////////////////////////////////////////////////////////////////////////////////////////
func generateFieldElement(rand *mathrand.Rand) Element {
const maskLow52Bits = (1 << 52) - 1
return Element{
rand.Uint64() & maskLow52Bits,
rand.Uint64() & maskLow52Bits,
rand.Uint64() & maskLow52Bits,
rand.Uint64() & maskLow52Bits,
rand.Uint64() & maskLow52Bits,
}
}
////////////////////////////////////////////////////////////////////////////////////////////////////////////////////////
// weirdLimbs can be combined to generate a range of edge-case field elements.
// 0 and -1 are intentionally more weighted, as they combine well.
var (
weirdLimbs51 = []uint64{
0, 0, 0, 0,
1,
19 - 1,
19,
0x2aaaaaaaaaaaa,
0x5555555555555,
(1 << 51) - 20,
(1 << 51) - 19,
(1 << 51) - 1, (1 << 51) - 1,
(1 << 51) - 1, (1 << 51) - 1,
}
weirdLimbs52 = []uint64{
0, 0, 0, 0, 0, 0,
1,
19 - 1,
19,
0x2aaaaaaaaaaaa,
0x5555555555555,
(1 << 51) - 20,
(1 << 51) - 19,
(1 << 51) - 1, (1 << 51) - 1,
(1 << 51) - 1, (1 << 51) - 1,
(1 << 51) - 1, (1 << 51) - 1,
1 << 51,
(1 << 51) + 1,
(1 << 52) - 19,
(1 << 52) - 1,
}
)
////////////////////////////////////////////////////////////////////////////////////////////////////////////////////////
func generateWeirdFieldElement(rand *mathrand.Rand) Element {
return Element{
weirdLimbs52[rand.Intn(len(weirdLimbs52))],
weirdLimbs51[rand.Intn(len(weirdLimbs51))],
weirdLimbs51[rand.Intn(len(weirdLimbs51))],
weirdLimbs51[rand.Intn(len(weirdLimbs51))],
weirdLimbs51[rand.Intn(len(weirdLimbs51))],
}
}
////////////////////////////////////////////////////////////////////////////////////////////////////////////////////////
func (Element) Generate(rand *mathrand.Rand, size int) reflect.Value {
if rand.Intn(2) == 0 {
return reflect.ValueOf(generateWeirdFieldElement(rand))
}
return reflect.ValueOf(generateFieldElement(rand))
}
////////////////////////////////////////////////////////////////////////////////////////////////////////////////////////
// isInBounds returns whether the element is within the expected bit size bounds
// after a light reduction.
func isInBounds(x *Element) bool {
return bits.Len64(x.l0) <= 52 &&
bits.Len64(x.l1) <= 52 &&
bits.Len64(x.l2) <= 52 &&
bits.Len64(x.l3) <= 52 &&
bits.Len64(x.l4) <= 52
}
////////////////////////////////////////////////////////////////////////////////////////////////////////////////////////
func TestMultiplyDistributesOverAdd(t *testing.T) {
multiplyDistributesOverAdd := func(x, y, z Element) bool {
// Compute t1 = (x+y)*z
t1 := new(Element)
t1.Add(&x, &y)
t1.Multiply(t1, &z)
// Compute t2 = x*z + y*z
t2 := new(Element)
t3 := new(Element)
t2.Multiply(&x, &z)
t3.Multiply(&y, &z)
t2.Add(t2, t3)
return t1.Equal(t2) == 1 && isInBounds(t1) && isInBounds(t2)
}
if err := quick.Check(multiplyDistributesOverAdd, quickCheckConfig(1024)); err != nil {
t.Error(err)
}
}
////////////////////////////////////////////////////////////////////////////////////////////////////////////////////////
func TestMul64to128(t *testing.T) {
a := uint64(5)
b := uint64(5)
r := mul(a, b)
if r.lo != 0x19 || r.hi != 0 {
t.Errorf("lo-range wide mult failed, got %d + %d*(2**64)", r.lo, r.hi)
}
a = uint64(18014398509481983) // 2^54 - 1
b = uint64(18014398509481983) // 2^54 - 1
r = mul(a, b)
if r.lo != 0xff80000000000001 || r.hi != 0xfffffffffff {
t.Errorf("hi-range wide mult failed, got %d + %d*(2**64)", r.lo, r.hi)
}
a = uint64(1125899906842661)
b = uint64(2097155)
r = mul(a, b)
r = addMul(r, a, b)
r = addMul(r, a, b)
r = addMul(r, a, b)
r = addMul(r, a, b)
if r.lo != 16888498990613035 || r.hi != 640 {
t.Errorf("wrong answer: %d + %d*(2**64)", r.lo, r.hi)
}
}
////////////////////////////////////////////////////////////////////////////////////////////////////////////////////////
func TestSetBytesRoundTrip(t *testing.T) {
f1 := func(in [32]byte, fe Element) bool {
fe.SetBytes(in[:])
// Mask the most significant bit as it's ignored by SetBytes. (Now
// instead of earlier so we check the masking in SetBytes is working.)
in[len(in)-1] &= (1 << 7) - 1
return bytes.Equal(in[:], fe.Bytes()) && isInBounds(&fe)
}
if err := quick.Check(f1, nil); err != nil {
t.Errorf("failed bytes->FE->bytes round-trip: %v", err)
}
f2 := func(fe, r Element) bool {
r.SetBytes(fe.Bytes())
// Intentionally not using Equal not to go through Bytes again.
// Calling reduce because both Generate and SetBytes can produce
// non-canonical representations.
fe.reduce()
r.reduce()
return fe == r
}
if err := quick.Check(f2, nil); err != nil {
t.Errorf("failed FE->bytes->FE round-trip: %v", err)
}
// Check some fixed vectors from dalek
type feRTTest struct {
fe Element
b []byte
}
var tests = []feRTTest{
{
fe: Element{358744748052810, 1691584618240980, 977650209285361, 1429865912637724, 560044844278676},
b: []byte{74, 209, 69, 197, 70, 70, 161, 222, 56, 226, 229, 19, 112, 60, 25, 92, 187, 74, 222, 56, 50, 153, 51, 233, 40, 74, 57, 6, 160, 185, 213, 31},
},
{
fe: Element{84926274344903, 473620666599931, 365590438845504, 1028470286882429, 2146499180330972},
b: []byte{199, 23, 106, 112, 61, 77, 216, 79, 186, 60, 11, 118, 13, 16, 103, 15, 42, 32, 83, 250, 44, 57, 204, 198, 78, 199, 253, 119, 146, 172, 3, 122},
},
}
for _, tt := range tests {
b := tt.fe.Bytes()
fe, _ := new(Element).SetBytes(tt.b)
if !bytes.Equal(b, tt.b) || fe.Equal(&tt.fe) != 1 {
t.Errorf("Failed fixed roundtrip: %v", tt)
}
}
}
////////////////////////////////////////////////////////////////////////////////////////////////////////////////////////
func swapEndianness(buf []byte) []byte {
for i := 0; i < len(buf)/2; i++ {
buf[i], buf[len(buf)-i-1] = buf[len(buf)-i-1], buf[i]
}
return buf
}
////////////////////////////////////////////////////////////////////////////////////////////////////////////////////////
func TestBytesBigEquivalence(t *testing.T) {
f1 := func(in [32]byte, fe, fe1 Element) bool {
fe.SetBytes(in[:])
in[len(in)-1] &= (1 << 7) - 1 // mask the most significant bit
b := new(big.Int).SetBytes(swapEndianness(in[:]))
fe1.fromBig(b)
if fe != fe1 {
return false
}
buf := make([]byte, 32)
buf = swapEndianness(fe1.toBig().FillBytes(buf))
return bytes.Equal(fe.Bytes(), buf) && isInBounds(&fe) && isInBounds(&fe1)
}
if err := quick.Check(f1, nil); err != nil {
t.Error(err)
}
}
////////////////////////////////////////////////////////////////////////////////////////////////////////////////////////
// fromBig sets v = n, and returns v. The bit length of n must not exceed 256.
func (v *Element) fromBig(n *big.Int) *Element {
if n.BitLen() > 32*8 {
panic("edwards25519: invalid field element input size")
}
buf := make([]byte, 0, 32)
for _, word := range n.Bits() {
for i := 0; i < bits.UintSize; i += 8 {
if len(buf) >= cap(buf) {
break
}
buf = append(buf, byte(word))
word >>= 8
}
}
v.SetBytes(buf[:32])
return v
}
////////////////////////////////////////////////////////////////////////////////////////////////////////////////////////
func (v *Element) fromDecimal(s string) *Element {
n, ok := new(big.Int).SetString(s, 10)
if !ok {
panic("not a valid decimal: " + s)
}
return v.fromBig(n)
}
////////////////////////////////////////////////////////////////////////////////////////////////////////////////////////
// toBig returns v as a big.Int.
func (v *Element) toBig() *big.Int {
buf := v.Bytes()
words := make([]big.Word, 32*8/bits.UintSize)
for n := range words {
for i := 0; i < bits.UintSize; i += 8 {
if len(buf) == 0 {
break
}
words[n] |= big.Word(buf[0]) << big.Word(i)
buf = buf[1:]
}
}
return new(big.Int).SetBits(words)
}
////////////////////////////////////////////////////////////////////////////////////////////////////////////////////////
func TestDecimalConstants(t *testing.T) {
sqrtM1String := "19681161376707505956807079304988542015446066515923890162744021073123829784752"
if exp := new(Element).fromDecimal(sqrtM1String); sqrtM1.Equal(exp) != 1 {
t.Errorf("sqrtM1 is %v, expected %v", sqrtM1, exp)
}
// d is in the parent package, and we don't want to expose d or fromDecimal.
// dString := "37095705934669439343138083508754565189542113879843219016388785533085940283555"
// if exp := new(Element).fromDecimal(dString); d.Equal(exp) != 1 {
// t.Errorf("d is %v, expected %v", d, exp)
// }
}
////////////////////////////////////////////////////////////////////////////////////////////////////////////////////////
func TestSetBytesRoundTripEdgeCases(t *testing.T) {
// TODO: values close to 0, close to 2^255-19, between 2^255-19 and 2^255-1,
// and between 2^255 and 2^256-1. Test both the documented SetBytes
// behavior, and that Bytes reduces them.
}
////////////////////////////////////////////////////////////////////////////////////////////////////////////////////////
// Tests self-consistency between Multiply and Square.
func TestConsistency(t *testing.T) {
var x Element
var x2, x2sq Element
x = Element{1, 1, 1, 1, 1}
x2.Multiply(&x, &x)
x2sq.Square(&x)
if x2 != x2sq {
t.Fatalf("all ones failed\nmul: %x\nsqr: %x\n", x2, x2sq)
}
var bytes [32]byte
_, err := io.ReadFull(rand.Reader, bytes[:])
if err != nil {
t.Fatal(err)
}
x.SetBytes(bytes[:])
x2.Multiply(&x, &x)
x2sq.Square(&x)
if x2 != x2sq {
t.Fatalf("all ones failed\nmul: %x\nsqr: %x\n", x2, x2sq)
}
}
////////////////////////////////////////////////////////////////////////////////////////////////////////////////////////
func TestEqual(t *testing.T) {
x := Element{1, 1, 1, 1, 1}
y := Element{5, 4, 3, 2, 1}
eq := x.Equal(&x)
if eq != 1 {
t.Errorf("wrong about equality")
}
eq = x.Equal(&y)
if eq != 0 {
t.Errorf("wrong about inequality")
}
}
////////////////////////////////////////////////////////////////////////////////////////////////////////////////////////
func TestInvert(t *testing.T) {
x := Element{1, 1, 1, 1, 1}
one := Element{1, 0, 0, 0, 0}
var xinv, r Element
xinv.Invert(&x)
r.Multiply(&x, &xinv)
r.reduce()
if one != r {
t.Errorf("inversion identity failed, got: %x", r)
}
var bytes [32]byte
_, err := io.ReadFull(rand.Reader, bytes[:])
if err != nil {
t.Fatal(err)
}
x.SetBytes(bytes[:])
xinv.Invert(&x)
r.Multiply(&x, &xinv)
r.reduce()
if one != r {
t.Errorf("random inversion identity failed, got: %x for field element %x", r, x)
}
zero := Element{}
x.Set(&zero)
if xx := xinv.Invert(&x); xx != &xinv {
t.Errorf("inverting zero did not return the receiver")
} else if xinv.Equal(&zero) != 1 {
t.Errorf("inverting zero did not return zero")
}
}
////////////////////////////////////////////////////////////////////////////////////////////////////////////////////////
func TestSelectSwap(t *testing.T) {
a := Element{358744748052810, 1691584618240980, 977650209285361, 1429865912637724, 560044844278676}
b := Element{84926274344903, 473620666599931, 365590438845504, 1028470286882429, 2146499180330972}
var c, d Element
c.Select(&a, &b, 1)
d.Select(&a, &b, 0)
if c.Equal(&a) != 1 || d.Equal(&b) != 1 {
t.Errorf("Select failed")
}
c.Swap(&d, 0)
if c.Equal(&a) != 1 || d.Equal(&b) != 1 {
t.Errorf("Swap failed")
}
c.Swap(&d, 1)
if c.Equal(&b) != 1 || d.Equal(&a) != 1 {
t.Errorf("Swap failed")
}
}
////////////////////////////////////////////////////////////////////////////////////////////////////////////////////////
func TestMult32(t *testing.T) {
mult32EquivalentToMul := func(x Element, y uint32) bool {
t1 := new(Element)
for i := 0; i < 100; i++ {
t1.Mult32(&x, y)
}
ty := new(Element)
ty.l0 = uint64(y)
t2 := new(Element)
for i := 0; i < 100; i++ {
t2.Multiply(&x, ty)
}
return t1.Equal(t2) == 1 && isInBounds(t1) && isInBounds(t2)
}
if err := quick.Check(mult32EquivalentToMul, quickCheckConfig(1024)); err != nil {
t.Error(err)
}
}
////////////////////////////////////////////////////////////////////////////////////////////////////////////////////////
func TestSqrtRatio(t *testing.T) {
// From draft-irtf-cfrg-ristretto255-decaf448-00, Appendix A.4.
type test struct {
u, v []byte
wasSquare int
r []byte
}
var tests = []test{
// If u is 0, the function is defined to return (0, TRUE), even if v
// is zero. Note that where used in this package, the denominator v
// is never zero.
{
decodeHex("0000000000000000000000000000000000000000000000000000000000000000"),
decodeHex("0000000000000000000000000000000000000000000000000000000000000000"),
1, decodeHex("0000000000000000000000000000000000000000000000000000000000000000"),
},
// 0/1 == 0²
{
decodeHex("0000000000000000000000000000000000000000000000000000000000000000"),
decodeHex("0100000000000000000000000000000000000000000000000000000000000000"),
1, decodeHex("0000000000000000000000000000000000000000000000000000000000000000"),
},
// If u is non-zero and v is zero, defined to return (0, FALSE).
{
decodeHex("0100000000000000000000000000000000000000000000000000000000000000"),
decodeHex("0000000000000000000000000000000000000000000000000000000000000000"),
0, decodeHex("0000000000000000000000000000000000000000000000000000000000000000"),
},
// 2/1 is not square in this field.
{
decodeHex("0200000000000000000000000000000000000000000000000000000000000000"),
decodeHex("0100000000000000000000000000000000000000000000000000000000000000"),
0, decodeHex("3c5ff1b5d8e4113b871bd052f9e7bcd0582804c266ffb2d4f4203eb07fdb7c54"),
},
// 4/1 == 2²
{
decodeHex("0400000000000000000000000000000000000000000000000000000000000000"),
decodeHex("0100000000000000000000000000000000000000000000000000000000000000"),
1, decodeHex("0200000000000000000000000000000000000000000000000000000000000000"),
},
// 1/4 == (2⁻¹)² == (2^(p-2))² per Euler's theorem
{
decodeHex("0100000000000000000000000000000000000000000000000000000000000000"),
decodeHex("0400000000000000000000000000000000000000000000000000000000000000"),
1, decodeHex("f6ffffffffffffffffffffffffffffffffffffffffffffffffffffffffffff3f"),
},
}
for i, tt := range tests {
u, _ := new(Element).SetBytes(tt.u)
v, _ := new(Element).SetBytes(tt.v)
want, _ := new(Element).SetBytes(tt.r)
got, wasSquare := new(Element).SqrtRatio(u, v)
if got.Equal(want) == 0 || wasSquare != tt.wasSquare {
t.Errorf("%d: got (%v, %v), want (%v, %v)", i, got, wasSquare, want, tt.wasSquare)
}
}
}
////////////////////////////////////////////////////////////////////////////////////////////////////////////////////////
func TestFeSquare(t *testing.T) {
asmLikeGeneric := func(a Element) bool {
t1 := a
t2 := a
feSquareGeneric(&t1, &t1)
feSquare(&t2, &t2)
if t1 != t2 {
t.Logf("got: %#v,\nexpected: %#v", t1, t2)
}
return t1 == t2 && isInBounds(&t2)
}
if err := quick.Check(asmLikeGeneric, quickCheckConfig(1024)); err != nil {
t.Error(err)
}
}
////////////////////////////////////////////////////////////////////////////////////////////////////////////////////////
func TestFeMul(t *testing.T) {
asmLikeGeneric := func(a, b Element) bool {
a1 := a
a2 := a
b1 := b
b2 := b
feMulGeneric(&a1, &a1, &b1)
feMul(&a2, &a2, &b2)
if a1 != a2 || b1 != b2 {
t.Logf("got: %#v,\nexpected: %#v", a1, a2)
t.Logf("got: %#v,\nexpected: %#v", b1, b2)
}
return a1 == a2 && isInBounds(&a2) &&
b1 == b2 && isInBounds(&b2)
}
if err := quick.Check(asmLikeGeneric, quickCheckConfig(1024)); err != nil {
t.Error(err)
}
}
////////////////////////////////////////////////////////////////////////////////////////////////////////////////////////
func decodeHex(s string) []byte {
b, err := hex.DecodeString(s)
if err != nil {
panic(err)
}
return b
}
////////////////////////////////////////////////////////////////////////////////////////////////////////////////////////