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// Copyright (c) 2019 The Go Authors. All rights reserved.
// Use of this source code is governed by a BSD-style
// license that can be found in the LICENSE file.
package edwards25519
import "sync"
// basepointTable is a set of 32 affineLookupTables, where table i is generated
// from 256i * basepoint. It is precomputed the first time it's used.
func basepointTable() *[32]affineLookupTable {
basepointTablePrecomp.initOnce.Do(func() {
p := NewGeneratorPoint()
for i := 0; i < 32; i++ {
basepointTablePrecomp.table[i].FromP3(p)
for j := 0; j < 8; j++ {
p.Add(p, p)
}
}
})
return &basepointTablePrecomp.table
}
var basepointTablePrecomp struct {
table [32]affineLookupTable
initOnce sync.Once
}
// ScalarBaseMult sets v = x * B, where B is the canonical generator, and
// returns v.
//
// The scalar multiplication is done in constant time.
func (v *Point) ScalarBaseMult(x *Scalar) *Point {
basepointTable := basepointTable()
// Write x = sum(x_i * 16^i) so x*B = sum( B*x_i*16^i )
// as described in the Ed25519 paper
//
// Group even and odd coefficients
// x*B = x_0*16^0*B + x_2*16^2*B + ... + x_62*16^62*B
// + x_1*16^1*B + x_3*16^3*B + ... + x_63*16^63*B
// x*B = x_0*16^0*B + x_2*16^2*B + ... + x_62*16^62*B
// + 16*( x_1*16^0*B + x_3*16^2*B + ... + x_63*16^62*B)
//
// We use a lookup table for each i to get x_i*16^(2*i)*B
// and do four doublings to multiply by 16.
digits := x.signedRadix16()
multiple := &affineCached{}
tmp1 := &projP1xP1{}
tmp2 := &projP2{}
// Accumulate the odd components first
v.Set(NewIdentityPoint())
for i := 1; i < 64; i += 2 {
basepointTable[i/2].SelectInto(multiple, digits[i])
tmp1.AddAffine(v, multiple)
v.fromP1xP1(tmp1)
}
// Multiply by 16
tmp2.FromP3(v) // tmp2 = v in P2 coords
tmp1.Double(tmp2) // tmp1 = 2*v in P1xP1 coords
tmp2.FromP1xP1(tmp1) // tmp2 = 2*v in P2 coords
tmp1.Double(tmp2) // tmp1 = 4*v in P1xP1 coords
tmp2.FromP1xP1(tmp1) // tmp2 = 4*v in P2 coords
tmp1.Double(tmp2) // tmp1 = 8*v in P1xP1 coords
tmp2.FromP1xP1(tmp1) // tmp2 = 8*v in P2 coords
tmp1.Double(tmp2) // tmp1 = 16*v in P1xP1 coords
v.fromP1xP1(tmp1) // now v = 16*(odd components)
// Accumulate the even components
for i := 0; i < 64; i += 2 {
basepointTable[i/2].SelectInto(multiple, digits[i])
tmp1.AddAffine(v, multiple)
v.fromP1xP1(tmp1)
}
return v
}
// ScalarMult sets v = x * q, and returns v.
//
// The scalar multiplication is done in constant time.
func (v *Point) ScalarMult(x *Scalar, q *Point) *Point {
checkInitialized(q)
var table projLookupTable
table.FromP3(q)
// Write x = sum(x_i * 16^i)
// so x*Q = sum( Q*x_i*16^i )
// = Q*x_0 + 16*(Q*x_1 + 16*( ... + Q*x_63) ... )
// <------compute inside out---------
//
// We use the lookup table to get the x_i*Q values
// and do four doublings to compute 16*Q
digits := x.signedRadix16()
// Unwrap first loop iteration to save computing 16*identity
multiple := &projCached{}
tmp1 := &projP1xP1{}
tmp2 := &projP2{}
table.SelectInto(multiple, digits[63])
v.Set(NewIdentityPoint())
tmp1.Add(v, multiple) // tmp1 = x_63*Q in P1xP1 coords
for i := 62; i >= 0; i-- {
tmp2.FromP1xP1(tmp1) // tmp2 = (prev) in P2 coords
tmp1.Double(tmp2) // tmp1 = 2*(prev) in P1xP1 coords
tmp2.FromP1xP1(tmp1) // tmp2 = 2*(prev) in P2 coords
tmp1.Double(tmp2) // tmp1 = 4*(prev) in P1xP1 coords
tmp2.FromP1xP1(tmp1) // tmp2 = 4*(prev) in P2 coords
tmp1.Double(tmp2) // tmp1 = 8*(prev) in P1xP1 coords
tmp2.FromP1xP1(tmp1) // tmp2 = 8*(prev) in P2 coords
tmp1.Double(tmp2) // tmp1 = 16*(prev) in P1xP1 coords
v.fromP1xP1(tmp1) // v = 16*(prev) in P3 coords
table.SelectInto(multiple, digits[i])
tmp1.Add(v, multiple) // tmp1 = x_i*Q + 16*(prev) in P1xP1 coords
}
v.fromP1xP1(tmp1)
return v
}
// basepointNafTable is the nafLookupTable8 for the basepoint.
// It is precomputed the first time it's used.
func basepointNafTable() *nafLookupTable8 {
basepointNafTablePrecomp.initOnce.Do(func() {
basepointNafTablePrecomp.table.FromP3(NewGeneratorPoint())
})
return &basepointNafTablePrecomp.table
}
var basepointNafTablePrecomp struct {
table nafLookupTable8
initOnce sync.Once
}
// VarTimeDoubleScalarBaseMult sets v = a * A + b * B, where B is the canonical
// generator, and returns v.
//
// Execution time depends on the inputs.
func (v *Point) VarTimeDoubleScalarBaseMult(a *Scalar, A *Point, b *Scalar) *Point {
checkInitialized(A)
// Similarly to the single variable-base approach, we compute
// digits and use them with a lookup table. However, because
// we are allowed to do variable-time operations, we don't
// need constant-time lookups or constant-time digit
// computations.
//
// So we use a non-adjacent form of some width w instead of
// radix 16. This is like a binary representation (one digit
// for each binary place) but we allow the digits to grow in
// magnitude up to 2^{w-1} so that the nonzero digits are as
// sparse as possible. Intuitively, this "condenses" the
// "mass" of the scalar onto sparse coefficients (meaning
// fewer additions).
basepointNafTable := basepointNafTable()
var aTable nafLookupTable5
aTable.FromP3(A)
// Because the basepoint is fixed, we can use a wider NAF
// corresponding to a bigger table.
aNaf := a.nonAdjacentForm(5)
bNaf := b.nonAdjacentForm(8)
// Find the first nonzero coefficient.
i := 255
for j := i; j >= 0; j-- {
if aNaf[j] != 0 || bNaf[j] != 0 {
break
}
}
multA := &projCached{}
multB := &affineCached{}
tmp1 := &projP1xP1{}
tmp2 := &projP2{}
tmp2.Zero()
// Move from high to low bits, doubling the accumulator
// at each iteration and checking whether there is a nonzero
// coefficient to look up a multiple of.
for ; i >= 0; i-- {
tmp1.Double(tmp2)
// Only update v if we have a nonzero coeff to add in.
if aNaf[i] > 0 {
v.fromP1xP1(tmp1)
aTable.SelectInto(multA, aNaf[i])
tmp1.Add(v, multA)
} else if aNaf[i] < 0 {
v.fromP1xP1(tmp1)
aTable.SelectInto(multA, -aNaf[i])
tmp1.Sub(v, multA)
}
if bNaf[i] > 0 {
v.fromP1xP1(tmp1)
basepointNafTable.SelectInto(multB, bNaf[i])
tmp1.AddAffine(v, multB)
} else if bNaf[i] < 0 {
v.fromP1xP1(tmp1)
basepointNafTable.SelectInto(multB, -bNaf[i])
tmp1.SubAffine(v, multB)
}
tmp2.FromP1xP1(tmp1)
}
v.fromP2(tmp2)
return v
}
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