b73a7c8249
Calling v.ScalarMult on a receiver v that is not the identity point results in an incorrect operation. This was fixed by setting v to the identity point in ScalarMult. A simple test was added to check this behaviour.
287 lines
8.8 KiB
Go
287 lines
8.8 KiB
Go
// Copyright (c) 2019 The Go Authors. All rights reserved.
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// Use of this source code is governed by a BSD-style
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// license that can be found in the LICENSE file.
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package edwards25519
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// ScalarBaseMult sets v = x * B, where B is the canonical generator, and
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// returns v.
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//
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// The scalar multiplication is done in constant time.
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func (v *Point) ScalarBaseMult(x *Scalar) *Point {
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// Write x = sum(x_i * 16^i) so x*B = sum( B*x_i*16^i )
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// as described in the Ed25519 paper
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//
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// Group even and odd coefficients
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// x*B = x_0*16^0*B + x_2*16^2*B + ... + x_62*16^62*B
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// + x_1*16^1*B + x_3*16^3*B + ... + x_63*16^63*B
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// x*B = x_0*16^0*B + x_2*16^2*B + ... + x_62*16^62*B
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// + 16*( x_1*16^0*B + x_3*16^2*B + ... + x_63*16^62*B)
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//
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// We use a lookup table for each i to get x_i*16^(2*i)*B
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// and do four doublings to multiply by 16.
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digits := x.signedRadix16()
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multiple := &affineCached{}
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tmp1 := &projP1xP1{}
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tmp2 := &projP2{}
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// Accumulate the odd components first
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v.Set(NewIdentityPoint())
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for i := 1; i < 64; i += 2 {
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basepointTable[i/2].SelectInto(multiple, digits[i])
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tmp1.AddAffine(v, multiple)
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v.fromP1xP1(tmp1)
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}
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// Multiply by 16
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tmp2.FromP3(v) // tmp2 = v in P2 coords
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tmp1.Double(tmp2) // tmp1 = 2*v in P1xP1 coords
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tmp2.FromP1xP1(tmp1) // tmp2 = 2*v in P2 coords
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tmp1.Double(tmp2) // tmp1 = 4*v in P1xP1 coords
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tmp2.FromP1xP1(tmp1) // tmp2 = 4*v in P2 coords
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tmp1.Double(tmp2) // tmp1 = 8*v in P1xP1 coords
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tmp2.FromP1xP1(tmp1) // tmp2 = 8*v in P2 coords
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tmp1.Double(tmp2) // tmp1 = 16*v in P1xP1 coords
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v.fromP1xP1(tmp1) // now v = 16*(odd components)
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// Accumulate the even components
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for i := 0; i < 64; i += 2 {
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basepointTable[i/2].SelectInto(multiple, digits[i])
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tmp1.AddAffine(v, multiple)
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v.fromP1xP1(tmp1)
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}
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return v
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}
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// ScalarMult sets v = x * q, and returns v.
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//
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// The scalar multiplication is done in constant time.
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func (v *Point) ScalarMult(x *Scalar, q *Point) *Point {
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checkInitialized(q)
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var table projLookupTable
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table.FromP3(q)
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// Write x = sum(x_i * 16^i)
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// so x*Q = sum( Q*x_i*16^i )
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// = Q*x_0 + 16*(Q*x_1 + 16*( ... + Q*x_63) ... )
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// <------compute inside out---------
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//
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// We use the lookup table to get the x_i*Q values
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// and do four doublings to compute 16*Q
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digits := x.signedRadix16()
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// Unwrap first loop iteration to save computing 16*identity
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multiple := &projCached{}
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tmp1 := &projP1xP1{}
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tmp2 := &projP2{}
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table.SelectInto(multiple, digits[63])
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v.Set(NewIdentityPoint())
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tmp1.Add(v, multiple) // tmp1 = x_63*Q in P1xP1 coords
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for i := 62; i >= 0; i-- {
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tmp2.FromP1xP1(tmp1) // tmp2 = (prev) in P2 coords
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tmp1.Double(tmp2) // tmp1 = 2*(prev) in P1xP1 coords
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tmp2.FromP1xP1(tmp1) // tmp2 = 2*(prev) in P2 coords
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tmp1.Double(tmp2) // tmp1 = 4*(prev) in P1xP1 coords
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tmp2.FromP1xP1(tmp1) // tmp2 = 4*(prev) in P2 coords
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tmp1.Double(tmp2) // tmp1 = 8*(prev) in P1xP1 coords
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tmp2.FromP1xP1(tmp1) // tmp2 = 8*(prev) in P2 coords
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tmp1.Double(tmp2) // tmp1 = 16*(prev) in P1xP1 coords
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v.fromP1xP1(tmp1) // v = 16*(prev) in P3 coords
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table.SelectInto(multiple, digits[i])
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tmp1.Add(v, multiple) // tmp1 = x_i*Q + 16*(prev) in P1xP1 coords
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}
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v.fromP1xP1(tmp1)
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return v
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}
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// MultiScalarMult sets v = sum(scalars[i] * points[i]), and returns v.
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//
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// Execution time depends only on the lengths of the two slices, which must match.
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func (v *Point) MultiScalarMult(scalars []*Scalar, points []*Point) *Point {
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if len(scalars) != len(points) {
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panic("edwards25519: called MultiScalarMult with different size inputs")
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}
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checkInitialized(points...)
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// Proceed as in the single-base case, but share doublings
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// between each point in the multiscalar equation.
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// Build lookup tables for each point
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tables := make([]projLookupTable, len(points))
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for i := range tables {
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tables[i].FromP3(points[i])
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}
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// Compute signed radix-16 digits for each scalar
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digits := make([][64]int8, len(scalars))
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for i := range digits {
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digits[i] = scalars[i].signedRadix16()
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}
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// Unwrap first loop iteration to save computing 16*identity
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multiple := &projCached{}
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tmp1 := &projP1xP1{}
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tmp2 := &projP2{}
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// Lookup-and-add the appropriate multiple of each input point
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for j := range tables {
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tables[j].SelectInto(multiple, digits[j][63])
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tmp1.Add(v, multiple) // tmp1 = v + x_(j,63)*Q in P1xP1 coords
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v.fromP1xP1(tmp1) // update v
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}
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tmp2.FromP3(v) // set up tmp2 = v in P2 coords for next iteration
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for i := 62; i >= 0; i-- {
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tmp1.Double(tmp2) // tmp1 = 2*(prev) in P1xP1 coords
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tmp2.FromP1xP1(tmp1) // tmp2 = 2*(prev) in P2 coords
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tmp1.Double(tmp2) // tmp1 = 4*(prev) in P1xP1 coords
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tmp2.FromP1xP1(tmp1) // tmp2 = 4*(prev) in P2 coords
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tmp1.Double(tmp2) // tmp1 = 8*(prev) in P1xP1 coords
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tmp2.FromP1xP1(tmp1) // tmp2 = 8*(prev) in P2 coords
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tmp1.Double(tmp2) // tmp1 = 16*(prev) in P1xP1 coords
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v.fromP1xP1(tmp1) // v = 16*(prev) in P3 coords
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// Lookup-and-add the appropriate multiple of each input point
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for j := range tables {
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tables[j].SelectInto(multiple, digits[j][i])
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tmp1.Add(v, multiple) // tmp1 = v + x_(j,i)*Q in P1xP1 coords
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v.fromP1xP1(tmp1) // update v
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}
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tmp2.FromP3(v) // set up tmp2 = v in P2 coords for next iteration
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}
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return v
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}
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// VarTimeDoubleScalarBaseMult sets v = a * A + b * B, where B is the canonical
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// generator, and returns v.
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//
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// Execution time depends on the inputs.
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func (v *Point) VarTimeDoubleScalarBaseMult(a *Scalar, A *Point, b *Scalar) *Point {
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checkInitialized(A)
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// Similarly to the single variable-base approach, we compute
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// digits and use them with a lookup table. However, because
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// we are allowed to do variable-time operations, we don't
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// need constant-time lookups or constant-time digit
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// computations.
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//
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// So we use a non-adjacent form of some width w instead of
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// radix 16. This is like a binary representation (one digit
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// for each binary place) but we allow the digits to grow in
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// magnitude up to 2^{w-1} so that the nonzero digits are as
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// sparse as possible. Intuitively, this "condenses" the
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// "mass" of the scalar onto sparse coefficients (meaning
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// fewer additions).
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var aTable nafLookupTable5
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aTable.FromP3(A)
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// Because the basepoint is fixed, we can use a wider NAF
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// corresponding to a bigger table.
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aNaf := a.nonAdjacentForm(5)
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bNaf := b.nonAdjacentForm(8)
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// Find the first nonzero coefficient.
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i := 255
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for j := i; j >= 0; j-- {
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if aNaf[j] != 0 || bNaf[j] != 0 {
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break
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}
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}
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multA := &projCached{}
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multB := &affineCached{}
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tmp1 := &projP1xP1{}
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tmp2 := &projP2{}
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tmp2.Zero()
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// Move from high to low bits, doubling the accumulator
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// at each iteration and checking whether there is a nonzero
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// coefficient to look up a multiple of.
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for ; i >= 0; i-- {
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tmp1.Double(tmp2)
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// Only update v if we have a nonzero coeff to add in.
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if aNaf[i] > 0 {
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v.fromP1xP1(tmp1)
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aTable.SelectInto(multA, aNaf[i])
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tmp1.Add(v, multA)
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} else if aNaf[i] < 0 {
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v.fromP1xP1(tmp1)
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aTable.SelectInto(multA, -aNaf[i])
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tmp1.Sub(v, multA)
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}
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if bNaf[i] > 0 {
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v.fromP1xP1(tmp1)
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basepointNafTable.SelectInto(multB, bNaf[i])
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tmp1.AddAffine(v, multB)
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} else if bNaf[i] < 0 {
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v.fromP1xP1(tmp1)
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basepointNafTable.SelectInto(multB, -bNaf[i])
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tmp1.SubAffine(v, multB)
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}
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tmp2.FromP1xP1(tmp1)
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}
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v.fromP2(tmp2)
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return v
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}
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// VarTimeMultiScalarMult sets v = sum(scalars[i] * points[i]), and returns v.
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//
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// Execution time depends on the inputs.
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func (v *Point) VarTimeMultiScalarMult(scalars []*Scalar, points []*Point) *Point {
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if len(scalars) != len(points) {
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panic("edwards25519: called VarTimeMultiScalarMult with different size inputs")
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}
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checkInitialized(points...)
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// Generalize double-base NAF computation to arbitrary sizes.
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// Here all the points are dynamic, so we only use the smaller
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// tables.
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// Build lookup tables for each point
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tables := make([]nafLookupTable5, len(points))
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for i := range tables {
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tables[i].FromP3(points[i])
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}
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// Compute a NAF for each scalar
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nafs := make([][256]int8, len(scalars))
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for i := range nafs {
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nafs[i] = scalars[i].nonAdjacentForm(5)
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}
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multiple := &projCached{}
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tmp1 := &projP1xP1{}
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tmp2 := &projP2{}
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tmp2.Zero()
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// Move from high to low bits, doubling the accumulator
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// at each iteration and checking whether there is a nonzero
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// coefficient to look up a multiple of.
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//
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// Skip trying to find the first nonzero coefficent, because
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// searching might be more work than a few extra doublings.
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for i := 255; i >= 0; i-- {
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tmp1.Double(tmp2)
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for j := range nafs {
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if nafs[j][i] > 0 {
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v.fromP1xP1(tmp1)
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tables[j].SelectInto(multiple, nafs[j][i])
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tmp1.Add(v, multiple)
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} else if nafs[j][i] < 0 {
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v.fromP1xP1(tmp1)
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tables[j].SelectInto(multiple, -nafs[j][i])
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tmp1.Sub(v, multiple)
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}
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}
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tmp2.FromP1xP1(tmp1)
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}
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v.fromP2(tmp2)
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return v
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}
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