aboutsummaryrefslogtreecommitdiffstats
path: root/contrib/go/_std_1.21/src/crypto/internal/edwards25519/scalarmult.go
blob: f7ca3cef993c0cc1de0bc6101cdac1fe5c7d3b17 (plain) (blame)
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
19
20
21
22
23
24
25
26
27
28
29
30
31
32
33
34
35
36
37
38
39
40
41
42
43
44
45
46
47
48
49
50
51
52
53
54
55
56
57
58
59
60
61
62
63
64
65
66
67
68
69
70
71
72
73
74
75
76
77
78
79
80
81
82
83
84
85
86
87
88
89
90
91
92
93
94
95
96
97
98
99
100
101
102
103
104
105
106
107
108
109
110
111
112
113
114
115
116
117
118
119
120
121
122
123
124
125
126
127
128
129
130
131
132
133
134
135
136
137
138
139
140
141
142
143
144
145
146
147
148
149
150
151
152
153
154
155
156
157
158
159
160
161
162
163
164
165
166
167
168
169
170
171
172
173
174
175
176
177
178
179
180
181
182
183
184
185
186
187
188
189
190
191
192
193
194
195
196
197
198
199
200
201
202
203
204
205
206
207
208
209
210
211
212
213
214
// 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
}