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
|
package stage1
import (
. "kesim.org/seal/nizk"
)
// Implements the proof and verification of statements of the following form:
// [ Z=g^(xy) && X=g^x && Y=g^y && C=g^(αβ) && A=g^α && B=g^β ]
// || [ Z=g^(xr) && X=g^x && R=g^r && C=g^(αβ+1) && A=g^α && B=g^β ]
// for given Z, X, Y, R, C, A and B
type Statement struct {
x *Scalar
y *Scalar
r *Scalar
α *Scalar
β *Scalar
plus bool
*Commitment
}
type Commitment struct {
A *Point
B *Point
C *Point
R *Point
X *Point
Y *Point
Z *Point
}
func NewStatement(x, y, r, α, β *Scalar, plus bool) *Statement {
return &Statement{
x: x,
y: y,
r: r,
α: α,
β: β,
plus: plus,
Commitment: commitment(x, y, r, α, β, plus),
}
}
func commitment(x, y, r, α, β *Scalar, plus bool) *Commitment {
var Z *Point
φ := α.Mul(β)
if plus {
Z = G.Exp(x.Mul(r))
φ = φ.Add(One)
} else {
Z = G.Exp(x.Mul(y))
}
return &Commitment{
Z: Z,
X: G.Exp(x),
Y: G.Exp(y),
R: G.Exp(r),
A: G.Exp(α),
B: G.Exp(β),
C: G.Exp(φ),
}
}
func (s *Statement) Commit() *Commitment {
return s.Commitment
}
type Proof struct {
Ch [2]*Scalar
Rho [2][2]*Scalar
}
func (s *Statement) Proof() *Proof {
var ε [2][4]*Point
var r1, r2, ρ1, ρ2, ω *Scalar
for _, s := range []**Scalar{&r1, &r2, &ρ1, &ρ2, &ω} {
*s = Curve.RandomScalar()
}
if s.plus {
ε[0][0] = G.Exp(r1).Mul(s.X.Exp(ω))
ε[0][1] = G.Exp(r2).Mul(s.A.Exp(ω))
ε[0][2] = s.Y.Exp(r1).Mul(s.Z.Exp(ω))
ε[0][3] = s.B.Exp(r2).Mul(s.C.Exp(ω))
ε[1][0] = G.Exp(ρ1)
ε[1][1] = G.Exp(ρ2)
ε[1][2] = s.R.Exp(ρ1)
ε[1][3] = s.B.Exp(ρ2)
} else {
ε[0][0] = G.Exp(r1)
ε[0][1] = G.Exp(r2)
ε[0][2] = s.Y.Exp(r1)
ε[0][3] = s.B.Exp(r2)
ε[1][0] = G.Exp(ρ1).Mul(s.X.Exp(ω))
ε[1][1] = G.Exp(ρ2).Mul(s.A.Exp(ω))
ε[1][2] = s.R.Exp(ρ1).Mul(s.Z.Exp(ω))
ε[1][3] = s.B.Exp(ρ2).Mul(s.C.Div(G).Exp(ω))
}
p := []*Point{G, s.A, s.B, s.C, s.R, s.X, s.Y, s.Z}
for _, e := range ε[0] {
p = append(p, e)
}
for _, e := range ε[1] {
p = append(p, e)
}
ch := Challenge(p...)
pr := &Proof{}
if s.plus {
pr.Ch[0] = ω
pr.Ch[1] = ch.Sub(ω)
pr.Rho[0][0] = r1
pr.Rho[0][1] = r2
pr.Rho[1][0] = ρ1.Sub(s.x.Mul(pr.Ch[1]))
pr.Rho[1][1] = ρ2.Sub(s.α.Mul(pr.Ch[1]))
} else {
pr.Ch[0] = ch.Sub(ω)
pr.Ch[1] = ω
pr.Rho[0][0] = r1.Sub(s.x.Mul(pr.Ch[0]))
pr.Rho[0][1] = r2.Sub(s.α.Mul(pr.Ch[0]))
pr.Rho[1][0] = ρ1
pr.Rho[1][1] = ρ2
}
return pr
}
func (c *Commitment) Verify(p *Proof) bool {
var ε [2][4]*Point
ε[0][0] = G.Exp(p.Rho[0][0]).Mul(c.X.Exp(p.Ch[0]))
ε[0][1] = G.Exp(p.Rho[0][1]).Mul(c.A.Exp(p.Ch[0]))
ε[0][2] = c.Y.Exp(p.Rho[0][0]).Mul(c.Z.Exp(p.Ch[0]))
ε[0][3] = c.B.Exp(p.Rho[0][1]).Mul(c.C.Exp(p.Ch[0]))
ε[1][0] = G.Exp(p.Rho[1][0]).Mul(c.X.Exp(p.Ch[1]))
ε[1][1] = G.Exp(p.Rho[1][1]).Mul(c.A.Exp(p.Ch[1]))
ε[1][2] = c.R.Exp(p.Rho[1][0]).Mul(c.Z.Exp(p.Ch[1]))
ε[1][3] = c.B.Exp(p.Rho[1][1]).Mul(c.C.Div(G).Exp(p.Ch[1]))
points := []*Point{G, c.A, c.B, c.C, c.R, c.X, c.Y, c.Z}
for _, e := range ε[0] {
points = append(points, e)
}
for _, e := range ε[1] {
points = append(points, e)
}
ch := Challenge(points...)
return p.Ch[0].Add(p.Ch[1]).Equal(ch)
}
|
