1package bn128
  2
  3import (
  4	"math/big"
  5
  6	"github.com/arnaucube/go-snark/fields"
  7)
  8
  9type G1 struct {
 10	F fields.Fq
 11	G [3]*big.Int
 12}
 13
 14func NewG1(f fields.Fq, g [2]*big.Int) G1 {
 15	var g1 G1
 16	g1.F = f
 17	g1.G = [3]*big.Int{
 18		g[0],
 19		g[1],
 20		g1.F.One(),
 21	}
 22	return g1
 23}
 24
 25func (g1 G1) Zero() [2]*big.Int {
 26	return [2]*big.Int{g1.F.Zero(), g1.F.Zero()}
 27}
 28func (g1 G1) IsZero(p [3]*big.Int) bool {
 29	return g1.F.IsZero(p[2])
 30}
 31
 32func (g1 G1) Add(p1, p2 [3]*big.Int) [3]*big.Int {
 33
 34	// https://en.wikibooks.org/wiki/Cryptography/Prime_Curve/Jacobian_Coordinates
 35	// https://github.com/zcash/zcash/blob/master/src/snark/libsnark/algebra/curves/alt_bn128/alt_bn128_g1.cpp#L208
 36	// http://hyperelliptic.org/EFD/g1p/auto-code/shortw/jacobian-0/addition/add-2007-bl.op3
 37
 38	if g1.IsZero(p1) {
 39		return p2
 40	}
 41	if g1.IsZero(p2) {
 42		return p1
 43	}
 44
 45	x1 := p1[0]
 46	y1 := p1[1]
 47	z1 := p1[2]
 48	x2 := p2[0]
 49	y2 := p2[1]
 50	z2 := p2[2]
 51
 52	z1z1 := g1.F.Square(z1)
 53	z2z2 := g1.F.Square(z2)
 54
 55	u1 := g1.F.Mul(x1, z2z2)
 56	u2 := g1.F.Mul(x2, z1z1)
 57
 58	t0 := g1.F.Mul(z2, z2z2)
 59	s1 := g1.F.Mul(y1, t0)
 60
 61	t1 := g1.F.Mul(z1, z1z1)
 62	s2 := g1.F.Mul(y2, t1)
 63
 64	h := g1.F.Sub(u2, u1)
 65	t2 := g1.F.Add(h, h)
 66	i := g1.F.Square(t2)
 67	j := g1.F.Mul(h, i)
 68	t3 := g1.F.Sub(s2, s1)
 69	r := g1.F.Add(t3, t3)
 70	v := g1.F.Mul(u1, i)
 71	t4 := g1.F.Square(r)
 72	t5 := g1.F.Add(v, v)
 73	t6 := g1.F.Sub(t4, j)
 74	x3 := g1.F.Sub(t6, t5)
 75	t7 := g1.F.Sub(v, x3)
 76	t8 := g1.F.Mul(s1, j)
 77	t9 := g1.F.Add(t8, t8)
 78	t10 := g1.F.Mul(r, t7)
 79
 80	y3 := g1.F.Sub(t10, t9)
 81
 82	t11 := g1.F.Add(z1, z2)
 83	t12 := g1.F.Square(t11)
 84	t13 := g1.F.Sub(t12, z1z1)
 85	t14 := g1.F.Sub(t13, z2z2)
 86	z3 := g1.F.Mul(t14, h)
 87
 88	return [3]*big.Int{x3, y3, z3}
 89}
 90
 91func (g1 G1) Neg(p [3]*big.Int) [3]*big.Int {
 92	return [3]*big.Int{
 93		p[0],
 94		g1.F.Neg(p[1]),
 95		p[2],
 96	}
 97}
 98func (g1 G1) Sub(a, b [3]*big.Int) [3]*big.Int {
 99	return g1.Add(a, g1.Neg(b))
100}
101func (g1 G1) Double(p [3]*big.Int) [3]*big.Int {
102
103	// https://en.wikibooks.org/wiki/Cryptography/Prime_Curve/Jacobian_Coordinates
104	// http://hyperelliptic.org/EFD/g1p/auto-code/shortw/jacobian-0/doubling/dbl-2009-l.op3
105	// https://github.com/zcash/zcash/blob/master/src/snark/libsnark/algebra/curves/alt_bn128/alt_bn128_g1.cpp#L325
106
107	if g1.IsZero(p) {
108		return p
109	}
110
111	a := g1.F.Square(p[0])
112	b := g1.F.Square(p[1])
113	c := g1.F.Square(b)
114
115	t0 := g1.F.Add(p[0], b)
116	t1 := g1.F.Square(t0)
117	t2 := g1.F.Sub(t1, a)
118	t3 := g1.F.Sub(t2, c)
119
120	d := g1.F.Double(t3)
121	e := g1.F.Add(g1.F.Add(a, a), a)
122	f := g1.F.Square(e)
123
124	t4 := g1.F.Double(d)
125	x3 := g1.F.Sub(f, t4)
126
127	t5 := g1.F.Sub(d, x3)
128	twoC := g1.F.Add(c, c)
129	fourC := g1.F.Add(twoC, twoC)
130	t6 := g1.F.Add(fourC, fourC)
131	t7 := g1.F.Mul(e, t5)
132	y3 := g1.F.Sub(t7, t6)
133
134	t8 := g1.F.Mul(p[1], p[2])
135	z3 := g1.F.Double(t8)
136
137	return [3]*big.Int{x3, y3, z3}
138}
139
140func (g1 G1) MulScalar(p [3]*big.Int, e *big.Int) [3]*big.Int {
141	// https://en.wikipedia.org/wiki/Elliptic_curve_point_multiplication#Double-and-add
142	// for more possible implementations see g2.go file, at the function g2.MulScalar()
143
144	q := [3]*big.Int{g1.F.Zero(), g1.F.Zero(), g1.F.Zero()}
145	d := g1.F.Copy(e)
146	r := p
147	for i := d.BitLen() - 1; i >= 0; i-- {
148		q = g1.Double(q)
149		if d.Bit(i) == 1 {
150			q = g1.Add(q, r)
151		}
152	}
153
154	return q
155}
156
157func (g1 G1) Affine(p [3]*big.Int) [2]*big.Int {
158	if g1.IsZero(p) {
159		return g1.Zero()
160	}
161
162	zinv := g1.F.Inverse(p[2])
163	zinv2 := g1.F.Square(zinv)
164	x := g1.F.Mul(p[0], zinv2)
165
166	zinv3 := g1.F.Mul(zinv2, zinv)
167	y := g1.F.Mul(p[1], zinv3)
168
169	return [2]*big.Int{x, y}
170}
171
172func (g1 G1) Equal(p1, p2 [3]*big.Int) bool {
173	if g1.IsZero(p1) {
174		return g1.IsZero(p2)
175	}
176	if g1.IsZero(p2) {
177		return g1.IsZero(p1)
178	}
179
180	z1z1 := g1.F.Square(p1[2])
181	z2z2 := g1.F.Square(p2[2])
182
183	u1 := g1.F.Mul(p1[0], z2z2)
184	u2 := g1.F.Mul(p2[0], z1z1)
185
186	z1cub := g1.F.Mul(p1[2], z1z1)
187	z2cub := g1.F.Mul(p2[2], z2z2)
188
189	s1 := g1.F.Mul(p1[1], z2cub)
190	s2 := g1.F.Mul(p2[1], z1cub)
191
192	return g1.F.Equal(u1, u2) && g1.F.Equal(s1, s2)
193}