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