Module irmaseal_curve::notes::design

Design of BLS12-381

Fixed Generators

Although any generator produced by hashing to $\mathbb{G}_1$ or $\mathbb{G}_2$ is safe to use in a cryptographic protocol, we specify some simple, fixed generators.

In order to derive these generators, we select the lexicographically smallest valid $x$-coordinate and the lexicographically smallest corresponding $y$-coordinate, and then scale the resulting point by the cofactor, such that the result is not the identity. This results in the following fixed generators:

1. $\mathbb{G}_1$
   • $x = 3685416753713387016781088315183077757961620795782546409894578378688607592378376318836054947676345821548104185464507$
   • $y = 1339506544944476473020471379941921221584933875938349620426543736416511423956333506472724655353366534992391756441569$
2. $\mathbb{G}_2$
   • $x = 352701069587466618187139116011060144890029952792775240219908644239793785735715026873347600343865175952761926303160 + 3059144344244213709971259814753781636986470325476647558659373206291635324768958432433509563104347017837885763365758 u$
   • $y = 1985150602287291935568054521177171638300868978215655730859378665066344726373823718423869104263333984641494340347905 + 927553665492332455747201965776037880757740193453592970025027978793976877002675564980949289727957565575433344219582 u$

This can be derived using the following sage script:

param = -0xd201000000010000
def r(x):
    return (x**4) - (x**2) + 1
def q(x):
    return (((x - 1) ** 2) * ((x**4) - (x**2) + 1) // 3) + x
def g1_h(x):
	return ((x-1)**2) // 3
def g2_h(x):
    return ((x**8) - (4 * (x**7)) + (5 * (x**6)) - (4 * (x**4)) + (6 * (x**3)) - (4 * (x**2)) - (4*x) + 13) // 9
q = q(param)
r = r(param)
Fq = GF(q)
ec = EllipticCurve(Fq, [0, 4])
def psqrt(v):
	assert(not v.is_zero())
	a = sqrt(v)
	b = -a
	if a < b:
		return a
	else:
		return b
for x in range(0,100):
	rhs = Fq(x)^3 + 4
	if rhs.is_square():
		y = psqrt(rhs)
		p = ec(x, y) * g1_h(param)
		if (not p.is_zero()) and (p * r).is_zero():
			print("g1 generator: {}".format(p))
			break
Fq2.<i> = GF(q^2, modulus=[1, 0, 1])
ec2 = EllipticCurve(Fq2, [0, (4 * (1 + i))])
assert(ec2.order() == (r * g2_h(param)))
for x in range(0,100):
	rhs = (Fq2(x))^3 + (4 * (1 + i))
	if rhs.is_square():
		y = psqrt(rhs)
		p = ec2(Fq2(x), y) * g2_h(param)
		if not p.is_zero() and (p * r).is_zero():
			print("g2 generator: {}".format(p))
			break