# Revision history [back]

The following works for me:

# parameters for BLS12-381
z = -0xd201000000010000    # this is z = -15132376222941642752
q = (z^4 - z^2 + 1)
p = ZZ( z + q*(z - 1)^2/3 )

h1 = ZZ( (z - 1)^2 / 3 )
h2 = ZZ( (z^8 - 4*z^7 + 5*z^6 - 4*z^4 + 6*z^3 - 4*z^2-4*z + 13) / 9 )

# we work over one and only one common field, L, with p^12 elements.
F = GF(p)
L.<a> = GF(p^12)
RF.<T> = PolynomialRing(F12)
j = (T^2 + 1).roots(ring=F12, multiplicities=0)[0]

# L contains F canonically.
# L also contains F[j] by construction

E0 = EllipticCurve(F  , [0, 4])
E1 = EllipticCurve(F12, [0, 4])
E2 = EllipticCurve(F12, [0, 4 + 4*j])

# Generators of G1 and G2 (from https://aandds.com/blog/bls.html)
y1 = 0x08b3f481e3aaa0f1a09e30ed741d8ae4fcf5e095d5d00af600db18cb2c04b3edd03cc744a2888ae40caa232946c5e7e1
g1 = E1( (x1, y1) )

+ 0x13E02B6052719F607DACD3A088274F65596BD0D09920B61AB5DA61BBDC7F5049334CF11213945D57E5AC7D055D042B7E * j )
+ 0x0606C4A02EA734CC32ACD2B02BC28B99CB3E287E85A763AF267492AB572E99AB3F370D275CEC1DA1AAA9075FF05F79BE * j )
g2 = E2( (x2, y2) )

phi = E2.isomorphism_to(E1)

p1 = 3 * g1
p2 = 7 * g2

k = 12
t = p + 1 - E0.order()

ate = p1.ate_pairing(phi(p2), q, k, t, p)


The code works over a field, where there is an isomorphism from one group to the other one, in fact from some curve to an other one. The crypto-papers are but so cryptic, that one is not able to isolate structure from the usual humanly chosen path and story that puts together data for the curves, CPU performances, possible attacks and recommended bits, and in between some (iso)morphism (that may be a hash table in part).

As a reference, i would link

https://crypto.stackexchange.com/questions/95836/isomorphic-mapping-of-bls12-381-g2-points-to-g1

for the part, where we need a bridge from $\Bbb G_2$ to $\Bbb G_1$.

In our case, $\Bbb G_1$ is $E_0(F)=E_0(\Bbb F_p)$, and $E_1$ is a curve that becomes isomorphic to $E_0$, when making a base change to $\Bbb F_{p^{12}}$. Our $k$ is this $12$. $\Bbb G_2$ is a subgroup (of same order as $\Bbb G_1$, when i correctly understood what happens) of $E_1(\Bbb F_{p^2})$.

The code above implements thus the pairing: $$\Bbb G_1\times \Bbb G_2= E_0(\Bbb F_p)\times E_1(\Bbb F_{p^2}) \to E_0(\Bbb F_{p^k})\times E_1(\Bbb F_{p^k}) \overset{1\times\varphi}\longrightarrow E_0(\Bbb F_{p^k})\times E_0(\Bbb F_{p^k}) \overset{\langle\ ,\ \rangle}\longrightarrow \Bbb F_{p^k}\ .$$ At the last step, we insert the $8$-pairing... So for short, $$(P_1,P_2)\to\langle\ P_1\ ,\ \varphi(P_2)\ \rangle\ .$$