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sage: verify Eisenstein integer connection for GLV constants

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Add assertions to verify that the GLV decomposition constants arise
from the Eisenstein integer factorization of the group order N.

The group order factors as N = pi * conj(pi) in Z[w], where pi = A - B*w
is an Eisenstein prime. The GLV eigenvalue LAMBDA = B/A mod N, which is
the image of w^2 under the isomorphism Z[w]/(pi) -> Z/NZ.
This commit is contained in:
Justsomebuddy
2026-01-14 11:26:00 +00:00
parent 4721e077b4
commit 29ac4d8491
1 changed files with 9 additions and 0 deletions
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sage/gen_split_lambda_constants.sage
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@@ -81,6 +81,15 @@ assert (A1 + A2)/2 < sqrt(N)
assert B1 < sqrt(N)
assert B2 < sqrt(N)
# Verify connection to Eisenstein integers Z[w] where w = (-1 + sqrt(-3))/2.
# The group order N factors as N = pi * conj(pi) in Z[w], where pi = A - B*w
# is an Eisenstein prime with norm A^2 + A*B + B^2. The GLV endomorphism
# eigenvalue LAMBDA equals B/A mod N, which is the image of w^2 under the
# isomorphism Z[w]/(pi) -> Z/NZ (since w -> A/B and (A/B)^2 = B/A in Z/NZ).
A_EIS, B_EIS = -B1, A1
assert A_EIS**2 + A_EIS*B_EIS + B_EIS**2 == N
assert Z(B_EIS / A_EIS) == LAMBDA
G1 = round((2**384)*B2/N)
G2 = round((2**384)*(-B1)/N)