Skip to content

sage: verify Eisenstein integer connection for GLV constants #1800

New issue

Have a question about this project? Sign up for a free GitHub account to open an issue and contact its maintainers and the community.

Sign up for GitHub

By clicking “Sign up for GitHub”, you agree to our terms of service and privacy statement. We’ll occasionally send you account related emails.

Already on GitHub? Sign in to your account

Merged
real-or-random merged 1 commit into from
Jan 21, 2026
Merged

sage: verify Eisenstein integer connection for GLV constants #1800

Changes from all commits
Commits
Show all changes
1 commit
Select commit
File filter

Filter by extension

Filter by extension
Conversations
Failed to load comments.
Loading
Jump to
Jump to file
Failed to load files.
Loading
Diff view
Diff view
9 changes: 9 additions & 0 deletions sage/gen_split_lambda_constants.sage
View file
Open in desktop
Original file line number Diff line number Diff line change
Expand Up @@ -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)

Expand Down