---
id: H02
title: 'End(E) = Z[ω] and the GLV basis is the Eisenstein ideal (π̄ − 1): closed form of the libsecp256k1 constants'
axis: A
kind: algebraic
statement: 'For secp256k1, t² − 4p = −3f² with f integral; End(E) = Z[ω]; the GLV lattice {(k1,k2): k1 + k2 λ ≡ 0 (mod n)}
  is the principal ideal (π̄−1)Z[ω] and a1 = (t−f)/2 − 1, b1 = −f, a2 = a1 − b1, b2 = a1, with a1² − a1 b1 + b1² = n.'
motivation: 'Research question A1: are there efficiently computable endomorphisms beyond the automorphisms? CM theory says
  no; the closed form explains the ''magic'' constants.'
assumptions: E ordinary, p ≡ 1 (mod 3), j = 0.
expected_contribution: Explanation of constants usually presented as output of the extended Euclidean algorithm; proof that
  no endomorphism outside Z[φ] exists.
related_work:
- gallant2001faster
- sica2002analysis
- galbraith2011endomorphisms
- silverman2009arithmetic
possible_counterexamples: If a1 ≠ (t−f)/2 − 1 exactly, the ideal description would be wrong (it was checked exactly).
verification_strategy: Exact integer check of the closed form; identity search Q1/Q3; extended-Euclid re-derivation.
computational_budget: seconds
status: evaluated
classification: known_reproduced
evidence:
- results/automated_discovery/identity_search.json
- results/baselines/secp256k1_facts.json
verdict: Closed form holds exactly; all 22 identities found by the search are explained by CM theory. Known in principle (ideal
  structure of the GLV lattice), the explicit identification with the published constants is a useful clarification, not new
  mathematics.
commands:
- .venv/bin/python experiments/automated_discovery/identity_search.py
---

# H02 — End(E) = Z[ω] and the GLV basis is the Eisenstein ideal (π̄ − 1): closed form of the libsecp256k1 constants

For secp256k1, t² − 4p = −3f² with f integral; End(E) = Z[ω]; the GLV lattice {(k1,k2): k1 + k2 λ ≡ 0 (mod n)} is the principal ideal (π̄−1)Z[ω] and a1 = (t−f)/2 − 1, b1 = −f, a2 = a1 − b1, b2 = a1, with a1² − a1 b1 + b1² = n.