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EP-0112: Galois Finite Fields & Cryptographic Arithmetic

Field Value
EP 0112
Title Galois Finite Fields & Cryptographic Arithmetic
Author Eran Rivlis & Antigravity
Status Final
Type Standards Track
Created 2026-08-01
Updated 2026-08-01

Abstract

This proposal specifies algebrax.galois, building upon QuotientMonoidAlgebraSemiring (EP-0100). It introduces Galois Finite Fields \(\text{GF}(p^m)\) as polynomial modulo quotient semirings, enabling sparse matrix operations over finite fields, supporting AES \(\text{GF}(2^8)\) field arithmetic, Reed-Solomon error correction matrices, and QAP polynomial evaluations for zero-knowledge proofs.

Deliverables

  1. Core Implementation: src/algebrax/galois.py (GaloisFieldSemiring, gf_matrix_mul, reed_solomon_generator).
  2. Unit Tests: tests/algebrax/test_galois.py (verifying field identity, AES MixColumns matrix multiplication, and error correction).
  3. Use Case Recipe: recipes/galois_field_cryptography.py & .ipynb.
  4. Graphical Laboratory View: View 23 (view_galois_finite_fields_group) in recipes/lab.py.