Notes on zero-knowledge proofs, cryptography and the math behind them.https://hecmas.github.io/https://hecmas.github.io/writing/stir/https://hecmas.github.io/writing/stir/Notes on STIR-style proximity testing, where each round cuts the degree by k but only halves the domain: the construction, how to handle repeated queries, and round-by-round soundness.Fri, 28 Aug 2026 00:00:00 GMThttps://hecmas.github.io/writing/deep-quotients-over-the-base-field/https://hecmas.github.io/writing/deep-quotients-over-the-base-field/Keep STARK DEEP quotients over the base field by dividing by the minimal polynomial of the out-of-domain point instead of X − z, so base-field columns never touch extension-field arithmetic.Thu, 27 Aug 2026 00:00:00 GMThttps://hecmas.github.io/writing/fast-ood-evaluation/https://hecmas.github.io/writing/fast-ood-evaluation/Computing a STARK's out-of-domain evaluations quickly with barycentric interpolation over the trace subgroup.Wed, 26 Aug 2026 00:00:00 GMThttps://hecmas.github.io/writing/how-zisk-enforces-cross-component-interaction-with-arguments/https://hecmas.github.io/writing/how-zisk-enforces-cross-component-interaction-with-arguments/How ZisK's components (CPU, memory, ALU…) stay consistent: cross-AIR arguments reduced to sumchecks via LogUp, with each component's partial sums combined through proof aggregation.Thu, 05 Feb 2026 00:00:00 GMThttps://hecmas.github.io/writing/on-constraint-satisfiability/https://hecmas.github.io/writing/on-constraint-satisfiability/Proving STARK constraints that only hold on part of the trace (boundaries, periodic rows, row selectors) by picking subdomains with sparse vanishing polynomials instead of adding selector columns.Mon, 12 Jan 2026 00:00:00 GMThttps://hecmas.github.io/writing/secure-challenge-derivation-in-zisk/https://hecmas.github.io/writing/secure-challenge-derivation-in-zisk/How ZisK lets many provers agree on a single Fiat–Shamir challenge: accumulate their commitments with LtHash, a lattice-based multiset hash whose security reduces to SIS.Wed, 20 Aug 2025 00:00:00 GMThttps://hecmas.github.io/writing/elliptic-curves-over-goldilocks/https://hecmas.github.io/writing/elliptic-curves-over-goldilocks/Building prime-order elliptic curves over a quintic extension of Goldilocks with 128-bit security and efficient hash-to-curve: EcGFp5 and EcMasFp5.Mon, 24 Feb 2025 00:00:00 GMThttps://hecmas.github.io/writing/overwrite-mode-in-sponge-based-permutations/https://hecmas.github.io/writing/overwrite-mode-in-sponge-based-permutations/Why the Polygon zkEVM's twist on Poseidon's overwrite mode, which reroutes outputs between permutation calls, is exactly as safe as the standard construction.Fri, 03 Nov 2023 00:00:00 GMThttps://hecmas.github.io/writing/computing-the-optimal-ate-pairing-over-the-bls12-381-curve/https://hecmas.github.io/writing/computing-the-optimal-ate-pairing-over-the-bls12-381-curve/A computation-first walkthrough of the optimal Ate pairing on BLS12-381: the extension-field tower, line equations, final exponentiation and cyclotomic tricks behind our zkASM implementation.Fri, 19 May 2023 00:00:00 GMThttps://hecmas.github.io/writing/computing-the-optimal-ate-pairing-over-the-bn254-curve/https://hecmas.github.io/writing/computing-the-optimal-ate-pairing-over-the-bn254-curve/The optimal Ate pairing on BN254, step by step: twist, line equations, the Miller loop's last two lines, final exponentiation and Frobenius shortcuts, as implemented in the Polygon zkEVM.Wed, 05 Apr 2023 00:00:00 GMThttps://hecmas.github.io/writing/books/https://hecmas.github.io/writing/books/Learning cutting-edge research areas through books.Fri, 14 Oct 2022 00:00:00 GMThttps://hecmas.github.io/writing/snark-degree/https://hecmas.github.io/writing/snark-degree/On adjusting the degree of SNARK-related polynomials to fit our needs, and why the randomness matters.Wed, 12 Oct 2022 00:00:00 GMT