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.
Writing
Notes on zero-knowledge proofs, cryptography and the math behind them.
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.
Fast OOD Evaluation
Computing a STARK's out-of-domain evaluations quickly with barycentric interpolation over the trace subgroup.
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.
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.
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.
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.
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.
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.
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.
Books for Cryptography and Blockchain
Learning cutting-edge research areas through books.
Degree Adjustment in SNARKs
On adjusting the degree of SNARK-related polynomials to fit our needs, and why the randomness matters.