educative · zk
𝒫𝔩𝔬𝔫𝒦’s many layers (selector polynomials, wiring permutations, quotient tests, random challenges and KZG commitments) can be overwhelming. Our zkSecurity tutorial uses a single running example to demystify them all. Build tables and interpolate low-degree BN254 polynomials, encode gate and wiring constraints, run deterministic and probabilistic zero-tests, then layer in randomness and KZG commitments to produce a full Fiat–Shamir proof. Grab the Jupyter Notebook (Sage or Cocalc), or work in your favorite language with our guided test cases.
Martín Ochoa
August 05, 2025
1 min
Read →
Circle STARKs · Part 3
In this blog post, we explore how to efficiently implement polynomial operations using Circle FFT in the context of STARKs, drawing parallels with the Cooley-Tukey FFT. We discuss how the Circle FFT handles bivariate polynomials over the circle group, replacing traditional multiplicative subgroups with twin-cosets. You'll discover the nuanced process of decomposing and recomposing polynomials using projection and squaring maps, leading to efficient computations. We also address the gap between the polynomial degree space and the space spanned by Circle FFT. This is a fascinating dive into the heart of polynomial computations in cryptography.
Varun Thakore
August 04, 2025
27 min
security · zk
We recently discovered a subtle but important soundness issue in Halo2, which we’ve named the query collision bug. It affects certain edge-case circuits and was present in widely used versions, including the main Zcash implementation and PSE’s fork. We disclosed the issue to the relevant teams, including Zcash, PSE, and Axiom, all of whom have since patched it. While no known production circuits were affected, the bug reveals a surprising vulnerability in the proving system that deserves attention.
Suneal Gong
July 09, 2025
10 min
