Paper 2024/278

Circle STARKs

Abstract

Traditional STARKs require a cyclic group of a smooth order in the field. This allows efficient interpolation of points using the FFT algorithm, and writing constraints that involve neighboring rows. The Elliptic Curve FFT (ECFFT, Part I and II) introduced a way to make efficient STARKs for any finite field, by using a cyclic group of an elliptic curve. We show a simpler construction in the lines of ECFFT over the circle curve $x^2 + y^2 = 1$. When $p + 1$ is divisible by a large power of $2$, this construction is as efficient as traditional STARKs and ECFFT. Applied to the Mersenne prime $p = 2^{31} − 1$, which has been recently advertised in the IACR eprint 2023:824, our preliminary benchmarks indicate a speed-up by a factor of $1.4$ compared to a traditional STARK using the Babybear prime $p = 2^{31} − 2^{27} + 1$.

Note: This version corrects minor typos in Section 5.3, and adds an optimized treatment of constraints with punctuated activation domains.