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Circle STARKs

4 parts June 2024 – November 2025 68 min in total Mathias Hall-Andersen · Varun Thakore
  1. Part 1

    Circle STARKs: Part I, Mersenne

    Discover the intriguing world of Circle STARKs and how they can supercharge zero-knowledge proofs. This blog post sets the stage for a fascinating series about utilizing Mersenne prime fields to achieve lightning-fast arithmetic operations in STARK systems. You'll explore recent breakthroughs that make these fields more practical despite their previous limitations, and you'll get a sneak peek at what’s to come, including delving into group structures and implementing circle FFTs. If you're keen on cryptography and zero-knowledge proofs, this series will unveil how modern advancements are pushing the boundaries of what's possible.

    Mathias Hall-Andersen June 03, 2024 9 min Read →
  2. Part 2

    Circle STARKs: Part II, Circles

    In this blog post, we dive into the fascinating world of Circle STARKs, exploring the algebra of complex numbers and how they can be extended to any field. We revisit the concept of the unit circle and its unique group structure, which allows for cool operations like squaring and doubling angles. You'll discover how these ideas apply to finite fields, creating intriguing structures like the twin-coset and standard position coset. The post leads us to understand vanishing polynomials, crucial in STARKs, and sets the stage for exploring the circle FFT in upcoming discussions. Perfect for anyone curious about cutting-edge cryptographic techniques!

    Varun Thakore, Mathias Hall-Andersen February 21, 2025 16 min Read →
  3. Part 3

    Circle STARKs: Part III, Circle FFT

    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 Read →
  4. Part 4

    Circle STARKs: Part IV, Arithmetizing Circles

    In this final part of our series on Circle STARKs, we tie together concepts from Mersenne prime fields, circle curves and bivariate polynomials to showcase a comprehensive Circle STARK protocol. We detail the process of arithmetization over the circle curve, introducing Circle FRI as a low-degree test and walk through the complete Circle STARK construction. Readers will explore how trace commitments, constraint batching and low-degree proofs combine for efficient verifiable computation, delving into the nuances of proof validity and security analysis.

    Varun Thakore November 17, 2025 16 min Read →
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