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Announcing the S-two Book

S-two Book

We are excited to announce that we partnered with Starkware to create the S-two book! (Link to repo: *.mdx files and *.md files)

S-two is Starkware's next-generation prover with state-of-the-art performance thanks to its implementation of Circle STARKs over the Mersenne31 prime field. It is also flexible enough to support proving custom circuits such as various VMs and ML inference.

We encourage anyone who is interested in the following topics to check it out:

  • Learning how to write AIRs using the S-two prover
  • Learning how the Cairo AIR is implemented in S-two
  • Learning how S-two implements Circle STARKs
Tip

If you need an introduction to Circle STARKs, check out our series of blog posts: Part I, Part II, Part III.

Keep reading
Recommended

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

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

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
More to explore

Soundness Failures in LaBRADOR Implementations from NTT -Friendly Rings

Post-quantum cryptography is making its way into production libraries, and correct implementation is far from trivial. We reviewed several LaBRADOR implementations on GitHub and found three of them broken, with soundness collapsing as far as a single bit, because of parameter choices that look perfectly reasonable in many lattice-based protocols; a power-of-2 modulus, a composite modulus, or NTT-friendly rings for efficient ring multiplication. The post goes through how each of those choices undermines the protocol's soundness argument, and why NTT-friendliness, of all things, turns out to be the wrong instinct if not handled carefully.

Georgios Raikos · April 30, 2026

10 Must-Read Papers That Shaped Modern Zero-Knowledge Proofs

Zero-knowledge proofs have come a long way in 40 years, thanks to groundbreaking work from many brilliant minds. We're taking you on a tour of the key milestones in this fascinating journey, from the foundational concepts in 1985 by Goldwasser, Micali, and Rackoff, to modern applications and innovations like STARKs and zkVMs. You'll get a taste of major papers and projects that have pushed these proofs from theoretical curiosity to practical and scalable solutions. Dive into the history, understand the evolution, and see how these cryptographic marvels continue to shape the future of secure computation!

ZK/SEC · November 12, 2024

Playing with LaBRADOR: Building Compact Lattice-based Proofs with Recursion

In this post, we explore LaBRADOR, a transparent, lattice-based proof system that achieves sublinear proof sizes through recursion. Built on the Module-SIS assumption, LaBRADOR lets a prover efficiently demonstrate knowledge of short vectors satisfying dot product constraints, general enough to express R1CS. The protocol’s key ideas include amortized openings, outer commitments, and a strategy to shrink the witness after each round of recursion. Together, these techniques make LaBRADOR a powerful, scalable framework for post-quantum zero-knowledge proofs.

Georgios Raikos · November 13, 2025