A new interactive tutorial on Sumcheck, Multilinear Extensions, and HyperPlonk with complete SageMath implementations and exercises. Go beyond the theory and understand how these protocols actually work by implementing them yourself.
In this blog post, we explore how to optimize the sumcheck protocol, particularly when working with values in a small field and randomness from a large field, as often needed in zkVMs. We introduce various algorithms aimed at reducing expensive operations, focusing on minimizing large multiplications. Starting from using simple evaluation tables to more sophisticated techniques like precomputing accumulators and leveraging Lagrange interpolation, we demonstrate how to efficiently organize computations to speed up proving times. Readers will gain insights into handling arithmetic operations within the sumcheck protocol and learn about optimizing specific cases in zero-knowledge proofs.
A hands-on comparison of formal verification frameworks for arithmetic circuits, evaluating those in the ACL2 Book (r1cs, PFCS), acl2-jolt, Garden (Rocq), zk-lean, sp1-lean, and Clean. Each framework is tested on reproducibility, available examples (from basic field elements to RISC-V VM instructions), and practical verification tasks including the IsZero and weighted-sum circuits. The evaluation includes both human and Claude Code's ability to work with each framework, revealing insights about installation difficulty, proof automation capabilities, and the maturity of publicly available examples. This post maps the current landscape of formally verified ZK circuits and discusses what's coming next in this rapidly evolving field.
This article breaks down how Bulletproofs enable range proofs: proofs that a hidden value lies within a range without revealing it. Starting from bit decomposition, it shows how to express and combine constraints into a single inner product, then make the proof zero-knowledge with blinding polynomials and commitments. By the end, you’ll understand how systems like Monero’s confidential transactions prove valid amounts while keeping values private.
In this post, we dive deep into the Inner Product Argument (IPA), the mathematical core of Bulletproofs. Starting from simple vector folding, we build up to a full zero-knowledge proof with Pedersen commitments, explore how the mysterious `L` and `R` terms appear, and finish with smart verifier optimizations. All illustrated with clear, runnable SageMath code.
Learn how Bulletproofs enables efficient zero-knowledge proofs without trusted setups by computing inner products in a verifiable way. This post breaks down the core folding technique that reduces large vectors to single elements through recursive compression, making proofs both compact and fast to verify. Used in Monero, Mina's Kimchi, and Zcash's Halo 2, Bulletproofs is a practical alternative to pairing-based schemes.
Learn the fundamentals of Σ-protocols through the classic Schnorr protocol, exploring the three-step dance of commit, challenge, and respond. This post walks through knowledge soundness and witness extraction, then shows how to compose Sigma proofs with AND/OR logic and Pedersen commitments. See working SageMath implementations, discover how Fiat-Shamir transforms interactive proofs into non-interactive signatures, and understand the deeper mathematical structure as proofs of knowledge of homomorphism pre-images.
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.
A series of recent papers just disproved the proximity gaps conjecture, which has everyone wondering if hash-based SNARKs are in trouble. We break down what actually happened using some helpful visuals. Think of it as understanding which parameter choices are safe versus which ones are now confirmed to be risky. The post walks you through the different security zones (proven safe, conjectured safe, and definitely not safe), explains how these new results connect to other open math problems, and shows what it means for real-world SNARKs in terms of proof sizes and performance trade-offs.
In Session 07 of "Proof is in the Pudding," we explore the other dark forest, the realm of offchain public keys. We dive into zkLogin, ZK Email, and ZKPassport, examining how these protocols handle authentication and privacy. We also discuss the issue of unlinkability in privacy protocols and why replacing traditional signature verifications with zero-knowledge proofs could unlock more interesting and powerful ZK products.
In the latest "Proof is in the Pudding" session, we team up with Archetype to break down the essentials of Data Availability Sampling. We dive into how rollups and Ethereum's DA system work, explore the role of DA chains, and touch on the basics of verifiable sharding. This introduction is perfect for anyone curious about the foundations of data availability sampling and how these concepts are playing out in the blockchain world.
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.
We're thrilled to share our collaboration with Starkware on the S-two book. If you're curious about writing AIRs with the S-two prover, implementing Cairo AIR in S-two, or how Circle STARKs are utilized, this post is for you. Dive in to explore these insights and deepen your understanding of these fascinating topics.
This post is the second part of our series on Circom pitfalls. While part 1 covered issues with assertions, hints, and aliasing, this one explores unsafe component usage and the subtle fact that Circom’s comparison operators work over signed integers.
This blog post explains the security intuition behind the FRI protocol, which proves that a function is close to a valid Reed-Solomon codeword. It introduces the "prover message graph," a layered structure that visualizes how correct and incorrect folds affect verification. We conclude that if too many folds are inconsistent, the verifier will likely reject, but if most are correct, the initial function must be close to a proper codeword.
In this blog post, we explore a clever design in Zcash's halo2 implementation for securing the Fiat-Shamir transformation. By using a mutable transcript, the process ensures that values are automatically absorbed, reducing potential bugs. You'll find explanations of the distinct roles of `write` and `read` functions for points and scalars, highlighting how this abstraction makes the prover-verifier interaction seamless and secure. If you're curious about the inner workings of cryptographic protocols, this is a fascinating read.
Paul Kocher's 1996 timing attack showed how microsecond differences in execution time could leak private keys from RSA implementations. This tutorial recreates the attack journey from clean operation counting through noisy wall-clock measurements to sophisticated engineering solutions. Learn the variance distinguisher, explore schoolbook modular arithmetic, and discover the measurement techniques that make practical timing attacks possible despite system noise.
