Recap

In Part I of this series, we explored efficient field arithmetic over Mersenne primes and a key challenge in instantiating the STARK protocol using them, specifically the absence of a smooth multiplicative subgroup.

In Part II, we examined how this issue can be addressed using the circle group. We introduced the structure of its cosets, i.e., twin-coset and standard position coset, which can be used to instantiate STARK protocols. We also discussed the space of polynomials defined over the circle group, and the construction of vanishing polynomial over these cosets.

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.

Over the past year, we’ve been diving into whether AI can effectively identify bugs in zero-knowledge circuits and applications, sparking questions about the future of auditing. This led us to develop SnarkSentinel, an AI-powered auditing tool. We’ll share what worked, what didn’t, and how our journey with AI could impact auditing. From early challenges with Circom to innovative methods like retrieval-augmented generation and agent-led probing, we’ll give you a peek into our findings, including both successes and setbacks with bug detection. Discover how AI might enhance or change the landscape of auditing and what this means for developers and security pros alike.

In the latest session of “Proof is in the Pudding,” we teamed up with Archetype to explore the basics of Trusted Execution Environments. Through a collaborative whiteboarding session, we break down key concepts and practical applications, making this tech topic accessible and engaging. Dive in to discover how these environments can enhance digital security in a straightforward way.

In June 2025, we uncovered a serious soundness issue in Solana’s ZK ElGamal Proof Program that could let attackers manipulate confidential token transfers undetected. We worked with the Anza team to quickly address the flaw by pausing and disabling vulnerable components. This post dives into the root cause, which was a subtle mistake in handling prover-generated challenges within sigma OR proofs, revealing broader lessons in zero-knowledge protocol security. If you’re interested in cryptographic protocol design, this could provide valuable insights.

Programming in Circom comes with its fair share of challenges. After reviewing numerous Circom codebases, we’ve identified certain anti-patterns that occur frequently. In this series, we’ll provide a comprehensive overview of these issues to help you avoid the most common pitfalls. Of course, this won’t be a complete list of every mistake possible (Circom has plenty of ways to trip you up). But the footguns we’ll cover are the ones that tend to catch developers off guard the most.

In this engaging blog post, we dive into the future of cryptography by exploring how lattice-based proof systems could fend off the quantum threat. We break down Greyhound, a promising lattice-based solution, explaining its transparent nature, efficient construction, and adaptability with today’s systems. Along the way, you’ll get a glimpse of key concepts like lattices, Ajtai commitments, and how Greyhound cleverly ensures security and efficiency. Whether you’re curious about the cutting-edge of cryptographic research or just want to understand the basics of quantum-resistant solutions, this post offers both depth and clarity.

In this blog post, we explore an optimization for Barrett reduction, a popular method for modular arithmetic. We discovered that the error bound can be tighter than traditionally thought, meaning you often only need a single subtraction instead of two. This tweak can speed up cryptographic operations like those in the RustCrypto library by 14% when applied to NIST curves, which is a game changer for both encryption and performance. Dive into the details to see how this simple change can lead to significant improvements in real-world applications!

In this blog post, we dive into the world of polynomial commitment schemes (PCS), which are crucial for constructing most practical SNARKs. We cover the basics of how PCS works, focusing on KZG10, known for its efficiency in proof size and verification time. You’ll learn about the essential properties of binding and hiding and explore technical concepts like homomorphism, batching, and unconditionally hiding. We break down various methods to achieve these features, offering insight into how PCS maintains the security and privacy of polynomials in cryptographic systems. Get ready to understand these powerful concepts and their applications in modern cryptography!

Hey there! We’re diving into how leveraging WebGPU can revolutionize client-side proving for privacy-preserving zero-knowledge apps. By tackling the usual time and memory bottlenecks with innovative techniques like using GPUs, we’re achieving significant performance gains. We’ll share our experiences using WebGPU with various frameworks, offering insights into optimizing operations and managing data transfer efficiently. Plus, learn about the challenges and exciting potential of integrating WebGPU into zero-knowledge frameworks. Join us to explore the future of privacy tech!

We’re diving into our new project called clean, aimed at creating an embedded DSL and formal verification framework for Zero Knowledge (ZK) circuits using Lean4. Imagine being able to not only define ZK circuits but also formally prove their correctness—sounds like a game-changer, right? We’ll walk you through our process of building a robust library of reusable, verified circuit gadgets, focusing on the importance of soundness and completeness. Plus, you’ll get a peek at some cool examples like 8-bit addition and how we’re tackling ZKVM design with techniques borrowed from Fibonacci sequences. It’s exciting stuff, and if you’re curious about how we’re paving the way for bug-free ZK circuits, this is a read you won’t want to miss!

We recently teamed up with Celo for an in-depth security audit of the Self project, exploring its innovative approach to on-chain identity using biometric passports and zero-knowledge proofs. During our three-week dive, we examined everything from cryptographic primitives to smart contract architecture and a unique proof delegation system using AWS Nitro Enclaves. The Celo team impressed us with their commitment and responsiveness, and we collaborated on refining the system with strategic improvements. Curious about the nitty-gritty details and our insights? Check out the full report!

We’re diving into the world of Bitcoin’s UTXO model and how recent advancements like BitVM can overcome its limitations, allowing for more complex computations without changing Bitcoin’s core. This blog post explores cutting-edge techniques like covenants, statefulness, and circuit models, showing how they enable intricate logic on Bitcoin. We’ll break down how these innovations make trustless cross-chain transactions possible, and highlight the potential of optimistic protocols to optimize the on-chain footprint. If you’re curious about the future of Bitcoin’s capabilities, this is the deep dive you need!

Curious about how zero-knowledge virtual machines (zkVMs) make computing more secure without the hassle? We delve into the fascinating world of zkVMs, where you can program in high-level languages like Rust or C and let cryptography handle the complexity. We’ll explore the evolution of these innovative systems through projects like Cairo and RISC Zero, while touching on the unique benefits and technical insights each brings. Plus, learn about groundbreaking techniques for optimizing zkVMs with projects like Jolt, and discover a range of other influential zkVM initiatives. Get ready for an enlightening journey into secure computation!

In our third whiteboard session with Archetype, we dive into the fascinating world of cryptographic protocols by breaking down the intricacies of the Fiat-Shamir security model and the GKR protocol. Whether you’re a cryptography enthusiast or just curious about how these complex mechanisms enhance security, this is a chance to explore the theories with us in a friendly and digestible way. Don’t miss the opportunity to expand your understanding of this cutting-edge topic!

Thinking about diving into the world of cryptography and cutting-edge tech? We’re on the lookout for bright minds to join us for internships in areas like ZK, MPC, and post-quantum cryptography. Our past interns have tackled exciting projects like exploring ZK circuit vulnerabilities and delving into RISC-V zkVMs. If you want a fast track to an interview, try out the zkBank challenge, or simply send us your resume. Come join us and see where the journey takes you!

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!

In this blog post, we dive into the fascinating world of zero-knowledge proofs using the MPC-in-the-Head transformation—a clever method that constructs proof systems from any secure multiparty computation protocol. Originally proposed in 2007, this transformation uses a creative approach involving “imaginary parties” to prove knowledge without revealing it. We explain how this technique can be applied to develop post-quantum signature schemes, providing insights into its practical implications and efficiency. By exploring these concepts, readers will uncover a unique intersection of cryptography and computer science.

In November 2024, we found a significant inflation bug in the Aleo mainnet that could have allowed token minting without proper checks. We immediately informed the Aleo team, who swiftly addressed the issue with no detected exploitation. This post dives into the inner workings of Aleo and explains how transitions and records operate, providing insight into how the vulnerability was discovered and resolved. It’s an intriguing look at blockchain security, zero-knowledge proofs, and the importance of thorough type checks to ensure robust protocol integrity.

In our latest blog post, we dig into the fascinating world of blockchain rollups, focusing on their security and how they help Ethereum scale while maintaining its core values of decentralization. We’ll break down the concepts of Optimistic and ZK-Rollups, discuss the importance of projects like L2BEAT in assessing rollup maturity, and introduce our formal model for ensuring rollup security. If you’re curious about how forced transactions, safe blacklisting, and upgradeability are shaping the future of Ethereum, this is a read you won’t want to miss.