LambdaClass Blog

Introduction It's been nearly 3 years since we started lambdaworks, our cryptographic library for zero-knowledge proofs. The first PR was merged on January 18th 2023 and since then we had over 575 PRs merged from almost 80 collaborators and counting. At that time, there was still a big

Introduction At LambdaClass, we've been exploring Lean 4 as a tool for formal verification in zero-knowledge proof systems and in formalizing the kernel of Concrete. As the ZK ecosystem matures, the stakes for correctness have never been higher: a single bug in a zkVM or proof system can

In this article, we will review recent developments in the analysis of the cryptographic security of one of the biggest pieces of the ZK engine: the FRI protocol, implementing interactive oracle proofs of proximity, IOPPs. A newly uploaded article by Eli Ben-Sasson, Dan Carmon, Swastik Kopparty, and Shubhangi Saraf, alongside

By Federico Carrone and Roberto Catalan Ethereum is emerging as a general purpose financial backend that reduces the cost and complexity of building financial services while improving their speed and security. For decades the internet accelerated communication but did not create a neutral system for defining ownership or enforcing obligations.

Modern economic systems rest on two foundations: tools that expand productive capacity and institutions that define who controls their output. The internet transformed how information moves, but it did not reconstruct the institutional machinery that governs ownership and exchange. Digital economic life therefore expanded without a durable system of rights,

Introduction In this post, we'll present an in-depth walkthrough of our implementation of the sumcheck optimizations proposed by Bagad, Dao, Domb, and Thaler (BDDT) in their paper. In previous posts, we've explained the main theoretical ideas (see part I and part II). Here, we dive deep

Introduction One of the key components of the Transformer architecture is the Attention layer, which is in charge of making every word (or more generally, every token) learn the context given by every other in a sequence, and was introduced in the seminal paper Attention is all you need. In

Introduction In this article we will continue our study and analysis of the work by Bagad, Dao, Domb and Thaler,''Speeding up SUMCHECK'' regarding optimizations for the SUMCHECK protocol applied to polynomials that are products of multilinear polynomials. It is now time to dive into specificity:

Introduction Ethereum has been quickly evolving to become the financial backend of the World. Research and development in zero-knowledge proofs have allowed Ethereum to scale with rollups, by batching transactions off-chain and then posting an update of the accounts together with a cryptographic proof attesting to the validity of the

Introduction In this article we review some of the optimizations for the SUMCHECK protocol as discussed in the recent article by Bagad, Dao, Domb and Thaler. The authors tackle the problem of disproportionate cost of field multiplications. In many SNARK applications, the sum-check protocol operates over extension fields, which are

Introduction In this article we briefly introduce a list of basic properties of multilinear polynomials that are convenient to have in mind when tackling a very interesting piece of work by Bagad, Dao, Domb and Thaler: "Speeding up SUM-CHECK proving". The authors focus on a specific setting for

Introduction The GKR (Goldwasser–Kalai–Rothblum) protocol provides an efficient way to verify computations over arithmetic circuits, avoiding re-execution and reducing the verifier’s work. In our previous post, GKR protocol: a step-by-step example, we explored how the protocol works in detail, focusing on its mathematical structure and walking through

Introduction Bitcoin was the first blockchain in history, enabling a peer-to-peer electronic cash system for the first time. Introduced in 2008 by Satoshi Nakamoto, it provided an elegant yet simple construction to enable people from across the world to store and exchange value over a permissionless and censorship-resistant network. An

Introduction At the heart of any Zero-Knowledge Proof (ZKP) system lies the concept of arithmetization, the process of transforming a computational problem into a mathematical problem that can be expressed and verified within a specific algebraic structure, such as polynomials or arithmetic circuits. Within the Binius framework, this arithmetization is

Warning: This post is more math heavy than other articles. Introduction In this article we continue our study of towers of binary fields, motivated by the proposal of Diamond and Posen for a complete protocol working over fields of characteristic 2, BINIUS. Previously we covered basic arithmetic of field elements

Introduction The development of general-purpose zkVMs has made writing verifiable applications easier, by allowing developers to write high-level code and then compiling it to RISC-V or another instruction set architecture (ISA) and running it on top of the virtual machine. The virtual machine then generates a succinct proof of execution