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Finite Fields and Modular Arithmetic for ZK Proofs This article is the third in a series. We present finite fields in the context of circuits for zero-knowledge proofs. The previous chapters are P vs...
Last updated on Mar 10, 2026P vs NP and its application to zero knowledge proofs The P = NP problem asks: "If we can quickly verify a solution to a problem is correct, can we also quickly compute the solution?" Most researchers...
Last updated on Aug 29, 202418 min read
Groth16 Explained The Groth16 algorithm enables a quadratic arithmetic program to be computed by a prover over elliptic curve points derived in a trusted setup, and quickly checked by a verifier. It...
Last updated on Mar 16, 2026R1CS to Quadratic Arithmetic Program over a Finite Field in Python To make the transformation from R1CS to QAP less abstract, let’s use a real example. Let’s say we are encoding the arithmetic...
Last updated on Aug 5, 2024Bilinear Pairings in Python, Solidity, and the EVM Sometimes also called bilinear mappings, bilinear pairings allow us to take three numbers, $a$, $b$, and $c$, where $ab = c$, encrypt them to become...
Last updated on Oct 19, 202412 min read
Abstract Algebra Abstract Algebra is the study of sets that have one or more operators on that set. For our purposes, we only care about sets where the operator is a binary operator. Given a set with...
Last updated on Sep 10, 2024Converting Algebraic Circuits to R1CS (Rank One Constraint System) This article is explains how to turn a set of arithmetic constraints into Rank One Constraint System (R1CS). The focus of this...
Last updated on Jan 22, 202617 min read
Quadratic Arithmetic Programs A quadratic arithmetic program is an arithmetic circuit, specifically a Rank 1 Constraint System (R1CS) represented as a set of polynomials. It is derived using Lagrange...
Last updated on Mar 22, 202616 min read
Invariant Testing in Foundry Introduction In this article, we will discuss invariants and how to perform an invariant test on Solidity smart contracts using Foundry test suites. Invariant testing is...
Last updated on Oct 22, 20256 min read
Solidity Mutation Testing Mutation testing is a method to check the quality of the test suite by intentionally introducing bugs into the code and ensuring the tests catch the bug. The kind of bugs...
Last updated on Oct 29, 20256 min read
Ethereum precompiled contracts Ethereum precompiles behave like smart contracts built into the Ethereum protocol. The nine precompiles live in addresses 0x01 to 0x09. The utility of precompiles falls...
Last updated on Oct 29, 202511 min read
How ERC721 Enumerable Works An Enumerable ERC721 is an ERC721 with added functionality that enables a smart contract to list all the NFTs an address owns. This article describes how functions and how...
Last updated on Oct 22, 2025