Browse all Zero Knowledge articles.
14 min read
Elliptic Curve Point Addition This article describes how elliptic curve addition works over real numbers. Cryptography uses elliptic curves over finite fields, but elliptic curves are easier to...
Last updated on Mar 10, 202618 min read
Elliptic Curves over Finite Fields What do elliptic curves in finite fields look like? It’s easy to visualize smooth elliptic curves, but what do elliptic curves over a finite field look like? The...
Last updated on Mar 10, 2026Finite 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, 202618 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, 202612 min read
Elementary Group Theory for Programmers !Group Theory Hero Image This article provides several examples of algebraic groups so that you can build an intuition for them. A group is a set with: a...
Last updated on Feb 25, 202611 min read
Homomorphisms by Example A homomorphism between two groups exists if a structure preserving map between the two groups exists. Suppose we have two algebraic data structures $(A,\square)$ and $(B,...
Last updated on Feb 25, 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 Jan 5, 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, 2024Building a Zero Knowledge Proof from an R1CS Given an arithmetic circuit encoded as a Rank 1 Constraint System, it is possible to create a ZK-proof of having a witness, albeit not a succinct one....
Last updated on Oct 31, 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, 202622 min read
Elementary Set Theory for Programmers Why another set theory tutorial? The target audience for this piece is the sort of folks who don’t care about abstract math unless they see a direct use-case for...
Last updated on Apr 25, 20253 min read
Trusted Setup A trusted setup is a mechanism ZK-SNARKs use to evaluate a polynomial at a secret value. Observe that a polynomial $f(x)$ can be evaluated by computing the inner product of the...
Last updated on Oct 10, 2024