Zero Knowledge

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...

R1CS 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...

Converting 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...

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...

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...

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...

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...

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...

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...

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...

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...

P 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...