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...

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...

Introduction to the Cairo Language Cairo is a programming language designed for provable, verifiable computation, particularly within the context of zero-knowledge systems like Starknet, a Layer 2...

Visual representation of the roots of unity The property that if $\omega$ is a $k$-th root of unity, then $\omega^i$ and $\omega^{i+k/2}$ are additive inverses may seem a little abstract — this...

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...

The Fundamental Theorem of Finite Cyclic Groups The Fundamental Theorem of Cyclic Groups provides guarantees about the existence of cyclic subgroups within a cyclic group. In the context of the...

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...

Roots of Unity in Finite Fields This article explains what Roots of Unity in a Finite Field are and how they are intertwined with multiplicative subgroups. The reader is expected to be familiar with...

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...