Browse all Zero Knowledge articles.
3 min read
Quin Selector The Quin Selector is a design pattern that allows us to use a signal as an index for an array of signals. As a prerequisite, we assume the reader has read the chapter on Conditional...
Last updated on Apr 15, 202521 min read
Range Proof A range proof in the context of inner product arguments is a proof that the scalar $v$ has been committed to $V$ and $v$ is less than $2^n$ for some non-negative integer $n$. This article...
Last updated on Nov 20, 202410 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 Nov 17, 2024Reducing the number of equality checks (constraints) through random linear combinations Random linear combinations are a common trick in zero knowledge proof algorithms to enable $m$ equality checks...
Last updated on Nov 13, 2024A Zero Knowledge Proof for the Inner Product An inner product argument is a proof that the prover carried out the inner product computation correctly. This chapter shows how to construct a zero...
Last updated on Nov 13, 202410 min read
Introduction to ZK Bulletproofs Bulletproofs are a zero knowledge inner product argument, which enable a prover to convince a verifier that they correctly computed an inner product. That is, the...
Last updated on Nov 4, 2024Bulletproofs ZKP: Zero Knowledge and Succinct Proofs for Inner Products Bulletproofs ZKPs allow a prover to prove knowledge of an inner product with a logarithmic-sized proof. Bulletproofs do not...
Last updated on Nov 4, 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, 2024What are Pedersen Commitments and How They Work Pedersen commitments allow us to encode arbitrarily large vectors with a single elliptic curve point, while optionally hiding any information about the...
Last updated on Oct 29, 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, 20243 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, 202425 min read
Arithmetic Circuits for ZK In the context of zero-knowledge proofs, an arithmetic circuit is a system of equations that models a problem in NP. A key point from our article on P vs NP is that any...
Last updated on Sep 28, 2024