Circom Template Parameters, Variables, Loops, If Statements, Assert This chapter covers essential syntax, which you'll see in most Circom programs. With Circom, we're able to define a Rank 1...

Indicate Then Constrain If we want to say that “ can be equal to 5 or 6” we can simply use the following constraint: However, suppose we want to say that “ is less than 5 or is greater than 17.” In...

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...

Reducing 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...

Bulletproofs 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...

Logarithmic sized proofs of commitment In a previous chapter, we showed that multiplying the sums of elements of the vectors $\mathbf{a}$ and $\mathbf{G}$ computes the sum of the outer product terms,...

Succinct proofs of a vector commitment If we have a Pedersen vector commitment $A$ which contains a commitment to a vector $\mathbf{a}$ as $A = a_1G_1 + a_2G_2+\dots + a_nG_n$ we can prove we know...

A 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...

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...

The intuition behind elliptic curve digital signatures (ECDSA) This article explains how the ECDSA (Elliptic Curve Digital Signature Algorithm) works as well as why it works. We will incrementally...