ERC-1155 Multi Token Standard The ERC-1155 standard describes how to create both fungible and non-fungible tokens then incorporate them into a single smart contract. This saves significant deployment...

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...

Inner Product Algebra In this article, we give some useful algebraic tricks for inner products that will be useful in deriving range proofs (and encoding circuits as inner products) later. Each rule...

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...

Storage Slots of Dynamic Types (Mappings, Arrays, Strings, Bytes) Dynamic-sized types in Solidity (sometimes referred to as complex types) are data types with variable size. They include mappings,...