Zero Knowledge
Browse all Zero Knowledge articles.
8 min read
32-Bit Emulation in ZK
32-Bit Emulation in ZK The default datatype in ZK is the field element, where all arithmetic is done modulo a large prime number. However, most “real” computation is done using 32, 64, or 256-bit...
3 min read
Quin Selector
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...
4 min read
Introduction to Stateful Computations in ZK
Introduction to Stateful Computations in ZK When carrying out iterative computations such as powers, factorials, or computing the Fibonacci sequence, we need to “stop the computation” after a certain...
24 min read
Modeling the Stack Data Structure in ZK
Modeling the Stack Data Structure in ZK This tutorial shows how to create a stack in Circom. Be warned — this chapter is long. However, the strategy for creating ZK proofs about stacks will be...
10 min read
Hello World Circom
Hello World Circom Introduction This chapter shows the relationship between Circom code and the Rank 1 Constraint System (R1CS) it compiles to. Understanding R1CSs is critical to understanding...
12 min read
Compute Then Constrain
Compute Then Constrain "Compute then constrain" is a design pattern in ZK circuits where an algorithm's correct output is first computed without constraints. The correctness of the solution is then...
6 min read
Indicate Then Constrain
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...
6 min read
Circom Template Parameters, Variables, Loops, If Statements, Assert
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...
21 min read
Range Proof
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...
6 min read
Reducing the number of equality checks (constraints) through random linear combinations
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...
6 min read
Bulletproofs ZKP: Zero Knowledge and Succinct Proofs for Inner Products
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...
9 min read
Logarithmic sized proofs of commitment
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,...