Browse all Zero Knowledge articles.
4 min read
Conditional Statements in Circom Circom is very strict with the usage of if-statements. The following rules must be followed: Signals cannot be used to alter the behavior of an if-statement. A signal...
8 min read
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...
4 min read
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...
6 min read
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 This chapter covers essential syntax, which you'll see in most Circom programs. With Circom, we're able to define a Rank 1...
12 min read
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...
10 min read
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...
21 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...
6 min read
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 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 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,...
18 min read
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...