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22 min read
Multiplicative Subgroups and Primitive Elements
Multiplicative Subgroups and Primitive Elements Introduction This chapter continues our study of group theory by exploring subgroups and generators. The concept of a primitive element will be...
8 min read
Square and Multiply Algorithm
Square and Multiply Algorithm The square and multiply algorithm computes integer exponents in $\mathcal{O}(\log n)$ (logarithmic time). The naive way to compute an exponent $x^n$ is to multiply $x$...
7 min read
Computing the Current Tick Given sqrtPriceX96
Computing the Current Tick Given sqrtPriceX96 In the previous chapters, we saw that the protocol stores the square root of the price instead of the price itself. Therefore, it is necessary to relate...
7 min read
Uniswap V3 Factory and the Relationship Between Tick Spacing and Fees
Uniswap V3 Factory and the Relationship Between Tick Spacing and Fees In early chapters, we introduced the concept of ticks, which discretize the price curve. A tick is a price defined by the formula...
4 min read
ZK Proof of Selection Sort
ZK Proof of Selection Sort Most computations of interest are generally “stateful” — that is, they need to go through a series of steps to produce the final result. Sometimes, we do not need to show...
9 min read
How a ZKVM Works
How a ZKVM Works A Zero-Knowledge Virtual Machine (ZKVM) is a virtual machine that can create a ZK-proof that verifies it executed a set of machine instructions correctly. This allows us to take a...
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...
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...
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...
6 min read
The Permutation Argument
The Permutation Argument A permutation argument is a proof that two lists hold the same elements, but possibly in a different order. For example, is a permutation of and vice-versa. The permutation...
4 min read
MD5 Hash In Circom
MD5 Hash In Circom In this tutorial, we will implement the MD5 hash in Circom both to compute the hash and to constrain in Circom that it was computed correctly. Although the MD5 hash function is not...
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...