Tickmath getSqrtRatioAtTick This article explains how the function in Uniswap V3 TickMath library works. The function takes a tick index and returns the square root price at that exact tick as a...

Circle FFT — Part 1: Building the Circle Domain Circle STARKs is a new zk-STARK scheme that has been implemented in Stwo and Plonky3, and it has been adopted by several zkVM projects. Its key...

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...

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 $x$...

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...

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...

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...

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...

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...

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...