Polynomial Commitments Via Pedersen Commitments A polynomial commitment is a mechanism by which a prover can convince a verifier a polynomial $p$ has an evaluation $y = p(x)$ at point $x$ without revealing anything about $p$. The sequence is as follows: The prover sends to the verifier a commitment $C$ to the polynomial, “locking in” […]
Homomorphisms A homomorphism between two groups means a structure preserving map between the two groups exists. Suppose we have two algebraic data structures $(A,\square)$ and $(B, \blacksquare)$, where the binary operator of $A$ is $\square$ and the binary operator of $B$ is $\blacksquare$. A homomorphism exists from $A$ to $B$ if and only if there […]
Elementary Group Theory for Programmers This article provides several examples of algebraic groups so that you can build an intuition for them. A group is a set with: a closed binary operator the binary operator is also associative an identity element every element having an inverse We also discussed abelian groups. An abelian group has […]
Abstract Algebra Abstract Algebra is the study of sets that have one or more operators on that set. For our purposes, we only care about binary operators. If we have sets and a binary operator on that set, we can categorize those sets based on how the binary operator behaves, and what elements are allowed […]
Try Catch and all the ways Solidity can revert This article describes all the kinds of errors that can happen when a smart contract is called, and how the Solidity Try / Catch block responds (or fails to respond) to each of them. To understand how Try / Catch works in Solidity, we must understand […]
The Beacon Proxy Pattern Explained A Beacon Proxy is a smart contract upgrade pattern where multiple proxies use the same implementation contract, and all the proxies can be upgraded in a single transaction. This article explains how this proxy pattern works. Prerequisites We are going to assume that you already know how a minimal proxy […]
Storage Slots in Solidity: Storage Allocation and Low-level assembly storage operations This article examines the storage architecture of the Ethereum Smart Contracts. It explains how variables are kept in the EVM storage and how to read and write to storage slots using low-level assembly (Yul). This information is a prerequisite to understanding how proxies in […]
AliasCheck and Num2Bits_strict in Circomlib An alias bug in Circom (or any ZK circuit language) occurs when a binary array of signals encodes a number larger than the field element can hold. We will refer to signals and field elements interchangeably in this article. We refer to the characteristic of the field as p. Loosely […]
Invariant Testing in Foundry Introduction In this article, we will discuss invariants and how to perform an invariant test on solidity smart contracts using foundry test suites. Invariant testing is another test methodology like unit test and fuzzing to verify the correctness of code. If you are unfamiliar with unit tests, please see our article […]