get_D() and get_y() in Curve StableSwap This article shows algebraically step-by-step how the code for get_D() and get_y() are derived from the StableSwap invariant. Given the StableSwap Invariant: $$ An^n\sum x_i +D=An^nD+\frac{D^{n+1}}{n^n\prod x_i} $$ There are two frequent math operations we wish to conduct with it: Compute $D$ given fixed values for $A$, and the […]
20 Common Solidity Beginner Mistakes Our intent is not to be patronizing towards developers early in their journey with this article. Having reviewed code from numerous Solidity developers, we’ve seen some mistakes occur more frequently and we list those here. By no means is this an exhaustive list of mistakes a Solidity developer can make. […]
The intuition behind elliptic curve digital signatures (ECDSA) This article explains how the ECDSA (Elliptic Curve Digital Signature Algorithm) works as well as why it works. We will incrementally “rediscover” the algorithm from first principles in this tutorial. Prerequisites We assume prior knowledge of Elliptic Curve Arithmetic Elliptic Curve Arithmetic in Finite Fields Digital Signature […]
Smart Contract Foundry Upgrades with the OpenZeppelin Plugin Upgrading a smart contract is a multistep and error-prone process, so to minimize the chances of human error, it is desirable to use a tool that automates the procedure as much as possible. Therefore, the OpenZeppelin Upgrade Plugin streamlines deploying, upgrading and managing smart contracts built with Foundry or […]
UUPS: Universal Upgradeable Proxy Standard (ERC-1822) The UUPS pattern is a proxy pattern where the upgrade function resides in the implementation contract, but changes the implementation address stored in the proxy contract via a delegatecall from the proxy. The high level mechanism is shown in the animation below: Similar to the Transparent Upgradeable Proxy, the […]
Trusted Setup A trusted setup is a mechanism ZK-SNARKs use to evaluate a polynomial at a secret value. Observe that a polynomial $f(x)$ can be evaluated by computing the inner product of the coefficients with successive powers of $x$: For example, if $f(x)=3x^3+2x^2+5x+10$, then the coefficients are $[3,2,5,10]$ and we can compute the polynomial as […]
The Schwartz-Zippel Lemma and its application to Zero Knowledge Proofs Nearly all ZK-Proof algorithms rely on the Schwartz-Zippel Lemma to achieve succintness. The Schwartz-Zippel Lemma states that if we are given two polynomials $p(x)$ and $q(x)$ with degree $d_p$ and $d_q$ respectively, and if $p(x) \neq q(x)$, then the number of points where $p(x)$ and […]
Building a Zero Knowledge Proof from an R1CS Given an arithmetic circuit encoded as a Rank 1 Constraint System, it is possible to create a ZK-proof of having a witness, albeit not a succinct one. This article describes how to accomplish that. A zero knowledge proof for a R1CS is accomplished by converting the witness […]
Langrange Interpolation with Python Lagrange interpolation is a technique for computing a polynomial that passes through a set of $n$ points. Interpolating a vector as a polynomial Examples A straight line through two points Consider that if we have two points, they can be interpolated with a line. For example, given $(1, 1)$ and $(2, […]