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Intermediate Signals and Sub-Component Circom’s primary purpose is to compile down to a Rank 1 Constraint System (R1CS), but its secondary purpose is to populate the witness. For most circuits, the...
Last updated on Jun 20, 20253 min read
Quadratic Constraints Circom Constraints A Rank 1 Constraint System has at most one multiplication between signals per constraint. This is called a "quadratic" constraint. Any constraint containing...
Last updated on Apr 15, 202511 min read
Introduction to ZK Circuits with Circom Circom is a programming language for creating Rank 1 Constraint Systems (R1CS) and populating the witness vector of the R1CS. The R1CS format is of interest...
Last updated on Apr 14, 20253 min read
Symbolic Variables in Circom A symbolic variable in Circom is a variable that has been assigned values from a signal. When a signal is assigned to a variable (thereby turning it into a symbolic...
Last updated on Apr 15, 20254 min read
Public and Private Inputs A public input in Circom is a signal in the witness that will be revealed to the verifier. For example, suppose we want to create a ZK proof that states: “we know the input...
Last updated on Apr 15, 20256 min read
Tick limits in Uniswap V3 The smallest tick in Uniswap v3 is -887,272 and the largest tick is 887,272. This chapter explains the rationale behind this range, which is based on finding the tick that...
Last updated on Oct 8, 20259 min read
Square Root Price in Uniswap V3 In Uniswap V2, the protocol tracks token reserves and derives the spot price, $p_x=y/x$, and total liquidity, $L=xy$, where $x$ and $y$ are the reserves of tokens X...
Last updated on Aug 20, 202513 min read
ERC-6909 Minimal Multi-Token Standard The ERC-6909 token standard is a streamlined alternative to the ERC-1155 token standard. The ERC-1155 standard introduced a multiple-token interface that allows...
Last updated on Mar 14, 202512 min read
Coding a Solidity rebase token A “rebase token” (sometimes “rebasing token”) is an ERC-20 token where the total supply, and the balances of token holders, can change without transfers, minting, or...
Last updated on Mar 13, 202525 min read
The Diamond Proxy Pattern Explained The Diamond Pattern (ERC-2535) is a proxy pattern where the proxy contract uses multiple implementation contracts simultaneously, unlike the Transparent...
Last updated on Feb 25, 202513 min read
Q Number Format Q number format is a notation for describing binary fixed-point numbers. A fixed-point number is a popular design pattern in Solidity for storing fractional values, since the language...
Last updated on Oct 8, 202510 min read
Introduction to Proxies Proxy contracts enable smart contracts to retain their state while allowing their logic to be upgraded. By default, smart contracts cannot be upgraded because deployed...
Last updated on Aug 20, 2025