Blog
The technical content top engineers rely on to level up.
6 min read
Reducing the number of equality checks (constraints) through random linear combinations
Reducing the number of equality checks (constraints) through random linear combinations Random linear combinations are a common trick in zero knowledge proof algorithms to enable $m$ equality checks...
4 min read
Inner Product Algebra
Inner Product Algebra In this article, we give some useful algebraic tricks for inner products that will be useful in deriving range proofs (and encoding circuits as inner products) later. Each rule...
6 min read
Bulletproofs ZKP: Zero Knowledge and Succinct Proofs for Inner Products
Bulletproofs ZKP: Zero Knowledge and Succinct Proofs for Inner Products Bulletproofs ZKPs allow a prover to prove knowledge of an inner product with a logarithmic-sized proof. Bulletproofs do not...
9 min read
Logarithmic sized proofs of commitment
Logarithmic sized proofs of commitment In a previous chapter, we showed that multiplying the sums of elements of the vectors $\mathbf{a}$ and $\mathbf{G}$ computes the sum of the outer product terms,...
18 min read
Succinct proofs of a vector commitment
Succinct proofs of a vector commitment If we have a Pedersen vector commitment $A$ which contains a commitment to a vector $\mathbf{a}$ as $A = a_1G_1 + a_2G_2+\dots + a_nG_n$ we can prove we know...
8 min read
A Zero Knowledge Proof for the Inner Product
A Zero Knowledge Proof for the Inner Product An inner product argument is a proof that the prover carried out the inner product computation correctly. This chapter shows how to construct a zero...
10 min read
Introduction to ZK Bulletproofs
Introduction to ZK Bulletproofs Bulletproofs are a zero knowledge inner product argument, which enable a prover to convince a verifier that they correctly computed an inner product. That is, the...
22 min read
Storage Slots of Dynamic Types (Mappings, Arrays, Strings, Bytes)
Storage Slots of Dynamic Types (Mappings, Arrays, Strings, Bytes) Dynamic-sized types in Solidity (sometimes referred to as complex types) are data types with variable size. They include mappings,...
16 min read
Assembly revert
Assembly revert Reverting transactions using inline assembly can be more gas-efficient than using the high-level Solidity or statement. In this guide, we’ll explore how the different types of reverts...
9 min read
get_D() and get_y() in Curve StableSwap
get_D() and get_y() in Curve StableSwap This article shows algebraically step-by-step how the code for and are derived from the StableSwap invariant. Given the StableSwap Invariant: $$ An^n\sum x_i...
15 min read
Smart Contract Foundry Upgrades with the OpenZeppelin Plugin
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...
12 min read
UUPS: Universal Upgradeable Proxy Standard (ERC-1822)
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...