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Roots of Unity raised to the k/2 power equals 1 or -1 Any $k$-th root of unity with even $k$ raised to the $k/2$ power will result in 1 or -1. This should not be confused with the similar-looking...
Last updated on Nov 12, 20255 min read
Square Roots of Roots of Unity The square root of a number $x$ is $y$ such that $y^2=x$. When $x$ is of the form $x^m$ and $m$ is even, then the square root is easy to compute: it’s simply $x^{m/2}$....
Last updated on Nov 12, 2025The square of a k-th root of unity is a k/2-th root of unity If we take the set of $k$-th roots of unity (with $k$ even) and square each element, the resulting set will be a set of half the size. The...
Last updated on Nov 12, 2025The Image Preservation Theorem for Multivalued Functions We’ll start this chapter on an unusual note — the NTT algorithm is quite simple and can be implemented in less than 20 lines of code. However,...
Last updated on Nov 12, 20254 min read
Vandermonde Matrices A Vandermonde matrix is a matrix that converts a polynomial from its coefficient representation into its value representation at a set of points. For a polynomial $f(x) = a_0...
Last updated on Nov 12, 202520 min read
Roots of Unity in Finite Fields This article explains what Roots of Unity in a Finite Field are and how they are intertwined with multiplicative subgroups. The reader is expected to be familiar with...
Last updated on Oct 29, 202511 min read
Interest Bearing Token Part 2 The interest-bearing extension adds the ability for a token mint to accrue interest over time. In our previous discussion of Token-2022, we introduced the...
Last updated on Oct 22, 202512 min read
Interest Bearing Token Part 1 The Token-2022 interest-bearing extension enables a token mint to automatically accrue interest on all token accounts for that specific mint. It uses an annual rate...
Last updated on Oct 22, 202514 min read
Ed25519 Signature Verification in Solana Verifying Ed25519 Signatures in Solana Anchor Programs This tutorial shows how to verify an off-chain Ed25519 signature in a Solana program. In Solana, custom...
Last updated on Oct 17, 202510 min read
Solana Instruction Introspection Instruction introspection enables a Solana program to read an instruction other than its own within the same transaction. Normally, a program can only read the...
Last updated on Oct 16, 202522 min read
Multiplication of Polynomials in Point Form Polynomial multiplication is widely used in zero-knowledge proofs and mathematical cryptography. But the brute force or traditional approach for...
Last updated on Oct 16, 20259 min read
The Solana Token 2022 Specification Token-2022 is a new backward-compatible version of the SPL Token program that supports additional features in the form of extensions. The bytecode for these...
Last updated on Oct 17, 2025