Fft on Yahel Uffenheimer

Algebraic Techniques for Fast Integer Multiplication

Sun, 18 Jan 2026 00:00:00 +0000

Prerequisites

Ring Theory: Homomorphisms, Ideals, Quotient Rings, Chinese Remainder Theorem (see Appendix for definitions).
Polynomials: Polynomial multiplication, division, convolutions.
Algorithm Analysis: Basic recursive complexity and $O$-notation.

Given two integers $a,b\in \mathbb{N}$, how fast can we multiply them?

Fast Fourier Transform over Finite Fields

Sun, 18 Jan 2026 00:00:00 +0000

Prerequisites

Ring Theory: Basic definitions (Rings, Fields, Polynomials).
Group Theory: Cyclic groups and generators.
Basic FFT: Familiarity with the standard divide-and-conquer strategy on $\mathbb{C}$ is helpful but not required.

The Fast Fourier Transform (FFT) is one of the most important algorithms in history. Usually, it is introduced over the field of complex numbers $\mathbb{C}$, relying on geometric intuition about roots of unity lying on the unit circle.