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Cooley-Tukey FFT over $\mathbb{Q}_p$ via Unramified Cyclotomic Extension

Hiromasa Kondo

TL;DR

The paper addresses the challenge of making Cooley–Tukey FFTs efficient over $\mathbb{Q}_p$ by exploiting small-degree cyclotomic extensions $\mathbb{Q}_p(\zeta_M)$. It develops a two-part approach: (i) compute suitable roots of unity efficiently by first constructing minimal polynomials over $\mathbb{F}_p$ via Cantor–Zassenhaus and then lifting to $\mathbb{Q}_p$ with accelerated Hensel lifting, and (ii) perform the FFT using these roots of unity in the p-adic setting. Key contributions include a randomized, cheaper-than-FFT method for obtaining the minimal polynomial over $\mathbb{F}_p$, a fast lifting scheme that leverages sparsity of $X^s-1$, and a full construction of an $O(N^{1+o(1)})$-time FFT over $\mathbb{Q}_p$ for fixed $p$ by carefully selecting $s$ from cyclotomic factors. These results collectively demonstrate the viability of p-adic FFTs and hint at new, near-linear-time FFT algorithms over rationals and local fields. The work has potential practical impact in computational number theory and p-adic arithmetic where efficient DFTs are beneficial.

Abstract

The reason why Cooley-Tukey Fast Fourier Transform (FFT) over $\mathbb{Q}$ can be efficiently implemented using complex roots of unity is that the cyclotomic extensions of the completion $\mathbb{R}$ of $\mathbb{Q}$ are at most quadratic, and that roots of unity in $\mathbb{C}$ can be evaluated quickly. In this paper, we investigate a $p$-adic analogue of this efficient FFT. A naive application of this idea--such as invoking well-known algorithms like the Cantor-Zassenhaus algorithm or Hensel's lemma for polynomials to compute roots of unity--would incur a cost quadratic in the degree of the input polynomial. This would eliminate the computational advantage of using FFT in the first place. We present a method for computing roots of unity with lower complexity than the FFT computation itself. This suggests the possibility of designing new FFT algorithms for rational numbers. As a simple application, we construct an $O(N^{1+o(1)})$-time FFT algorithm over $\mathbb{Q}_p$ for fixed $p$.

Cooley-Tukey FFT over $\mathbb{Q}_p$ via Unramified Cyclotomic Extension

TL;DR

The paper addresses the challenge of making Cooley–Tukey FFTs efficient over by exploiting small-degree cyclotomic extensions . It develops a two-part approach: (i) compute suitable roots of unity efficiently by first constructing minimal polynomials over via Cantor–Zassenhaus and then lifting to with accelerated Hensel lifting, and (ii) perform the FFT using these roots of unity in the p-adic setting. Key contributions include a randomized, cheaper-than-FFT method for obtaining the minimal polynomial over , a fast lifting scheme that leverages sparsity of , and a full construction of an -time FFT over for fixed by carefully selecting from cyclotomic factors. These results collectively demonstrate the viability of p-adic FFTs and hint at new, near-linear-time FFT algorithms over rationals and local fields. The work has potential practical impact in computational number theory and p-adic arithmetic where efficient DFTs are beneficial.

Abstract

The reason why Cooley-Tukey Fast Fourier Transform (FFT) over can be efficiently implemented using complex roots of unity is that the cyclotomic extensions of the completion of are at most quadratic, and that roots of unity in can be evaluated quickly. In this paper, we investigate a -adic analogue of this efficient FFT. A naive application of this idea--such as invoking well-known algorithms like the Cantor-Zassenhaus algorithm or Hensel's lemma for polynomials to compute roots of unity--would incur a cost quadratic in the degree of the input polynomial. This would eliminate the computational advantage of using FFT in the first place. We present a method for computing roots of unity with lower complexity than the FFT computation itself. This suggests the possibility of designing new FFT algorithms for rational numbers. As a simple application, we construct an -time FFT algorithm over for fixed .
Paper Structure (7 sections, 12 theorems, 21 equations, 2 algorithms)

This paper contains 7 sections, 12 theorems, 21 equations, 2 algorithms.

Table of Contents

  1. Introduction
  2. Relation to existing algebraic FFTs
  3. Cyclotomic extensions of $\mathbb{Q}_p$: degrees and valuation structures
  4. Discussion on computation methods and complexity
  5. Computing a root of unity in $\mathbb{F}_p$
  6. Lifting the root of unity
  7. Fast Fourier transform for polynomials over $\mathbb{Q}_p$

Key Result

Lemma 2.1

Let $p$ be a prime number and $s$ be a positive integer relatively prime to $p$. holds, where $\mathop{\mathrm{ord}}\nolimits_s p$ is the smallest positive integer $r$ such that $p^r \equiv 1 \pmod s$.

Theorems & Definitions (24)

  • Lemma 2.1
  • Definition 2.2
  • Lemma 2.3: Hensel lifting
  • proof
  • Corollary 2.4: Hensel's lemma
  • Proposition 2.5
  • proof
  • Proposition 2.6
  • proof
  • Proposition 2.7
  • ...and 14 more