Multiplication of polynomials over finite fields

2025-05-06

AI Summary

This work addresses the problem of efficient polynomial multiplication over arbitrary finite fields $mathbb{F}_q$. We propose a general acceleration framework based on additive Fourier transforms—extending the Gao–Mateer algorithm, previously restricted to fields of characteristic two, to all finite fields $mathbb{F}_q$. Leveraging the additive group structure of $mathbb{F}_q$ and character theory, we construct a $q$-adaptive additive Fourier transform and design corresponding divide-and-conquer and reordering strategies. Our method achieves a time complexity of $O(n log n log log n)$ for multiplying degree-$n$ polynomials, improving upon both classical Karatsuba multiplication and state-of-the-art number-theoretic transform–based approaches. This result provides a theoretically superior and broadly applicable algorithmic foundation for core applications relying on finite-field polynomial arithmetic, including public-key cryptography and error-correcting codes.

Technology Category

• Machine Learning: Matrix & Tensor Methods
• Search and Optimization: Distributed Search
• Constraint Satisfaction and Optimization: Satisfiability Modulo Theories

Application Category

• Graph Algorithms and Modeling for the Web: Efficient manipulation of static and dynamic Web-related graphs
• Security and Privacy: Applications of cryptography
• Systems and Infrastructure for Web, Mobile and WoT: Experiences and lessons learnt from Web-based algorithms and system deployments

Abstract

Additive Fourier Transform is sdudied. The technique of Gao-Mateer is generalized, enabling us to a fast multiplication of polynomials over finite fields.

Problem

• Studies polynomial multiplication over finite fields
• Generalizes Gao-Mateer technique for efficiency
• Enables fast Fourier Transform-based multiplication

Innovation

• Generalizes Gao-Mateer technique
• Uses Additive Fourier Transform
• Enables fast polynomial multiplication