Simultaneous Modular Reduction and Kronecker Substitution for Small Finite Fields

TL;DR

This paper introduces algorithms that combine modular reduction and Kronecker substitution to efficiently perform polynomial operations over small finite fields using machine integer or floating point arithmetic, reducing conversions and improving performance.

Contribution

The paper proposes a novel integrated approach for modular polynomial multiplication that minimizes conversions and leverages machine arithmetic for efficiency.

Findings

Significant performance gains in polynomial multiplication

Efficient coefficient recovery methods

Practical improvements in finite field linear algebra

Abstract

We present algorithms to perform modular polynomial multiplication or modular dot product efficiently in a single machine word. We pack polynomials into integers and perform several modular operations with machine integer or floating point arithmetic. The modular polynomials are converted into integers using Kronecker substitution (evaluation at a sufficiently large integer). With some control on the sizes and degrees, arithmetic operations on the polynomials can be performed directly with machine integers or floating point numbers and the number of conversions can be reduced. We also present efficient ways to recover the modular values of the coefficients. This leads to practical gains of quite large constant factors for polynomial multiplication, prime field linear algebra and small extension field arithmetic.