The Discrete Cosine Transform over Prime Finite Fields

TL;DR

This paper introduces the Finite Field Discrete Cosine Transform (FFDCT) using finite field trigonometry over GF(p), enabling efficient computation especially with Mersenne primes and radix-2 algorithms.

Contribution

It defines the FFDCT over GF(p), explores its properties, and presents fast algorithms for Mersenne primes, extending digital transform techniques to finite fields.

Findings

FFDCT defined over GF(p) with blocklengths dividing (p+1)/2

Special case of Mersenne FFDCT with power-of-two blocklengths

Radix-2 fast algorithms enable efficient computation

Abstract

This paper examines finite field trigonometry as a tool to construct trigonometric digital transforms. In particular, by using properties of the k-cosine function over GF(p), the Finite Field Discrete Cosine Transform (FFDCT) is introduced. The FFDCT pair in GF(p) is defined, having blocklengths that are divisors of (p+1)/2. A special case is the Mersenne FFDCT, defined when p is a Mersenne prime. In this instance blocklengths that are powers of two are possible and radix-2 fast algorithms can be used to compute the transform.