Fast Digital Convolutions using Bit-Shifts

TL;DR

This paper introduces a novel exact digital convolution transform that uses only bit-shifts and additions, eliminating multiplications and quantization errors, suitable for arbitrary power-of-two lengths and cryptographic applications.

Contribution

It presents a new transform analogous to the DFT, based on cyclic integers and Carmichael's Theorem, enabling efficient, exact convolutions without numerical overflow.

Findings

Transform is exact and free from quantization errors.

Supports arbitrary power-of-two lengths with no multiplications.

Compatible with FFT algorithms for fast computation.

Abstract

An exact, one-to-one transform is presented that not only allows digital circular convolutions, but is free from multiplications and quantisation errors for transform lengths of arbitrary powers of two. The transform is analogous to the Discrete Fourier Transform, with the canonical harmonics replaced by a set of cyclic integers computed using only bit-shifts and additions modulo a prime number. The prime number may be selected to occupy contemporary word sizes or to be very large for cryptographic or data hiding applications. The transform is an extension of the Rader Transforms via Carmichael's Theorem. These properties allow for exact convolutions that are impervious to numerical overflow and to utilise Fast Fourier Transform algorithms.