A New Modular Division Algorithm and Applications

TL;DR

This paper introduces a new parallel modular division algorithm called PPI, which improves efficiency for large inputs and enables applications like exact division, digit modulus, and p-adic expansions.

Contribution

The paper presents the PPI algorithm, a novel parallel modular division method combining classical techniques with a new carry propagation, optimized for systolic parallelization.

Findings

PPI algorithm matches existing algorithms in performance but offers better parallelization for large inputs.

Parallelized PPI enables efficient exact division and digit modulus operations.

Application of PPI to rational number periods and p-adic expansions demonstrates its versatility.

Abstract

The present paper proposes a new parallel algorithm for the modular division $u/v\bmod \beta^s$, where $u,\; v,\; \beta$ and $s$ are positive integers $(\beta\ge 2)$. The algorithm combines the classical add-and-shift multiplication scheme with a new propagation carry technique. This "Pen and Paper Inverse" ({\em PPI}) algorithm, is better suited for systolic parallelization in a "least-significant digit first" pipelined manner. Although it is equivalent to Jebelean's modular division algorithm~\cite{jeb2} in terms of performance (time complexity, work, efficiency), the linear parallelization of the {\em PPI} algorithm improves on the latter when the input size is large. The parallelized versions of the {\em PPI} algorithm leads to various applications, such as the exact division and the digit modulus operation (dmod) of two long integers. It is also applied to the determination of the periods of rational numbers as well as their $p$-adic expansion in any radix $\beta \ge 2$.