On a Modification of the Agrawal-Biswas Primality Test

TL;DR

This paper introduces a modified version of the Agrawal-Biswas primality test that, when combined with Miller-Rabin, offers increased accuracy at the cost of slower performance.

Contribution

It presents a novel variant of the Agrawal-Biswas algorithm that enhances primality testing accuracy by integrating with Miller-Rabin.

Findings

The variant can be used with Miller-Rabin for more accurate primality testing.

The new algorithm is slower than Miller-Rabin but offers improved reliability.

It extends the applicability of existing primality tests with a novel modification.

Abstract

We present a variant of the Agrawal-Biswas algorithm, a Monte Carlo algorithm which tests the primality of an integer $N$ by checking whether or not $(x+a)^N$ and $x^N + a$ are equivalent in a residue ring of $\mathbb{Z}/N\mathbb{Z}[x]$. The variant that we present is also a randomization of Lenstra jr. and Pomerance's improvement to the Agrawal-Kayal-Saxena deterministic primality test. We show that our variant of the Agrawal-Biswas algorithm can be used with the Miller-Rabin primality test to yield an algorithm which is slower than the Miller-Rabin test but relatively more accurate.