A primality test for $Kp^\ell - 1$ numbers

TL;DR

This paper introduces a new deterministic primality test for numbers of the form Kp^l - 1 using an algebraic framework over quadratic fields, improving efficiency with only one modular exponentiation.

Contribution

It generalizes the Miller-Rabin test to quadratic fields and establishes an analogue of Korselt's criterion for these numbers, with practical efficiency demonstrated via SageMath.

Findings

Successfully tested primality of numbers in quadratic fields within milliseconds

Achieves a computational complexity of approximately log^2 N for the primality test

Provides a deterministic test requiring only a single modular exponentiation

Abstract

We develop an algebraic framework over arbitrary quadratic fields $L = \mathbb{Q}(\sqrt{D})$ to generalize the Miller-Rabin primality test. Consequently, we present a deterministic primality test for integers of the form $N = K p^{\ell} - 1$ that requires only a single modular exponentiation and achieves a computational complexity of $\tilde{\mathcal{O}}(\log^2 N)$. Furthermore, we also establish an analogue of Korselt's criterion within this setting. Finally, computational data generated using SageMath confirm its efficiency, successfully establishing the primality of numbers in the associated quadratic field within milliseconds.