A Totient Function Associated with Variants of Groups

TL;DR

This paper introduces a new totient function related to Euler's and Schemmel's functions, motivated by cryptographic applications, and explores its evaluation and number-theoretic properties.

Contribution

It defines a novel totient function linked to group variants and provides methods for its evaluation, expanding the theoretical framework for cryptographic group analysis.

Findings

New totient function related to Euler and Schemmel functions

Evaluation methods for the new totient function

Potential generalizations of the function

Abstract

Motivated by an application of semigroup variants to the discrete log problem in groups and related cryptographic applications, we introduce a new kind of totient function, related to both Euler's function and a generalisation of Euler's function introduced in 1869 by Schemmel. We focus on the problem of how to evaluate this function, and the number theory involved, while non-trivial and at times slightly technical, is reasonably accessible to a wide audience. It should also become clear that there are obvious generalisations of his new function that the interested reader might like to pursue.