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A Totient Function Associated with Variants of Groups

James Renshaw

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.

A Totient Function Associated with Variants of Groups

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.
Paper Structure (3 sections, 13 theorems, 32 equations)

This paper contains 3 sections, 13 theorems, 32 equations.

Table of Contents

  1. Introduction
  2. The totient function
  3. Discrete Logs and variants of groups

Key Result

Theorem 1

Let $G$ be a finite group of even order $n$ and let $g,x\in G$ and $e\in U_{n}$. Then the number of distinct pairs $(y,f)$ with $y\in G, f\in U_n$ such that $\left(gy\right)^{f-1} = \left(gx\right)^{e-1}$ is at least equal to $T(n)$. In particular if $p=2q+1$ is a safe prime and $G=U_{p}$ then the n

Theorems & Definitions (13)

  • Theorem 1
  • Theorem 2
  • Theorem 3
  • Corollary 6
  • Theorem 7
  • Theorem 8
  • Theorem 9
  • Lemma 10
  • Lemma 11
  • Lemma 12
  • ...and 3 more