On a cyclic structure of generators modulo primes

TL;DR

This paper introduces a new concept called the set of missing generators for primitive elements in cyclic groups modulo primes, analyzes their structure for special primes, and relates this to factoring RSA numbers under certain assumptions.

Contribution

It defines the set of missing generators, characterizes their structure for specific primes, and links the problem of factoring RSA numbers to computing a particular function T(p).

Findings

The cardinality of the set of missing generators is established for all odd primes.

For primes of a special form, the missing generators form a partition with a digraph structure of unicycles.

Factoring RSA numbers is shown to be computationally equivalent to computing T(p) under a specific prime-generating assumption.

Abstract

In this paper, we introduce a new notion called the \textit{set of missing generators} $\mathcal{M}(g)$ for a generator (or primitive element) $g$ of the cyclic group $\mathbb{Z}_p^*$, where $p$ is an odd prime. The cardinality of $\mathcal{M}(g)$ is established for all odd primes $p$. For primes $p$ of the form $2^iq_1^{j_1}q_2^{j_2}+1$, the collection $V_p = \{ \mathcal{M}(g):g\in \mathcal{G} \}$ forms an equinumerous partition of $\mathcal{G}$ (the set of all generators of $\mathbb{Z}_p^*$), and a digraph defined on the vertex set $V_p$ is a disjoint collection of unicycles of the same size. Thus, for every such prime, an unique triplet $(c,n,e)$ of integers, describing the structure of the digraph of missing generators, can be associated. With the help of cyclic structure, we present a macroscopic additive property of generators of $\mathbb{Z}_p^*$. Further, we show that factoring RSA numbers is computationally equivalent to computing $T(p)$, under the assumption that there exists an absolute constant $k$ such that the set $\{2^iN^j+1: 1\leq i,j<\log^k N\}$ contains a prime for any given odd $N$.