On a cyclic structure of generators modulo primes

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 $N$.