On the Number of Regular Integers Modulo $n$ and Its Significance for Cryptography
Klaus Dohmen, Mandy Lange-Geisler
TL;DR
This paper analyzes the number $\varrho(n)$ of regular integers modulo $n$, i.e., solutions to $m^2 x \equiv m \pmod{n}$, and proves Morgado's formula $\varrho(n)=\sum_{d\mid^{\ast} n}\varphi(d)$ for all $n$ using four self-contained combinatorial proofs. The approaches run from an equivalence-relations argument and a purely bijective encoding to a reduced-fractions bijection and an inclusion-exclusion count, illustrating different combinatorial perspectives on the same identity. The results also yield a natural demonstration of the multiplicativity of $\varrho(n)$ and connect the quantity to the encryption-decryption probability in a generalized multi-prime, multi-power RSA scheme, showing the practical impact of the sequence $A055653$. The work closes with remarks on sharper bounds and asymptotics studied in related work.
Abstract
We present four combinatorial proofs of Morgado's formula for the number $\varrho(n)$ of non-congruent regular integers modulo $n$, corresponding to sequence A055653 in the On-Line Encyclopedia of Integer Sequences (OEIS), where an integer $m$ is said to be regular modulo $n$ if the congruence $m^2 x \equiv m \pmod{n}$ has a solution $x\in\mathbb{Z}$. To illustrate the significance of the sequence and Morgado's formula, we relate them to a recent multi prime, multi-power generalization of the RSA cryptosystem.