A note on balancing sequences and application to cryptography

TL;DR

This paper establishes a lower bound on the distribution of balancing non-Wieferich primes in arithmetic progressions under the abc conjecture and explores their applications in cryptography.

Contribution

It provides the first lower bound for balancing non-Wieferich primes in progressions, linking number theory with cryptographic applications.

Findings

At least on the order of log x such primes exist up to x

Distribution of these primes in specific residue classes

Potential cryptographic uses of balancing sequences

Abstract

In this paper, we prove the lower bound for the number of balancing non-Wieferich primes in arithmetic progressions. More precisely, for any given integer $r\geq2$ there are $\gg\log x$ balancing non-Wieferich primes $p\leq x$ such that $p\equiv\pm1 \pmod{r}$, under the assumption of the $abc$ conjecture for the number field $\mathbb{Q}(\sqrt{2})$. Further, we discuss some applications of balancing sequences in cryptography.