A note on balancing sequences and application to cryptography

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 $r\ge 2$ there are $\gg \log x$ balancing non-Wieferich primes $p\le x$ such that $p\equiv \pm 1(\mathrm{mod}r)$, 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.

18 pages, some applications of balancing sequences are given