  We prove the lower bound for the number of Lucas non-Wieferich primes in
arithmetic progressions. More precisely, for any given integer $k\geq 2$ there
are $\gg \log x$ Lucas non-Wieferich primes $p\leq x$ such that
$p\equiv\pm1\pmod{k}$, assuming the $abc$ conjecture for number fields.
Further, we discuss some applications of Lucas sequences in Cryptography.
