Efficient Lifting of Discrete Logarithms Modulo Prime Powers

Abstract

We present a deterministic algorithm that, given a prime $p$ and a solution $x \in \mathbb Z$ to the discrete logarithm problem $a^x \equiv b \pmod p$ with $p\nmid a$, efficiently lifts it to a solution modulo $p^k$, i.e., $a^x \equiv b \pmod {p^k}$, for any fixed $k \geq 1$. The algorithm performs $k(\lceil \log_2 p\rceil +2)+O(\log p)$ multiplications modulo $p^k$ in the worst case, improving upon prior lifting methods by at least a factor of 8.