Quantum algorithms for computing short discrete logarithms and factoring RSA integers
Martin Ekerå, Johan Håstad
TL;DR
The paper advances quantum algorithm design by generalizing Ekerå's short discrete logarithm method to allow multiple executions, enabling a lattice-based classical post-processing step to recover the discrete logarithm from partial quantum results. This approach reduces quantum resource requirements, since exponents in the short DL setting are substantially smaller than in Shor's full factoring/finding algorithms, and it enables practical RSA factoring and order finding under side information. The authors show how RSA factoring can be recast as a short DL problem and how the overall algorithm can leverage a tradeoff between the number of quantum runs and classical computation. The work highlights a path toward quantum-cryptanalytic capabilities that demand smaller quantum devices, and discusses potential generalizations to other quantum algorithms such as Shor's factoring framework.
Abstract
In this paper we generalize the quantum algorithm for computing short discrete logarithms previously introduced by Ekerå so as to allow for various tradeoffs between the number of times that the algorithm need be executed on the one hand, and the complexity of the algorithm and the requirements it imposes on the quantum computer on the other hand. Furthermore, we describe applications of algorithms for computing short discrete logarithms. In particular, we show how other important problems such as those of factoring RSA integers and of finding the order of groups under side information may be recast as short discrete logarithm problems. This immediately gives rise to an algorithm for factoring RSA integers that is less complex than Shor's general factoring algorithm in the sense that it imposes smaller requirements on the quantum computer. In both our algorithm and Shor's algorithm, the main hurdle is to compute a modular exponentiation in superposition. When factoring an n bit integer, the exponent is of length 2n bits in Shor's algorithm, compared to slightly more than n/2 bits in our algorithm.