Quantum algorithms for computing short discrete logarithms and factoring RSA integers
This work addresses the challenge of making quantum algorithms more practical for cryptography by reducing hardware demands, though it is incremental as it builds on existing quantum methods.
The paper tackles the problem of reducing quantum computing requirements for factoring RSA integers and computing discrete logarithms by generalizing a quantum algorithm for short discrete logarithms, resulting in an algorithm with a modular exponentiation exponent length of slightly more than n/2 bits compared to 2n bits in Shor's algorithm.
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.