An initial step toward a quantum annealing approach to the discrete logarithm problem

Abstract

The discrete logarithm problem over a multiplicative group of integer modulo n is the key ingredient in the ElGamal encryption system. This problem has been proven to be tractable in polynomial time on a quantum computer by Shor's algorithm. This algorithm makes use of basic quantum gates, circuits, measurements and it has been experimented in the circuit-gate framework of quantum computing. On the other hand, the adiabatic framework of quantum computing with the quantum annealers such as the D-Wave machine that simulates the adiabatic process has become very popular, showing its potential for making a practical pathway to achieve quantum speedup. Furthermore, much progress has been reported recently on the tractability of the integer factoring problem, which is the other widely-used encryption method, on quantum annealers. In this context, it is important to explore the tractability of the discrete logarithm problem too using quantum annealers. In this work we present a first step made in this direction, by converting the discrete logarithm problem over a multiplicative group into the problem of minimizing a binary quadratic form, a standard problem format acceptable to a quantum annealer. Our formulation was tested for small-scale problem instances using the quantum-classical hybrid platform PyQUBO and we discuss the computational challenges and propose several potential improvements to the formulation.