Transformation of Elliptic Curve Discrete Logarithm Problem to QUBO Using Direct Method in Quantum Annealing Applications

Abstract

This paper investigates how to reduce the elliptic curve discrete logarithm problem over prime fields to the quadratic unconstrained binary optimization (QUBO) problem in order to obtain as few logical qubits as possible. In the best case scenario, if n is the bitlength of a characteristic of prime field Fp, approximately 3n³ logical qubits are required for such a reduction in the Edwards curve case. We present a practical attack on an elliptic curve discrete logarithm problem over the 3-bit prime field F7 for an elliptic curve with the subgroup of order 8. We solved this problem using the D-Wave Advantage QPU. To the best of the authors' knowledge, no one has made, so far, a practical attack on the elliptic curve discrete logarithm over a prime field using the direct quantum method.