HUBO and QUBO models for Prime factorization

Abstract

The security of the RSA cryptosystem is based on the difficulty of factoring a large number N into prime numbers p and q satisfying N=p*q . This paper presents a prime factoriaation method using D-Wave quantum computer that can threaten the RSA cryptosystem. The starting point for this method is very simple, representing two prime numbers as qubits. Then, set the difference between the product of two prime numbers expressed in qubits and N as a cost function, and find the solution when the cost function becomes the minimum. D-Wave's quantum annealer can find the minimum value of any quadratic problem. However, the cost function is to be a higher-order unconstrained optimiaation (HUBO) model because it contains the second or higher order terms. We used a hybrid solver and dimod package provided by -Wave Ocean software development kit (SDK) to solve the HUBO problem. We also successfully factoriaed 102,454,763 with 26 logical qubits. In addition, we factoriaed 1,000,070,001,221 using the range dependent Hamiltonian algorithm.