Progress in the prime factorization of large numbers

Baonan Wang, Dan Zhang

Schuurman, Lynn K.N.

Xinyi Guo, Geguang Miao, Shinichi Nishizawa +1 more

Factorization of a large semiprime is a hard problem on classical computers, and quantum annealing is one way to solve the prime factorization problem by converting to a combinatorial optimization problem. A conversion method using the block-wise sum of partial products has been proposed but that works well up to numbers with 21-bit. To enhance the input bit width of the prime factorization, the paper introduces three proposals for prime factorization based on quantum annealing. First one is to divide the original problem into several sub-problems and solve them using multiple annealings. Second and third ones are analytical variable reduction of LSB and MSB parts. The proposals have been implemented on Python for Fixstars Amplify Annealing Engine (AE) and can factorize numbers with 46-bit or less. The proposals can also factorize numbers up to 57-bit but with a lot of trials for each number.

Daniel Eppens

Anas Ahmad Abudaqa, Nujud Alyami, Mostefa Kara +2 more

Abstract Many modern asymmetric encryption methods rely on prime numbers, as they have distinctive properties. For instance, the security of RSA cryptosystem relies on the computational difficulty of factoring a large composite number in its prime factors, a problem that remains challenging for classical computers but potentially solvable using quantum algorithms. On the other hand, generating large prime numbers is also challenging due to their irregular distribution among integers, necessitat- ing the use of primality testing algorithms to verify candidate primes. In this paper, we intensively review and classify various classical and quantum algorithms for factorization and primality testing, highlighting their advantages, limitations, speed/accuracy tradeoffs, time complexities, along with a brief summary. Furthermore, we apply and compare these algorithms to gain practical insights and conduct a comprehensive performance comparison. The insights from this paper show that while quantum factoring algorithms, particularly Shor’s algorithm and its refinements, have introduced significant advancements over their classical counterparts, quantum primality testing algorithms have not demonstrated comparable advantages.

Kyungtaek Jun, Hyunju Lee

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