Integer factorization, a fundamental problem in computational mathematics, holds critical significance for modern cryptography, particularly in RSA encryption. Traditional approaches such as the number field sieve face exponential complexity limitations, while Shor’s quantum algorithm remains impractical due to hardware constraints. This study proposes a universal algorithm for integer factorization based on Ising machines by transforming the problem into a Quadratic Unconstrained Binary Optimization (QUBO) formulation. The algorithm introduces an optimal reduction formula to optimize the parameter ranges of local field coefficients ( h ) and coupling coefficients ( J ) in the Ising model. Additionally, a non-uniform column grouping method is employed to resolve the conflict between coefficient ranges and carry auxiliary quantum bits(qubits), minimizing the number of auxiliary qubits with minimal compromise on coefficient ranges. Using this approach, we successfully factorized the 22-bit integer 2,093,809 with only 118 qubits. Extrapolating to existing photonic Ising machines with 100,000 qubits, our method demonstrates the potential to factorize 631-bit integers, highlighting its promise for efficient large-scale integer factorization. All results presented in this paper are obtained from simulations on Fixstars Amplify and D-Wave simulators.
Integer factorization, a fundamental problem in computational mathematics, holds critical significance for modern cryptography, particularly in RSA encryption. Traditional approaches such as the number field sieve face exponential complexity limitations, while Shor’s quantum algorithm remains impractical due to hardware constraints. This study proposes a universal algorithm for integer factorization based on Ising machines by transforming the problem into a Quadratic Unconstrained Binary Optimization (QUBO) formulation. The algorithm introduces an optimal reduction formula to optimize the parameter ranges of local field coefficients ( h ) and coupling coefficients ( J ) in the Ising model. Additionally, a non-uniform column grouping method is employed to resolve the conflict between coefficient ranges and carry auxiliary quantum bits(qubits), minimizing the number of auxiliary qubits with minimal compromise on coefficient ranges. Using this approach, we successfully factorized the 22-bit integer 2,093,809 with only 118 qubits. Extrapolating to existing photonic Ising machines with 100,000 qubits, our method demonstrates the potential to factorize 631-bit integers, highlighting its promise for efficient large-scale integer factorization. All results presented in this paper are obtained from simulations on Fixstars Amplify and D-Wave simulators.
The difficulty in factoring large integers is a security basis of Rivest-Shamir-Adleman public key cryptography. Quantum annealing, a completely different method as compared with that of Shor’s algorithm, can transform an integer factorization problem into a combinatorial optimization problem using the quantum tunneling effects of D-wave quantum annealing to leap from local minima. In this paper, a new distributed integer factorization algorithm based on quantum annealing, which transforms an arbitrary integer into a stable Ising model executable using a D-wave quantum computer, is proposed. The stability and range of the local field coefficient h and the coupling term coefficient J of the Ising model are important factors that affect the success rate of integer factorization. Compared with the algorithm of Purdue University’s Jiang et al., the proposed method reduces the number of logical bits and the parameters of h and J by more than 60% and 40%, respectively, and the ranges of the coefficients of the Ising model are stable. Compared with the algorithm of Lockheed Martin Warren, the parameters of the proposed method are reduced in the orders from 106 to 102 of magnitude when the Ising model is guaranteed to be stable. In addition, to prove the correctness of the algorithm, Warren traversed factorization within 1000 integers. The proposed method traverses all integers within 10000 and all of them are successfully factored. The experimental results of this paper exceed the results of Shor’s algorithm, Jiang’s algorithm and the maximum published factorization scale of Lockheed Martin Warren’s algorithm.
This paper provides a new (second) way, which is completely different from Shor's algorithm, to show the optimistic potential of a D-Wave quantum computer for deciphering RSA and successfully factoring all integers within 10000. Our method significantly reduced the local field coefficient [Formula: see text] and coupling term coefficient [Formula: see text] by more than 33% and 26%, respectively, of those of Ising model, which can further improve the stability of qubit chains and improve the upper bound of integer factorization. In addition, our results obtained the best index (20-bit integer (1028171)) of quantum computing for deciphering RSA via the quantum computing software environment provided by D-Wave. Furthermore, Shor's algorithm requires approximately 40 qubits to factor the integer 1028171, which is far beyond the capacity of universal quantum computers. Thus, post quantum cryptography should further consider the potential of the D-Wave quantum computer for deciphering the RSA cryptosystem in future.
This work is the first to determine that a real quantum computer (including generalized and specialized) can decipher million-scale RSA relying solely on quantum algorithms, showing the real attack potential of D-Wave machines. The influence of different column widths on RSA factorization results is studied on the basis of a multiplication table, and the optimal column method is determined by traversal experiments. The traversal experiment of integer factorization within 10 000 shows that the local field and coupling coefficients are 75%-93% lower than the research of Shanghai University in 2020 and more than 85% lower than that of Purdue University in 2018. Extremely low Ising model parameters are crucial to reducing the hardware requirements, prompting factoring 1 245 407 on the D-Wave 2000Q real machine. D-Wave advantage already has more than 5000 qubits and will be expanded to 7000 qubits during 2023–2024, with remarkable improvements in decoherence and topology. This machine is expected to promote the solution of large-scale combinatorial optimization problems. One of the contributions of this paper is the discussion of the long-term impact of D-Wave on the development of post-quantum cryptography standards.
Integer factorization, a cryptographic cornerstone, remains exponentially hard for classical computers, while quantum approaches, though theoretically polynomial-time, still face hardware limitations. We propose an Ising-accelerated lattice sieve method using a photonic Ising machine to solve the NP-hard closest vector problem efficiently. Our approach requires only sub-linear growth in spin number and stable dynamic range of the interaction matrix as integer bit-length increases, minimizing hardware demands. We construct a programmable spatial photonic Ising machine based on a Dammann grating. Through multi-path simulated annealing experiments on the constructed SPIM, we experimentally achieved the complete factorization of a 32-bit integer and demonstrated the identification of multiple distinct smooth relation pairs required for factorizing a 100-bit integer. To further validate the generality and scalability of the proposed method, we performed classical computer simulations of the multi-path annealing process, achieving factorization of integers up to 120 bits and establishing a benchmark for lattice sieve-based factorization.
We have developed a framework to convert an arbitrary integer factorization problem to an executable Ising model by first writing it as an optimization function then transforming the k-bit coupling (k ≥ 3) terms to quadratic terms using ancillary variables. Our resource-efficient method uses [Formula: see text] binary variables (qubits) for finding the factors of an integer N. We present how to factorize 15, 143, 59989, and 376289 using 4, 12, 59, and 94 logical qubits, respectively. This method was tested using the D-Wave 2000Q for finding an embedding and determining the prime factors for a given composite number. The method is general and could be used to factor larger integers as the number of available qubits increases, or combined with other ad hoc methods to achieve better performances for specific numbers.
The majority of scholars believe that Shor’s algorithm is a unique and powerful quantum algorithm for RSA cryptanalysis, so current postquantum cryptography research has largely considered only the potential threats of Shor’s algorithm. This paper verifies the feasibility of deciphering RSA public key cryptography based on D-Wave, which is the second most effective RSA attack method after Shor’s algorithm. This paper proposes the influence of different column methods on the final integer factorization, puts forward a new dimension reduction formula, simplifies the integer factorization model based on quantum annealing, simulates it with the qbsolv quantum computing software environment provided by D-Wave, and factors the integer 1630729 (an 11-bit prime factor multiplied by an 11-bit prime factor). The research results show that choosing an appropriate number of columns and column width in the binary integer factorization multiplication table is very important for studying the optimization ability of the quantum annealing algorithm. In fact, Science, Nature, IEEE Spectrum, and the National Academies of Sciences (NAS) are consistent in asserting that the practical application of general-purpose quantum computers is far in the future. Therefore, although D-Wave computers were initially mainly purchased by Lockheed Martin, Google, etc., for purposes such as image processing, machine learning, combinatorial optimization, and software verification, post quantum cryptography research should further consider the potential of the D-Wave quantum computer in deciphering RSA cryptosystems in the future, and a discussion of this potential is one of the contributions of this paper.