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arXiv:1902.01448 · Scientific Reports

Factoring semi-primes with (quantum) SAT-solvers

Michele Mosca, Sebastian R. Verschoor

Published May 14, 2022• 17 citations
arXivHTMLPDF

Abstract

The computational difficulty of factoring large integers forms the basis of security for RSA public-key cryptography. The best-known factoring algorithms for classical computers run in sub-exponential time. The integer factorization problem can be reduced to the Boolean Satisfiability problem (SAT). While this reduction has proved to be useful for studying SAT solvers, large integers have not been factored via such a reduction. Shor's quantum factoring algorithm factors integers in expected polynomial time. Large-scale fault-tolerant quantum computers capable of implementing Shor's algorithm are not yet available, preventing relevant benchmarking experiments. Recently, several authors have attempted quantum factorizations via reductions to SAT or similar NP-hard problems. While this approach may shed light on algorithmic approaches for quantum solutions to NP-hard problems, in this paper we study and question its practicality. We find no evidence that this is a viable path toward factoring large numbers, even for scalable fault-tolerant quantum computers, as well as for various quantum annealing or other special purpose quantum hardware.

DOI
10.1038/s41598-022-11687-7

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The computational difficulty of factoring large integers forms the basis of security for RSA public-key cryptography. The best-known factoring algorithms for classical computers run in sub-exponential time. The integer factorization problem can be reduced to the Boolean Satisfiability problem (SAT). While this reduction has proved to be useful for studying SAT solvers, large integers have not been factored via such a reduction. Shor's quantum factoring algorithm factors integers in expected polynomial time. Large-scale fault-tolerant quantum computers capable of implementing Shor's algorithm are not yet available, preventing relevant benchmarking experiments. Recently, several authors have attempted quantum factorizations via reductions to SAT or similar NP-hard problems. While this approach may shed light on algorithmic approaches for quantum solutions to NP-hard problems, in this paper we study and question its practicality. We find no evidence that this is a viable path toward factoring large numbers, even for scalable fault-tolerant quantum computers, as well as for various quantum annealing or other special purpose quantum hardware.
This paper explores a quantum-inspired technique for factoring semi-prime integers, a problem central to the security of RSA cryptography. While Shor's algorithm offers a polynomial-time solution on a quantum computer, posing a significant threat to RSA, current quantum hardware limitations hinder its practical implementation. This study simulates Shor's algorithm classically, replacing the quantum period-finding subroutine—the source of the quantum speedup—with a classical method to analyze its performance. The experiments factored semi-primes ranging from 2 to 8 digits, demonstrating that while the technique works, the execution time increases exponentially with the size of the number, escalating from seconds for small integers to an order of minutes for an 8-digit number. This highlights the inefficiency of classical period-finding compared to the quantum approach. The results validate the theoretical framework of Shor's algorithm but underscore the impracticality of classical simulations for cryptographic-scale integers. This work also explores the necessity of advancing quantum computing to realize scalable factorization while classical methods remain confined to pedagogical or small-scale applications. The study emphasizes the urgency of hybrid quantum-classical optimizations and reinforces Shor's algorithm as pivotal for future cryptographic vulnerability assessments and quantum-ready solutions.
One of the most well known mathematically hard problems in number theory is the integer factorization problem, roughly stated that decomposition of a composite number into its prime factors. In modern cryptography, RSA encryption algorithm whose security is based on integer factorization problem is highly practical, widespread and up to date no classical algorithm having polynomial running time for the factorization of large numbers is known. In 1994, Peter Shor proposed an efficient algorithm on quantum computer. In this paper, we  mention about the fundamentals of  Shor's quantum algorithm illustrating a concrete example.
There have been several efforts to apply quantum SAT solving methods to factor large integers. While these methods may provide insight into quantum SAT solving, to date they have not led to a convincing path to integer factorization that is competitive with the best known classical method, the Number Field Sieve. Many of the techniques tried involved directly encoding multiplication to SAT or an equivalent NP-hard problem and looking for satisfying assignments of the variables representing the prime factors. The main challenge in these cases is that, to compete with the Number Field Sieve, the quantum SAT solver would need to be superpolynomially faster than classical SAT solvers. In this paper the use of SAT solvers is restricted to a smaller task related to factoring: finding smooth numbers, which is an essential step of the Number Field Sieve. We present a SAT circuit that can be given to quantum SAT solvers such as annealers in order to perform this step of factoring. If quantum SAT solvers achieve any asymptotic speedup over classical brute-force search for smooth numbers, then our factoring algorithm is faster than the classical NFS.
Many cryptographic algorithms depend on computational complexity assumptions. Notorious cases are the RSA algorithm for public key criptography or the Diffie-Hellman key exchange protocol, to publicly agree on a common secret key. Both algorithms are known to be broken by quantum computing as well as those that can be reduced to a discrete logarithm problem. These are key algorithms in our digital society and are at the basis of everyday tasks, specially those that rely on digital signatures. The RSA algorithm, in particular, is probably the most used algorithm and is its assumed security the one that guards most of the e-commerce nowadays. In this case, it is the time complexity of finding the prime factors of a large number, that grows worse than polinomially with the size of the number, the main guardian of our cyberinfrastructure. The fact that a quantum computer can solve this problem in polynomial time using Shor's algorithm is seen as a potentially major disruption and has prompted security agencies to advice the progressive deprecation of these algorithms.
The complexity of public key algorithms like RSA and variants of Elgamal is based on the involvement of exponentiation in encryption and decryption. RSA is based on integer factorization and variants of Elgamal or based on discrete logarithms and are formed as hard problems. To achieve the perceived level of security, the required key length for public key cryptosystem needs to be determined. Quantum computers are capable of conducting an exponential number of simultaneous computations because of which they can find factors in polynomial time. Shor’s algorithm is popular algorithm for factoring huge integers in polynomial time on quantum computers. This paper presents RSA algorithm followed by various possible attacks for factoring and demonstrates Shor’s algorithm for factoring and discuss results for interpretations.
In this world of massive communication networks, data security and confidentiality are of crucial importance for maintaining secured private communication and protecting information against eavesdropping attacks. Existing cryptosystems provide data security and confidentiality by the use of encryption and signature algorithms for secured communication. Classical computers use cryptographic algorithms that use the product of two large prime numbers for generating public and private keys. These classical algorithms are based on the fact that integer factorization is a non-deterministic polynomial-time (NP) problem and requires super-polynomial time making it impossible for large enough integers. Shor’s algorithm is a well-known algorithm for factoring large integers in polynomial time and takes only O(b3) time and O(b) space on b-bit number inputs. Shor’s algorithm poses a potential threat to the current security system with the ongoing advancements of Quantum computers. This paper discusses how Shor’s algorithm will be able to break integer factorization-based cryptographic algorithms, for example, Rivest–Shamir–Adleman (RSA) and Rabin Algorithms. As a proof of concept, experimental analysis of Quantum Shor’s algorithm on existing public-key cryptosystems using IBM Quantum Experience is performed for factorizing integers of moderate length (seven bits) due to limitations of thirty-two qubits in present IBM quantum computers. In a nutshell, this work will demonstrate how Shor’s algorithm poses threat to confidentiality and authentication services.