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arXiv:1511.04385

Factoring Safe Semiprimes with a Single Quantum Query

Frédéric Grosshans, Thomas Lawson, François Morain, Benjamin Smith

Published Nov 13, 2015• 9 citations
arXivHTMLPDF

Abstract

Shor's factoring algorithm (SFA), by its ability to efficiently factor large numbers, has the potential to undermine contemporary encryption. At its heart is a process called order finding, which quantum mechanics lets us perform efficiently. SFA thus consists of a \emph{quantum order finding algorithm} (QOFA), bookended by classical routines which, given the order, return the factors. But, with probability up to $1/2$, these classical routines fail, and QOFA must be rerun. We modify these routines using elementary results in number theory, improving the likelihood that they return the factors. The resulting quantum factoring algorithm is better than SFA at factoring safe semiprimes, an important class of numbers used in cryptography. With just one call to QOFA, our algorithm almost always factors safe semiprimes. As well as a speed-up, improving efficiency gives our algorithm other, practical advantages: unlike SFA, it does not need a randomly picked input, making it simpler to construct in the lab; and in the (unlikely) case of failure, the same circuit can be rerun, without modification. We consider generalizing this result to other cases, although we do not find a simple extension, and conclude that SFA is still the best algorithm for general numbers (non safe semiprimes, in other words). Even so, we present some simple number theoretic tricks for improving SFA in this case.

DOI
10.48550/arxiv.1511.04385

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Shor's factoring algorithm (SFA), by its ability to efficiently factor large numbers, has the potential to undermine contemporary encryption. At its heart is a process called order finding, which quantum mechanics lets us perform efficiently. SFA thus consists of a \emph{quantum order finding algorithm} (QOFA), bookended by classical routines which, given the order, return the factors. But, with probability up to 1/21/21/2, these classical routines fail, and QOFA must be rerun. We modify these routines using elementary results in number theory, improving the likelihood that they return the factors. The resulting quantum factoring algorithm is better than SFA at factoring safe semiprimes, an important class of numbers used in cryptography. With just one call to QOFA, our algorithm almost always factors safe semiprimes. As well as a speed-up, improving efficiency gives our algorithm other, practical advantages: unlike SFA, it does not need a randomly picked input, making it simpler to construct in the lab; and in the (unlikely) case of failure, the same circuit can be rerun, without modification. We consider generalizing this result to other cases, although we do not find a simple extension, and conclude that SFA is still the best algorithm for general numbers (non safe semiprimes, in other words). Even so, we present some simple number theoretic tricks for improving SFA in this case.
Quantum computing has raised significant concerns regarding the security of classical cryptographic systems, particularly RSA, as it depends on the computational difficulty of factoring large semiprime numbers.In this study, we demonstrate the practical application of Shor's algorithm by successfully factorizing a semiprime integer, up to 30 bits (857830637 = 29167 X 29411) using the QuantumRings simulator.Our work highlights the effectiveness of Shor's algorithm in solving the factorization problem in polynomial time, which will significantly improve over the classical method for a large number.The results suggest that with advances in quantum hardware, such as increasing qubit counts and improving error correction, breaking larger RSA keys (e.g., RSA-2048) may soon become feasible.This seriously threatens current cryptographic systems, emphasizing the need to adopt post-quantum cryptography.Our findings aim to raise awareness among researchers, policymakers, and industry leaders of the importance of preparing for a quantum-safe future.
Shor's factoring algorithm is one of the most anticipated applications of quantum computing. However, the limited capabilities of today's quantum computers only permit a study of Shor's algorithm for very small numbers. Here we show how large GPU-based supercomputers can be used to assess the performance of Shor's algorithm for numbers that are out of reach for current and near-term quantum hardware. First, we study Shor's original factoring algorithm. While theoretical bounds suggest success probabilities of only 3-4 %, we find average success probabilities above 50 %, due to a high frequency of "lucky" cases, defined as successful factorizations despite unmet sufficient conditions. Second, we investigate a powerful post-processing procedure, by which the success probability can be brought arbitrarily close to one, with only a single run of Shor's quantum algorithm. Finally, we study the effectiveness of this post-processing procedure in the presence of typical errors in quantum processing hardware. We find that the quantum factoring algorithm exhibits a particular form of universality and resilience against the different types of errors. The largest semiprime that we have factored by executing Shor's algorithm on a GPU-based supercomputer, without exploiting prior knowledge of the solution, is 549755813701 = 712321 * 771781. We put forward the challenge of factoring, without oversimplification, a non-trivial semiprime larger than this number on any quantum computing device.
Abstract The task of Shor’s factorization algorithm (SFA) is to find a non-trivial factor of a given composite number N. In the algorithm, it is explicitly required that the constructor f (x) = a x mod N cycle is an even number, otherwise it returns to the algorithm header to recalculate. According to the analysis of the principle of SFA, the characteristics of RSA public key cryptosystem and a large number of calculation results, the random selection of a value is related to the obtained period. Aiming at this defect, a new optimization scheme is proposed on the basis of the previous optimization, and the odd cycle obtained when a is a perfect square number randomly becomes effective. When the arbitrarily chosen value of a obtains a period that is a multiple of 3, by modifying the decomposition method, the requirements for the period are relaxed without affecting the algorithm complexity. Using the classic algorithm, we experimentally prove the effectiveness of the improved algorithm, and the new algorithm significantly reduces the probability of repeated operations.
Shor’s factoring algorithm is one of the most anticipated applications of quantum computing. However, the limited capabilities of today’s quantum computers only permit a study of Shor’s algorithm for very small numbers. Here, we show how large GPU-based supercomputers can be used to assess the performance of Shor’s algorithm for numbers that are out of reach for current and near-term quantum hardware. First, we study Shor’s original factoring algorithm. While theoretical bounds suggest success probabilities of only 3–4%, we find average success probabilities above 50%, due to a high frequency of “lucky” cases, defined as successful factorizations despite unmet sufficient conditions. Second, we investigate a powerful post-processing procedure, by which the success probability can be brought arbitrarily close to one, with only a single run of Shor’s quantum algorithm. Finally, we study the effectiveness of this post-processing procedure in the presence of typical errors in quantum processing hardware. We find that the quantum factoring algorithm exhibits a particular form of universality and resilience against the different types of errors. The largest semiprime that we have factored by executing Shor’s algorithm on a GPU-based supercomputer, without exploiting prior knowledge of the solution, is 549,755,813,701 = 712,321 × 771,781. We put forward the challenge of factoring, without oversimplification, a non-trivial semiprime larger than this number on any quantum computing device.
The security of messages encoded via the widely used RSA public key encryption system rests on the enormous computational effort required to find the prime factors of a large number N using classical (i.e., conventional) computers. In 1994, however, Peter Shor showed that for sufficiently large N a quantum computer would be expected to perform the factoring with much less computational effort. This paper endeavors to explain, in a fashion comprehensible to the non-expert readers of this journal: (i) the RSA encryption protocol; (ii) the various quantum computer manipulations constituting the Shor algorithm; (iii) how the Shor algorithm performs the factoring; and (iv) the precise sense in which a quantum computer employing Shor's algorithm can be said to accomplish the factoring of very large numbers with less computational effort than a classical computer can. It is made apparent that factoring NNN generally requires many successive runs of the algorithm. The careful analysis herein reveals, however, that the probability of achieving a successful factorization on a single run is about twice as large as commonly quoted in the literature.
The security of messages encoded via the widely used RSA public key encryption system rests on the enormous computational effort required to find the prime factors of a large number N using classical (conventional) computers. In 1994 Peter Shor showed that for sufficiently large N, a quantum computer could perform the factoring with much less computational effort. This paper endeavors to explain, in a fashion comprehensible to the nonexpert, the RSA encryption protocol; the various quantum computer manipulations constituting the Shor algorithm; how the Shor algorithm performs the factoring; and the precise sense in which a quantum computer employing Shor’s algorithm can be said to accomplish the factoring of very large numbers with less computational effort than a classical computer. It is made apparent that factoring N generally requires many successive runs of the algorithm. Our analysis reveals that the probability of achieving a successful factorization on a single run is about twice as large as commonly quoted in the literature.