RSA cryptography is based on the difficulty of factoring large integers, which is an NP-hard(and hence intractable) problem for a classical computer. However, Shor's algorithm shows that its complexity is polynomi...RSA cryptography is based on the difficulty of factoring large integers, which is an NP-hard(and hence intractable) problem for a classical computer. However, Shor's algorithm shows that its complexity is polynomial for a quantum computer, although technical difficulties mean that practical quantum computers that can tackle integer factorizations of meaningful size are still a long way away. Recently, Jiang et al. proposed a transformation that maps the integer factorization problem onto the quadratic unconstrained binary optimization(QUBO) model. They tested their algorithm on a D-Wave 2000 Q quantum annealing machine, raising the record for a quantum factorized integer to 376289 with only 94 qubits. In this study, we optimize the problem Hamiltonian to reduce the number of qubits involved in the final Hamiltonian while maintaining the QUBO coefficients in a reasonable range, enabling the improved algorithm to factorize larger integers with fewer qubits. Tests of our improved algorithm using D-Wave's hybrid quantum/classical simulator qbsolv confirmed that performance was improved, and we were able to factorize 1005973, a new record for quantum factorized integers, with only 89 qubits. In addition, our improved algorithm can tolerate more errors than the original one. Factoring 1005973 using Shor's algorithm would require about 41 universal qubits,which current universal quantum computers cannot reach with acceptable accuracy. In theory, the latest IBM Q System OneTM(Jan. 2019) can only factor up to 10-bit integers, while the D-Wave have a thousand-fold advantage on the factoring scale. This shows that quantum annealing machines, such as those by D-Wave, may be close to cracking practical RSA codes, while universal quantum-circuit-based computers may be many years away from attacking RSA.展开更多
In this paper is demonstrated a method for reduction of integer factorization problem to an analysis of a sequence of modular elliptic equations. As a result, the paper provides a non-deterministic algorithm that comp...In this paper is demonstrated a method for reduction of integer factorization problem to an analysis of a sequence of modular elliptic equations. As a result, the paper provides a non-deterministic algorithm that computes a factor of a semi-prime integer n=pq, where prime factors p and q are unknown. The proposed algorithm is based on counting points on a sequence of at least four elliptic curves y2=x(x2+b2)(modn) , where b is a control parameter. Although in the worst case, for some n the number of required values of parameter b that must be considered (the number of basic steps of the algorithm) substantially exceeds four, hundreds of computer experiments indicate that the average number of the basic steps does not exceed six. These experiments also confirm all important facts discussed in this paper.展开更多
The hardness of the integer factoring problem(IFP)plays a core role in the security of RSA-like cryptosystems that are widely used today.Besides Shor’s quantum algorithm that can solve IFP within polynomial time,quan...The hardness of the integer factoring problem(IFP)plays a core role in the security of RSA-like cryptosystems that are widely used today.Besides Shor’s quantum algorithm that can solve IFP within polynomial time,quantum annealing algorithms(QAA)also manifest certain advantages in factoring integers.In experimental aspects,the reported integers that were successfully factored by using the D-wave QAA platform are much larger than those being factored by using Shor-like quantum algorithms.In this paper,we report some interesting observations about the effects of QAA for solving IFP.More specifically,we introduce a metric,called T-factor that measures the density of occupied qubits to some extent when conducting IFP tasks by using D-wave.We find that T-factor has obvious effects on annealing times for IFP:The larger of T-factor,the quicker of annealing speed.The explanation of this phenomenon is also given.展开更多
This paper introduces a new fog-assisted cloud storage which can achieve much higher throughput compared to the traditional cloud-only storage architecture by reducing the traffics toward the cloud storage. The fog-st...This paper introduces a new fog-assisted cloud storage which can achieve much higher throughput compared to the traditional cloud-only storage architecture by reducing the traffics toward the cloud storage. The fog-storage service providers are transparency to end-users and therefore, no modification on the end-user devices is necessary. This new system is featured with(1) a stronger audit scheme which is naturally coupled with the proposed architecture and does not suffer from the replay attack and(2) a transparent and efficient compensation mechanism for the fog-storage service providers. We provide rigorous theoretical analysis on the correctness and soundness of the proposed system. To the best of our knowledge, this is the first paper to discuss about a storage data audit scheme for fog-assisted cloud storage as well as the compensation mechanism for the service providers of the fog-storage service providers.展开更多
基金supported by the National Natural Science Foundation of China(Grant Nos.61332019,61572304,61572034,and 61272096)the Grant of the Special Zone Project of National Defense Innovation
文摘RSA cryptography is based on the difficulty of factoring large integers, which is an NP-hard(and hence intractable) problem for a classical computer. However, Shor's algorithm shows that its complexity is polynomial for a quantum computer, although technical difficulties mean that practical quantum computers that can tackle integer factorizations of meaningful size are still a long way away. Recently, Jiang et al. proposed a transformation that maps the integer factorization problem onto the quadratic unconstrained binary optimization(QUBO) model. They tested their algorithm on a D-Wave 2000 Q quantum annealing machine, raising the record for a quantum factorized integer to 376289 with only 94 qubits. In this study, we optimize the problem Hamiltonian to reduce the number of qubits involved in the final Hamiltonian while maintaining the QUBO coefficients in a reasonable range, enabling the improved algorithm to factorize larger integers with fewer qubits. Tests of our improved algorithm using D-Wave's hybrid quantum/classical simulator qbsolv confirmed that performance was improved, and we were able to factorize 1005973, a new record for quantum factorized integers, with only 89 qubits. In addition, our improved algorithm can tolerate more errors than the original one. Factoring 1005973 using Shor's algorithm would require about 41 universal qubits,which current universal quantum computers cannot reach with acceptable accuracy. In theory, the latest IBM Q System OneTM(Jan. 2019) can only factor up to 10-bit integers, while the D-Wave have a thousand-fold advantage on the factoring scale. This shows that quantum annealing machines, such as those by D-Wave, may be close to cracking practical RSA codes, while universal quantum-circuit-based computers may be many years away from attacking RSA.
文摘In this paper is demonstrated a method for reduction of integer factorization problem to an analysis of a sequence of modular elliptic equations. As a result, the paper provides a non-deterministic algorithm that computes a factor of a semi-prime integer n=pq, where prime factors p and q are unknown. The proposed algorithm is based on counting points on a sequence of at least four elliptic curves y2=x(x2+b2)(modn) , where b is a control parameter. Although in the worst case, for some n the number of required values of parameter b that must be considered (the number of basic steps of the algorithm) substantially exceeds four, hundreds of computer experiments indicate that the average number of the basic steps does not exceed six. These experiments also confirm all important facts discussed in this paper.
基金the National Natural Science Foundation of China(NSFC)(Grant No.61972050)the Open Foundation of StateKey Laboratory ofNetworking and Switching Technology(Beijing University of Posts and Telecommunications)(SKLNST-2020-2-16).
文摘The hardness of the integer factoring problem(IFP)plays a core role in the security of RSA-like cryptosystems that are widely used today.Besides Shor’s quantum algorithm that can solve IFP within polynomial time,quantum annealing algorithms(QAA)also manifest certain advantages in factoring integers.In experimental aspects,the reported integers that were successfully factored by using the D-wave QAA platform are much larger than those being factored by using Shor-like quantum algorithms.In this paper,we report some interesting observations about the effects of QAA for solving IFP.More specifically,we introduce a metric,called T-factor that measures the density of occupied qubits to some extent when conducting IFP tasks by using D-wave.We find that T-factor has obvious effects on annealing times for IFP:The larger of T-factor,the quicker of annealing speed.The explanation of this phenomenon is also given.
基金supported by Institute for Information & Communications Technology Promotion (IITP) grant funded by the Korea government (MSIT) (No. 20166-00599, a study on functional signature and its applications)supported in part by the Soonchunhyang University Research Fund
文摘This paper introduces a new fog-assisted cloud storage which can achieve much higher throughput compared to the traditional cloud-only storage architecture by reducing the traffics toward the cloud storage. The fog-storage service providers are transparency to end-users and therefore, no modification on the end-user devices is necessary. This new system is featured with(1) a stronger audit scheme which is naturally coupled with the proposed architecture and does not suffer from the replay attack and(2) a transparent and efficient compensation mechanism for the fog-storage service providers. We provide rigorous theoretical analysis on the correctness and soundness of the proposed system. To the best of our knowledge, this is the first paper to discuss about a storage data audit scheme for fog-assisted cloud storage as well as the compensation mechanism for the service providers of the fog-storage service providers.
