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 is a logical continuation of my recently-published paper. Security of modern communication based on RSA cryptographic protocols and their analogues is as crypto-immune as integer factorization (iFac) is dif...This paper is a logical continuation of my recently-published paper. Security of modern communication based on RSA cryptographic protocols and their analogues is as crypto-immune as integer factorization (iFac) is difficult. In this paper are considered enhanced algorithms for the iFac that are faster than the algorithm proposed in the previous paper. Among these enhanced algorithms is the one that is based on the ability to count the number of integer solutions on quadratic and bi-quadratic modular equations. Therefore, the iFac complexity is at most as difficult as the problem of counting. Properties of various modular equations are provided and confirmed in numerous computer experiments. These properties are instrumental in the proposed factorization algorithms, which are numerically illustrated in several examples.展开更多
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.
文摘This paper is a logical continuation of my recently-published paper. Security of modern communication based on RSA cryptographic protocols and their analogues is as crypto-immune as integer factorization (iFac) is difficult. In this paper are considered enhanced algorithms for the iFac that are faster than the algorithm proposed in the previous paper. Among these enhanced algorithms is the one that is based on the ability to count the number of integer solutions on quadratic and bi-quadratic modular equations. Therefore, the iFac complexity is at most as difficult as the problem of counting. Properties of various modular equations are provided and confirmed in numerous computer experiments. These properties are instrumental in the proposed factorization algorithms, which are numerically illustrated in several examples.
