Quantum algorithms for factorization problem

Zhiqi Liu, Shihui Zheng, Xingyu Yan +2 more

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.

Samuel Čarnoký

Turgut Hanoymak, Akram Chehrazi

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.

Song Yuan Yan

Shah Muhammad Hamdi, Syed Tauhid Zuhori, Firoz Mahmud +1 more

Factoring large integers has been one of the most difficult problems in the history of mathematics and computer science. There was no efficient solution of this problem until Shor's algorithm emerged. Shor's algorithm is a polynomial time factoring algorithm which works on a quantum computer. Quantum computing is a new paradigm of computing that uses quantum mechanical phenomena in solving problems. The computers we are using right now are called classical computer. The most efficient classical factoring algorithm is General Number Field Sieve (GNFS). GNFS also cannot factor integers in polynomial time. In this paper, we compared these two algorithms in factoring integer in a standalone system.

Satoshi Iriyama, Masanori Ohya, Igor Vasil'evich Volovich