Abstract—Prime Factorization based on Quantum approach in two phases has been performed. The first phase has been achieved at Quantum computer and the second phase has been achieved at the classic computer (Post Processing). At the second phase the goal is to estimate the period r of equation 1 N r x ≡ and to find the prime factors of the composite integer N in classic computer. In this paper we present a method based on Randomized Approach for estimation the period r with a satisfactory probability and the composite integer N will be factorized therefore with the Randomized Approach even the gesture of the period is not exactly the real period at least we can find one of the prime factors of composite N. Finally we present some important points for designing an Emulator for Quantum Computer Simulation. Keywords—Quantum Prime Factorization, Randomized Algorithms, Quantum Computer Simulation, Quantum Computation. I. INTRODUCTION ACTORIZING large integers has been an important problem from past till now especially in the state of RSA technique [1, 2]. In the RSA data encryption technique it is needed to find a very large integer that equals to multiplication of two large prime numbers. Many types of Prime Factorization methods presented but none of them even the methods based on parallelism could factorize a composite integer at a polynomial time [3-8]. Quantum Approach powered by Quantum Parallelism, Entanglement [9] and etc can solving the algorithmic problems in polynomial time for instance Prime Factorization Quantum Algorithm solved by Peter Shor [10-13]. If the composite integer N is known the goal is solving (1). Assume x is a random variable such that gcd (N, x) =1: 1 ≡ x N r (1) Assume r is even then we have: 0 ) 1 )( 1 ( 0 1 2 2 N r r N r x x x ≡ + − ⇒ ≡ − (2) It means than at least one of the P1 or P2 in (3) is a prime factor of composite N, although some of the answers may be 1 or/and the N (the trivial factors): Prime Factorizing of composite N to be achieved through two phases: A. First Phase The simultaneously calculation of ModN x t such that N t ≤ ≤ 0 and then performing the Quantum Fourier Transformation at the Quantum Computer. B. Second Phase The period estimation that satisfies Equation 1 at the classic computer. At first by the reason of the Quantum Computer is not accessible anywhere widely I’ve designed a Simulator for Quantum Computer including two Quantum Register such that the basic concepts of Quantum Computer for instance: Quantum bit or Qubit(This word was invented first time by Schumacher in 1993), Quantum Register, Entanglement, Quantum Measurement (a destructive phenomena),...[14-16] have been simulated and then the necessary Unitary Functions for example Quantum Furrier Transform[16-18] has been Simulated too. But the only difficulty is the Estimation of Period of (1) therefore we present a method based on Randomized Approach for estimating the period. II. THE IMPLEMENTATION OF QUANTUM ALGORITHM If the composite number N have been assumed we are going to find an integer q such that: q=2 L and L ∈ then we must have: 2 2 2 N q N < < (4) At this computer two Entangled Quantum Registers named 1 and 2 has been used such that a and b are the binary vectors of Quantum Register 1 and 2.The system state at each time can be express as: Mir Shahriar Emami, and Mohammad Reza Meybodi A Post Processing Method for Quantum Prime Factorization Algorithm based on Randomized Approach F ⎟ ⎟ ⎠ ⎞ ⎜ ⎜ ⎝ ⎛ + = ⎟ ⎟ ⎠ ⎞ ⎜ ⎜ ⎝ ⎛ − = N ModN x P N ModN x P r r , ) 1 ( gcd 2 , ) 1 ( gcd 1 2 2 (3) Mir Shahriar Emami is Member of the Board of Islamic Azad University, Roudehen Branch, Technical and Engineering Faculty, Computer Engineering Group, Roudehen, Iran. Mohammad Reza Meybodi is the Professor of Computer Engineering and Information Technology Department, Amirkabir University of Technology, Tehran, Iran. INTERNATIONAL JOURNAL OF COMPUTER, INFORMATION, AND SYSTEMS SCIENCE, AND ENGINEERING VOLUME 1 NUMBER 4 2007 ISSN 1307-2331 IJCISSE VOLUME 1 NUMBER 4 ISSN 1307-2331 238 © 2007 WASET.ORG