Improvement of a System for Prime Factorization Based on Optical Interferometer Kouichi Nitta, Nobuto Katsuta, and Osamu Matoba Department of Computer Science and Systems Engineering, Graduate School of Engineering, Kobe University Rokkodai 1-1, Nada, Kobe 657-8501, Japan nitta@kobe-u.ac.jp, katsuta@cs.brian.cs.kobe-u.ac.jp, matoba@kobe-u.ac.jp Abstract. An optical system to execute prime factorization is modified to improve its processing performance. In the improvement, some deletions in interference measurements are permitted. As results, it is experimentally verified that the improved method can provide correct results in factorization of 30bits integer. Keywords: Parallel processing, plane wave, optical interference, modulo operation, prime factorization, spatial parallelism. 1 Introduction Recently, some solutions based on optical signal processing have been proposed for various problems requiring hard computational tasks [1-4]. For example, two different systems for the travelling salesman problem (TSP) are presented Refs. [1] and [2]. In Ref. [1], the TSP is solved with an optical correlator. On the other hand, white light interferometry is utilized for the problem in the system shown in Ref. [2]. In such a situation, we also research an optical method for prime factorization [5]. In the method, modulo multiplication is executed in parallel by a simple Michelson interferometer. Modulo multiplication is an important operation in an algorithm for prime factorization. The characteristics of the optical method are based on spatial parallelism of light. In our previous works, some advantaged features of the optical method are clarified. Optical hardware for parallel processing is achieved with only a single light source and a single light modulator. Requirement to accuracy of alignment in the optical set up is not so critical in prime factorization. This feature is caused by properties of the algorithm. We have presented some reports based on the research. Two-dimensional (2D) parallel processing of the method is demonstrated in Ref. [6]. This processing form is considered to be suitable for optical implementation. Also, an improved method has been proposed to treat larger scale targets [7]. S. Dolev and M. Oltean (Eds.): OSC 2009, LNCS 5882, pp. 124 129, 2009. Springer-Verlag Berlin Heidelberg 2009
Improvement of a System for Prime Factorization Based on Optical Interferometer 125 In this research, another improvement is reported. This method provides correct results for prime factorization though there are deletions in a data set for postprocessing. It is shown that this method is useful for large scale processing. 2 Modulo Operation Based on Optical Interference and Phase Modulation An algorithm for prime factorization utilizes a specific function shown in Eq. (1). f (x) = a x mod N (1) This function is called modulo exponentiation. In Eq. (1), N is a target integer of prime factorization and it is given as a product of two prime factors. a is also an integer which is satisfied with Eqs. (2) and (3), respectively. 1 < a < N (2) gcd(a,n) =1 (3) f(x) is known to be a periodic function and the period is an integer. In case that the period is an even number, elements of prime factors can be extracted with post processing. Details of the algorithm are equivalent to Shor s quantum method presented in Refs. [8, 9]. Because f(x) is derived with a sequence of parallel processing for modulo multiplication, this processing is an important operation in prime factorization. Modulo multiplication is defined as Eq. (4). g(x) = yx modn (4) An optical method to derive g(x) is based on phase modulation of optical signals. Considering a light wave represented as Eq. (5), U(φ) = exp( i2πφ) (5) The optical filed corresponding to g(x) is obtained by settingφ = yx/n. In accordance with the principle, a system for parallel processing is constructed with a Michelson interferometer. Figure 1 shows a schematic diagram of the optical system. In the system, tilt angle of the mirror device is given as Eq. (6). θ(y) = 1 yλ 2 sin 1 DN (6) In the equation, λand D show wavelength of the light source and pixel pitch of sensor array, respectively. As result, desired results of parallel processing can be obtained. Fig. 2 shows relations between pixels and phase of optical signals.
126 K. Nitta, N. Katsuta, and O. Matoba Fig. 1. Schematic diagram of optical set up for parallel modulo multiplication Fig. 2. Diagram to show relations between pixels of detector array and detected optical signals with phase modulation 3 Optical Hardware and Improved Solutions In our method for prime factorization, l i (= a 2i modn) is derived from i=0 to n-1 as preprocessing. n shows a bit length of N. Next, interference measurements are executed by setting. After n times measurements, a data set corresponding to f(x) is extracted for post-processing to obtain the period of f(x). In the original method [5], Eq. (7) should be satisfied. l i 2 i+1 1 < M (1 i n) (7) Here, M shows the number of detector array. As a result, two-dimensional parallel system can treat only less than 12 bits integers for factorization.
Improvement of a System for Prime Factorization Based on Optical Interferometer 127 Fig. 3. An example of profile corresponding to f(x) in cases of (a) the original method, (b) results obtained by optical measurements with some deletions, and (c) data set after interpolation Therefore, the method described as following is proposed for larger scale processing. In the method, data set corresponding to f(x) is described in accordance with the same procedure as that of the original method. If the condition shown as Eq. (7) is not satisfied at x=x, we set 0 at f(x ). An example of the data sets obtained by the original and proposed methods is shown Fig. 3. From this procedure, we may achieve prime factorization of larger integer. Autocorrelation of f(x) Fig. 4. Experimental results of verification at N=537625733 and a=305208178
128 K. Nitta, N. Katsuta, and O. Matoba We verify the proposed method experimentally. In the experiment, the optical hardware for 2D parallel processing [6] is used. Figure 4 describes an example of verification. Profile of the figure shows autocorrelation of f(x) derived with postprocessing. In the results of Fig. 3, N and a are set to be 537625733 (=23801 23293) and 305208178, respectively. From the profile, it is found that the period is 8. Finally, two correct factors can be obtained from the period. We can demonstrate that the proposed improvement executes prime factorization in 30 bits integer. Note that the improvement is not verified to achieve all integers which are less than 30bits. 4 Comments In terms of optical supercomputing, a feature of our method is to utilize spatial parallelism of light. In 1980's, many methods based on spatial parallelism have been developed in the research field of digital optical computing. In these methods, symbolic substitution [10], perfect shuffle [11], systolic array [12], and so on [13], are implemented on optical systems. Optical array logic has also been proposed as paradigm for general purpose of parallel processing [14]. Our research is much inspired by these methods. We describe essential difference between our method and these previous ones as following. In the previous methods, at first, many emitters are required to prepare a set of input data for target parallel processing. Additionally, photo-detector array is necessary to obtain output results. On the other hand, our method utilizes only one light source because a set of the input can be generated by collimation of the light beam and tilt of the plane wave obtained by the collimation. Also, large scale parallel processing can be achieved with less number of detectors. This is because the algorithm for prime factorization needs not a set of individual results obtained by optical parallel processing but only characteristics of profile in the set. Our method permits slight alignment errors whereas previous optical methods generally require highly accurate for alignment. We have estimated and discussed on the robustness of our method against the errors in Ref. [15]. Consequently, spatial parallelism is considered to be useful for optical supercomputing. 5 Summary We have presented an improvement in a system for prime factorization based on optical parallel processing. In the improvement, large scale factorization is executed by permitting some deletions in the profile of f(x). We have experimentally verified usefulness of the improved method. References 1. Shaked, N.T., Messika, S., Dolev, S., Rosen, J.: Optical solutions for bound NP-complete problems. Appl. Opt. 46, 711 724 (2007) 2. Haist, T., Osten, W.: An optical solution for the traveling salesman problem. Opt. Exp. 15, 10473 10482 (2007)
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