Exponential Sums Algorithm based on Optical Interference: Factorization of arbitrary large numbers in a single run Vincenzo Tamma 1,2 , Heyi Zhang 1 , Xuehua He 1 , Augusto Garuccio 2 , and Yanhua Shih 1 1 Department of Physics, University of Maryland, Baltimore County, Baltimore, Maryland 21250, USA, 2 Dipartimento Interateneo di Fisica, Universit` a degli Studi di Bari, 70100 Bari, Italy This article presents a new factorization algorithm based on the implementation of exponential sums using optical interference and exploiting the spectrum of the light source. Such a goal is achievable with the use of two different kinds of optical interferometers with variable optical paths: a liquid crystal grating and a generalized symmetric Michelson interferometer. This algorithm allows, for the first time, to find, in a single run, all the factors of an arbitrary large number N . The factorization of large numbers N is a lot more difficult than the reverse operation of multiplying large prime numbers. This difficulty is at the basis of encryp- tion systems [1]. A more recent approach to factorization, proposed by W. Schleich, exploits the periodic properties of truncated exponential sums [2, 3, 4] of order j : A (M,j) N (‘)= 1 M +1 M X m=0 exp  -2πim j N ‘  , (1) where M +1 is the number of phase terms in the sum (M is called the truncation parameter), N is the number to be factored, and j and l are positive integers, with j> 1 and 1 ≤ ‘ ≤ √ N , respectively. For j = 2, the truncated exponential sum reduces to a truncated Gauss sum [2]. If ‘ is a factor of N , all the terms interfere constructively and the truncated exponential sum assumes its maximum value, i.e. 1. On the other hand, if ‘ is not a factor of N , the truncated exponential sum assumes a value less than one, because of the destructive interference caused by the rapid oscillation of the phases terms of order j in Eq. (1). Obviously, the more terms involved in the sum (i.e. the larger the truncation parameter M ), the better we can distinguish between factors and non factors. It turns out that one needs at least M ∼ 2j √ N terms in order to discriminate factors from non factors [4]. Truncated Gauss sums have been reproduced using dif- ferent techniques [5, 6, 7, 8]. Unfortunately, all these past experimental realizations present two common problems. The first one is that there is only one knob (one phys- ical parameter) to vary the global ratio N/‘: the ratio N/‘ is known before the experiment is run. The second problem is that it is necessary to run the experiment for each possible trial factor ‘, to find out which ones are the factors. We wish to present a new approach, based on optical interference, which allows, for the first time, to obtain all the factors of a large number N in a single run, for any value of N . Such an approach is generalizable to the re- production of exponential sums of any order j , with sub- sequent reduction of the number of resources compared to the past realizations. Moreover, in this procedure, we will not take into ac- count the term with m = 0 in the truncated exponential op 1 op 2 op M Polychromatic plane wave Spectrometer op 3 FIG. 1: Theoretical model of a generic M-path optical inter- ferometer, where opm ≡ m j Nu is the length of the m th optical path with m =1, 2, ..., M. The optical paths are represented by arrows, which length increase quadratically, in the case of Gauss sums (j = 2). The incoming electromagnetic field is given by a polychromatic plane wave and the spectrum of the outgoing field, given by the interference of all the M optical paths, is measured by a spectrometer. sum (1), because it corresponds to a null phase, no mat- ter if the trial factor ‘ is a factor or not. Our procedure is based on two basic simple ideas. The first idea consists of the use of an M -path optical inter- ferometer. Such an interferometer needs to be able to reproduce, for a definite wavelength λ, M spatial modes with phase terms of order j and add them coherently in order to reproduce the truncated exponential sums. The second basic idea consists of exploiting the spectrum of the incoming light in order to reproduce all the possible trial factors in a single run for any value of N , measuring the intensity of the outgoing light as a function of the wavelength. We will now explain, in detail, the physical working principle behind the two main ideas stated above. In general, an M -path optical interferometer (see Fig. 1), interacting with an incoming polychromatic plane wave, allows the coherent superposition of M electro- magnetic phase terms (modes). The optical phase of the generic term, associated with the wavelength λ and with the m th optical path, with m =1, ..., M , is given by: φ m (λ) . =2π op m λ , (2) where op m . = n m d m (3) arXiv:0811.1595v1 [quant-ph] 10 Nov 2008