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ORTHOGONAL ARRAYS APPLICATION TO PSEUDORANDOM NUMBERS GENERATION AND OPTIMIZATION PROBLEMS A.G.Chefranov †‡, T.A.Mazurova ‡, I.D.Sidorov ‡, T.S.Letia †Eastern Mediterranean University, Gazimagusa, North Cyprus ‡Taganrog State University of Radio-Engineering, Taganrog, Russia Technical University of Cluj-Napoca, Cluj-Napoca, Romania

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Orthogonal Arrays Orthogonal arrays (OA) are widely used in many areas. In OA of strength t with L levels having L t rows and f columns with elements from {0,..,L-1} each combination of t columns contains without repetition all L t combinations of numbers {0,..,L-1}. OA may be used for generation of pseudorandom sequences using enumeration of combinations of columns and writing their rows in line. This results in very long not repeated sequences of numbers even in the case of rather small values of L and t. Thus, for L=7 and t=4 period length is equal to 10 7537. This approach requires generation of full OA; in the mentioned above case it contains 7 4 rows each of 7 numbers from {0,..,6}. For effective enumeration of permutations for this algorithm special partially factorial numbers representation is introduced. Also, directly formula for generating elements of OA may be used for pseudorandom numbers generation. Estimation of distances between OA rows is made showing uniform distribution of these vectors. This may be the ground for using OA in optimization

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Formula for OA Generation Elements of OA with prime L, strength t, L columns: (1)

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OA-PRNGA1 1. Choose L,t and some permutation of OA rows and permutation of some t columns as a seed of algorithm 2. Run through OA rows inside selected columns outputting OA elements as pseudorandom numbers. To improve statistical characteristics of generated sequence some transformation may be applied to it (for example, 0-s from this sequence are deleted, values are taken modulus 2, resulting bit sequence is transformed by grouping 2 neighboring bits, 00-s and 11-s are deleted, 01-s and 10-s are converted into 0-s and 1-s respectively, and then XORing of neighboring bytes). 3. Take next combination of this or next t columns, repeat step 2 4. Take next permutation of OA rows, repeat steps 2,3

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Partially Factorial-Based Numbers Representation n is a length of permutation, xi – numbers, which can be uniquely obtained from permutation, and which uniquely define respective permutation (2)

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Enumeration of Permutations and Calculation of xi Let we have permutation a1a2…an-1. Substituting n serially on all possible (numerated as 0,.., n-1) positions in this permutation we get n new permutations: na1a2…an-1, a1na2…an-1, …, a1a2…nan-1, a1a2…an-1n. To enumerate all permutations of 3 elements 0,1,2: 0 (all permutations of 1 element are ready), 10, 01 (all permutations of 2 elements are got), 210, 120, 102 (got by substitution of 2 into 1st permutation of size 2), 201, 021, 012 (all permutations are got). Value xi may be defined as place for insertion chosen in this algorithm for i-th element while getting particular permutation, for example, if we take permutation 012 then x0 = 0, x1 =1, x2 =2, and according to (2) x=3*x1+1*x2=5, for permutation 201 we get x0 = 0, x1 =1, x2 =0, x= 3*x1+1*x2=3, value of x0 will always be equal to 0.

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OA-PRNGA2 1. Choose L,t, some OA row number N and some combination of 2L numbers from {0,..,L-1} mix[2,L] as a seed. 2. Calculate next L OA row numbers and mix them with array mix, outputting L numbers taken as L outputs of generator. This mixture may be taken as where out[i] stands for i-th output pseudorandom number obtained from current OA row, i-th element of which is denoted as OArow[i] 3. Take next row cyclically by L t, repeat step 2.

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Measures Used for Comparison number of 1-s in generated bit sequence (length>=20000 bit), statistics is where k=2, must have asymptotic distribution as χ 2 with k-1 levels of freedom; number of m-bit vectors, statistics is given by: where Ni is a frequency of appearance of i-th vector in their sequence of k vectors; must have asymptotic distribution as χ 2 with 2 m -1levels of freedom;

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Measures Used for Comparison-1 number of sequences of 1-s and 0-s, statistics is where Bi is number of series of 0-s of length i, Gi is number of series of 1-s of length i, Ei=(n-i+3)/(2i+2) is expected number of series, k is maximal integer i for which Ei > 5; must have asymptotic distribution as χ 2 with 2k-2 levels of freedom;

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Measures Used for Comparison-2 coefficient of sequential correlation showing dependency of the next symbol on the previous one, calculated as Value of this statistics must be near 0 in 95% of occasions, n>2

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Comparison Results Sequences of 2500 and 250 000 bytes were used OA-PRNGA1, OA-PRNGA2, RC4, Standard Delphi Generator were compared Approximately same characteristics Timing for OA-PRNGA1 is slightly worse P-4 1.7Ghz 256 M was used

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UNIFORMITY OF OA ROWS DISTRIBUTION AND OA APPLICATION TO OPTIMIZATION PROBLEMS Theorem. Distance between any two rows of OA with L levels, L is a prime, L+1 columns, strength t=2, where is =

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Investigated for Optimization Functions Rosenbrock’s saddle and its generalization: Himmelblau’s function and its generalizations

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OA-Based Optimization Algorithms Essence of these algorithms is in mapping of initial ranges for each variable onto set of levels, calculation of function in the points shown by respective columns of OA, selection of the best point, and remapping of twice shrunk range on the set of levels. There may be used different approaches to select next point: by averaging of function’s values along each of dimensions separately, and choosing the optimal one as combination of separate optimums (AvgOA), or find best function value and choose corresponding set of variables as next optimum (DirOA); for remapping, next optimal point components may retain their relative positions as in the previous act of mapping (AvgOA, DirOA), or they become the center of new mapped range (CenOA).

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Results of Algorithms Investigation Algorithms DirOA, AvgOA, CenOA together with Monte-Carlo method (MCM) were considered Number of functions’ arguments 2-19 Number of OA levels up to 19 Number of runs for MCM up to 1 000 000 For little number of variables OA-based optimization algorithms have nearly the same productivity as MCM, but for larger number of variables (5-19) OA- based algorithms give significantly better results in quality and in timings as well.

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Conclusion Applicability of two proposed OA-based algorithms for pseudorandom numbers generating with extremely large periods was shown by comparison with well known algorithms. Special partially factorial numbers representation was introduced for effective enumeration of permutations for generated pseudorandom sequences. Such sequences may be used as stream ciphers for encryption in networks and for random number generation while modeling. Uniformity of scattering of points represented by OA rows is estimated. OA-based optimization algorithms iteratively reducing scope of search with the help of OA were considered. Relatively low computational complexity and sufficient accuracy especially in the case of large number of variables allows using them for optimization in control processes.

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