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Randomness properties of some vector sequences generated by multivariate polynomial iterations

Gürkan Balıkçıoğlu, Pınar
We examine the randomness properties of the sequences generated by the multivariate polynomial iterations method proposed by Ostafe and Shparlinski, by using the six different choices of polynomials given by the same authors. Our analysis is based on two approaches: distributions of the periods and linear complexities of the produced vector sequences. We define the efficiency parameters, PE for “period efficiency” and LCE for “linear complexity efficiency”, so that the actual values of the period and linear complexity of a sequence can be easily compared with those of the ideal cases. For each polynomial choice, in order to obtain the period distribution of the generated vector sequences, we perform an exhaustive search for prime field sizes up to 13; and observe that the probability of attaining a maximum-period sequence is extremely low. Linear complexities of the sequences are also computed exhaustively for prime field sizes up to 13 and the multivariate polynomial iterations with the proposed polynomial choices are observed to generate sequences with having high linear complexities quite seldomly. We then concentrate on the largest period sequences produced by each choice, and investigate the linear complexity of those sequences for a given polynomial choice, at a specific field size p and number of polynomials m. We observe that an increase of p or m does not bring any improvement on the randomness of the generated sequences. Finally, we analyze the linear complexity of Ostafe’s sequences by fixing the period but leaving the choice of m and other initial values random,as in real life. Although computational constraints limit our exhaustive search results in the first part to relatively small values of p and m; the last part of our study lets us use higher values of p and m, to justify the projection that Ostafe’s method with the proposed polynomial choices is not a promising way of implementing pseudo-random number generators.