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Pseudorandom number generators are required to generate pseudorandom numberswhich have good statistical properties as well as unpredictability incryptography. An m-sequence is a linear feedback shift register sequence withmaximal period over a finite field. M-sequences have good statisticalproperties, however we must nonlinearize m-sequences for cryptographicpurposes. A geometric sequence is a sequence given by applying a nonlinearfeedforward function to an m-sequence. Nogami, Tada and Uehara proposed ageometric sequence whose nonlinear feedforward function is given by theLegendre symbol, and showed the period, periodic autocorrelation and linearcomplexity of the sequence. Furthermore, Nogami et al. proposed ageneralization of the sequence, and showed the period and periodicautocorrelation. In this paper, we first investigate linear complexity of thegeometric sequences. In the case that the Chan--Games formula which describeslinear complexity of geometric sequences does not hold, we show the new formulaby considering the sequence of complement numbers, Hasse derivative andcyclotomic classes. Under some conditions, we can ensure that the geometricsequences have a large linear complexity from the results on linear complexityof Sidel'nikov sequences. The geometric sequences have a long period and largelinear complexity under some conditions, however they do not have the balanceproperty. In order to construct sequences that have the balance property, wepropose interleaved sequences of the geometric sequence and its complement.Furthermore, we show the periodic autocorrelation and linear complexity of theproposed sequences. The proposed sequences have the balance property, and havea large linear complexity if the geometric sequences have a large one.

ieice transactions on fundamentals of electronics, communications and computer sciences/ieice transactions on fundamentals of electronics, communications and computer science

Linear Complexity of Geometric Sequences Defined by Cyclotomic Classes and Balanced Binary Sequences Constructed by the Geometric Sequences