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A pseudorandom number generator is widely used in cryptography. Acryptographic pseudorandom number generator is required to generatepseudorandom numbers which have good statistical properties as well asunpredictability. An m-sequence is a linear feedback shift register sequencewith maximal period over a finite field. M-sequences have good statisticalproperties, however we must nonlinearize m-sequences for cryptographicpurposes. A geometric sequence is a binary sequence given by applying anonlinear feedforward function to an m-sequence. Nogami, Tada and Ueharaproposed a geometric sequence whose nonlinear feedforward function is given bythe Legendre symbol. They showed the geometric sequences have good propertiesfor the period, periodic autocorrelation and linear complexity. However, thegeometric sequences do not have the balance property. In this paper, weintroduce geometric sequences of two types and show some properties ofinterleaved sequences of the geometric sequences of two types. Theseinterleaved sequences have the balance property and double the period of thegeometric sequences by the interleaved structure. Moreover, we show correlationproperties and linear complexity of the interleaved sequences. A key of ourobservation is that the second type geometric sequence is the complement of theleft shift of the first type geometric sequence by half-period positions.

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

Interleaved sequences of geometric sequences binarized with Legendre symbol of two types