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

A pseudorandom number generator is widely used in cryptography. A cryptographic pseudorandom number generator is required to gener- ate pseudorandom numbers which have good statistical properties as well as unpredictability. An m-sequence is a linear feedback shift register se- quence with maximal period over a finite field. M-sequences have good statistical properties, however we must nonlinearize m-sequences for cryp- tographic purposes. A geometric sequence is a binary sequence given by applying a nonlinear feedforward function to an m-sequence. Nogami, Tada and Uehara proposed a geometric sequence whose nonlinear feedfor- ward function is given by the Legendre symbol. They showed the geomet- ric sequences have good properties for the period, periodic autocorrelation and linear complexity. However, the geometric sequences do not have the balance property. In this paper, we introduce geometric sequences of two types and show some properties of interleaved sequences of the geomet- ric sequences of two types. These interleaved sequences have the balance property and double the period of the geometric sequences by the in- terleaved structure. Moreover, we show correlation properties and linear complexity of the interleaved sequences. A key of our observation is that the second type geometric sequence is the complement of the left shift of the first type geometric sequence by half-period positions.