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Pseudorandom number generators are required to generate pseudorandom numbers which have good statistical properties as well as unpredictability in cryptography. An m-sequence is a linear feedback shift register sequence with maximal period over a finite field. M-sequences have good statistical properties, however we must nonlinearize m-sequences for cryptographic purposes. A geometric sequence is a sequence given by applying a nonlinear feedforward function to an m-sequence. Nogami, Tada and Uehara proposed a geometric sequence whose nonlinear feedforward function is given by the Legendre symbol, and showed the period, periodic autocorrelation and linear complexity of the sequence. Furthermore, Nogami et al. proposed a generalization of the sequence, and showed the period and periodic autocorrelation. In this paper, we first investigate linear complexity of the geometric sequences. In the case that the Chan--Games formula which describes linear complexity of geometric sequences does not hold, we show the new formula by considering the sequence of complement numbers, Hasse derivative and cyclotomic classes. Under some conditions, we can ensure that the geometric sequences have a large linear complexity from the results on linear complexity of Sidel'nikov sequences. The geometric sequences have a long period and large linear complexity under some conditions, however they do not have the balance property. In order to construct sequences that have the balance property, we propose interleaved sequences of the geometric sequence and its complement. Furthermore, we show the periodic autocorrelation and linear complexity of the proposed sequences. The proposed sequences have the balance property, and have a large linear complexity if the geometric sequences have a large one., Comment: 20 pages, 3 figures. arXiv admin note: text overlap with arXiv:1709.05163

A pseudorandom number generator is widely used in cryptography. A cryptographic pseudorandom number generator is required to generate pseudorandom numbers which have good statistical properties as well as unpredictability. An m-sequence is a linear feedback shift register sequence with maximal period over a finite field. M-sequences have good statistical properties, however we must nonlinearize m-sequences for cryptographic 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 feedforward function is given by the Legendre symbol. They showed the geometric 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 geometric sequences of two types. These interleaved sequences have the balance property and double the period of the geometric sequences by the interleaved 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., Comment: 16 pages, 11 figures