Interleaved Sequences of Geometric Sequences Binarized with Legendre Symbol of Two Types

Kazuyoshi TSUCHIYA  Yasuyuki NOGAMI  Satoshi UEHARA  

IEICE TRANSACTIONS on Fundamentals of Electronics, Communications and Computer Sciences   Vol.E100-A   No.12   pp.2720-2727
Publication Date: 2017/12/01
Online ISSN: 1745-1337
DOI: 10.1587/transfun.E100.A.2720
Type of Manuscript: Special Section PAPER (Special Section on Information Theory and Its Applications)
Category: Sequences
pseudorandom number generator,  geometric sequence,  balance property,  interleaved sequence,  

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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.