Statistical Properties of Modulo-p Added p-Ary Sequences

Binary sequences are the most fundamental random numbers and have been extensively used in several applications such as spread-spectrum CDMA communications and cryptosystems. M-sequences, Kasami sequences, and Gold sequences, all of which can be generated by linear feedback shift registers (LFSRs), are well known as conventional binary sequences [1]. It is also well known that chaos phenomena can be used to generate random numbers and have been studied by many researchers, some of which are also engaged in binary sequences called chaotic binary sequences [2]. Since modulo-2 addition is one of fundamental operations for binary variables, we have studied statistical properties of modulo-2 added binary sequences [3]. We have shown that if one sequence is balanced and i.i.d. (independent and identically distributed), then the modulo-2 added sequence is also balanced and i.i.d., which is independent of the other binary sequence. Furthermore, we have also given some conditions to generate two modulo-2 added sequences which are completely uncorrelated to each other from a single chaotic real-valued sequence. In this paper, we discuss statistical properties of sequences obtained by modulo-p addition of two p-ary sequences, that is, we generalize some results for modulo-2 added binary sequences to the p-ary case. First, we theoretically evaluate statistics of sequences obtained by modulop addition of two general p-ary random variables. Under an assumption, we show that if one sequence is k-distributed, then the modulo-p added sequence is also k-distributed, which is independent of the other sequence. Next, we consider statistics of modulo-p added chaotic p-ary sequences generated by one-dimensional chaotic maps. Our theoretical evaluation based on the theory of chaotic dynamical systems [2],[4] shows that if one sequence is balanced and i.i.d., then the modulo-p added sequence is also balanced and i.i.d., which is independent of the other chaotic p-ary sequence. Furthermore, some conditions for generating two modulo-p added sequences which are completely independent of each other from a single chaotic real-valued sequence are also given.