Multi-Valued Sequences Generated by Power Residue Symbols over Odd Characteristic Fields

Begum NASIMA  Yasuyuki NOGAMI  Satoshi UEHARA  Robert H. MOLEROS-ZARAGOZA  

Publication
IEICE TRANSACTIONS on Fundamentals of Electronics, Communications and Computer Sciences   Vol.E100-A   No.4   pp.922-929
Publication Date: 2017/04/01
Online ISSN: 1745-1337
Type of Manuscript: Special Section PAPER (Special Section on Signal Design and Its Applications in Communications)
Category: Sequences
Keyword: 
geometric sequence,  multi-valued sequence,  odd characteristic,  primitive polynomial,  Legendre symbol,  trace,  

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Summary: 
This paper proposes a new approach for generating pseudo random multi-valued (including binary-valued) sequences. The approach uses a primitive polynomial over an odd characteristic prime field $ {p}$, where p is an odd prime number. Then, for the maximum length sequence of vectors generated by the primitive polynomial, the trace function is used for mapping these vectors to scalars as elements in the prime field. Power residue symbol (Legendre symbol in binary case) is applied to translate the scalars to k-value scalars, where k is a prime factor of p-1. Finally, a pseudo random k-value sequence is obtained. Some important properties of the resulting multi-valued sequences are shown, such as their period, autocorrelation, and linear complexity together with their proofs and small examples.