A Multi-Value Sequence Generated by Power Residue Symbol and Trace Function over Odd Characteristic Field

Yasuyuki NOGAMI  Satoshi UEHARA  Kazuyoshi TSUCHIYA  Nasima BEGUM  Hiroto INO  Robert H. MOLEROS-ZARAGOZA  

Publication
IEICE TRANSACTIONS on Fundamentals of Electronics, Communications and Computer Sciences   Vol.E99-A   No.12   pp.2226-2237
Publication Date: 2016/12/01
Online ISSN: 1745-1337
DOI: 10.1587/transfun.E99.A.2226
Type of Manuscript: Special Section PAPER (Special Section on Information Theory and Its Applications)
Category: Sequences
Keyword: 
multi-value sequence,  odd characteristic,  primitive polynomial,  power residue symbol,  trace,  cross-correlation,  

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Summary: 
This paper proposes a new multi-value sequence generated by utilizing primitive element, trace, and power residue symbol over odd characteristic finite field. In detail, let p and k be an odd prime number as the characteristic and a prime factor of p-1, respectively. Our proposal generates k-value sequence T={ti | ti=fk(Tr(ωi)+A)}, where ω is a primitive element in the extension field $F{p}{m}$, Tr(⋅) is the trace function that maps $F{p}{m} ightarrow {p}$, A is a non-zero scalar in the prime field $ {p}$, and fk(⋅) is a certain mapping function based on k-th power residue symbol. Thus, the proposed sequence has four parameters as p, m, k, and A. Then, this paper theoretically shows its period, autocorrelation, and cross-correlation. In addition, this paper discusses its linear complexity based on experimental results. Then, these features of the proposed sequence are observed with some examples.