A Geometric Sequence Binarized with Legendre Symbol over Odd Characteristic Field and Its Properties

Yasuyuki NOGAMI  Kazuki TADA  Satoshi UEHARA  

IEICE TRANSACTIONS on Fundamentals of Electronics, Communications and Computer Sciences   Vol.E97-A   No.12   pp.2336-2342
Publication Date: 2014/12/01
Online ISSN: 1745-1337
DOI: 10.1587/transfun.E97.A.2336
Type of Manuscript: Special Section PAPER (Special Section on Information Theory and Its Applications)
Category: Sequence
geometric sequence,  odd characteristic,  primitive polynomial,  Legendre symbol,  trace,  

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Let p be an odd characteristic and m be the degree of a primitive polynomial f(x) over the prime field Fp. Let ω be its zero, that is a primitive element in F*pm, the sequence S={si}, si=Tr(ωi) for i=0,1,2,… becomes a non-binary maximum length sequence, where Tr(·) is the trace function over Fp. On this fact, this paper proposes to binarize the sequence by using Legendre symbol. It will be a class of geometric sequences but its properties such as the period, autocorrelation, and linear complexity have not been discussed. Then, this paper shows that the generated binary sequence (geometric sequence by Legendre symbol) has the period n=2(pm-1)/(p-1) and a typical periodic autocorrelation. Moreover, it is experimentally observed that its linear complexity becomes the maximum, that is the period n. Among such experimental observations, especially in the case of m=2, it is shown that the maximum linear complexity is theoretically proven. After that, this paper also demonstrates these properties with a small example.