A Geometric Sequence Binarized with Legendre Symbol over Odd Characteristic Field and Its Properties
Summary :
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.
- Publication
- IEICE TRANSACTIONS on Fundamentals Vol.E97-A No.12 pp.2336-2342
- Publication Date
- 2014/12/01
- Publicized
- 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
Authors
Yasuyuki NOGAMI
Okayama University
Kazuki TADA
Okayama University
Satoshi UEHARA
The University of Kitakyushu
Keyword
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Yasuyuki NOGAMI, Kazuki TADA, Satoshi UEHARA, "A Geometric Sequence Binarized with Legendre Symbol over Odd Characteristic Field and Its Properties" in IEICE TRANSACTIONS on Fundamentals,
vol. E97-A, no. 12, pp. 2336-2342, December 2014, doi: 10.1587/transfun.E97.A.2336.
Abstract: 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.
URL: https://global.ieice.org/en_transactions/fundamentals/10.1587/transfun.E97.A.2336/_p
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@ARTICLE{e97-a_12_2336,
author={Yasuyuki NOGAMI, Kazuki TADA, Satoshi UEHARA, },
journal={IEICE TRANSACTIONS on Fundamentals},
title={A Geometric Sequence Binarized with Legendre Symbol over Odd Characteristic Field and Its Properties},
year={2014},
volume={E97-A},
number={12},
pages={2336-2342},
abstract={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.},
keywords={},
doi={10.1587/transfun.E97.A.2336},
ISSN={1745-1337},
month={December},}
Copy
TY - JOUR
TI - A Geometric Sequence Binarized with Legendre Symbol over Odd Characteristic Field and Its Properties
T2 - IEICE TRANSACTIONS on Fundamentals
SP - 2336
EP - 2342
AU - Yasuyuki NOGAMI
AU - Kazuki TADA
AU - Satoshi UEHARA
PY - 2014
DO - 10.1587/transfun.E97.A.2336
JO - IEICE TRANSACTIONS on Fundamentals
SN - 1745-1337
VL - E97-A
IS - 12
JA - IEICE TRANSACTIONS on Fundamentals
Y1 - December 2014
AB - 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.
ER -
English
