Linear Complexity of Geometric Sequences Defined by Cyclotomic Classes and Balanced Binary Sequences Constructed by the Geometric Sequences
Summary :
Pseudorandom number generators are required to generate pseudorandom numbers which have good statistical properties as well as unpredictability in cryptography. An m-sequence is a linear feedback shift register sequence with maximal period over a finite field. M-sequences have good statistical properties, however we must nonlinearize m-sequences for cryptographic purposes. A geometric sequence is a sequence given by applying a nonlinear feedforward function to an m-sequence. Nogami, Tada and Uehara proposed a geometric sequence whose nonlinear feedforward function is given by the Legendre symbol, and showed the period, periodic autocorrelation and linear complexity of the sequence. Furthermore, Nogami et al. proposed a generalization of the sequence, and showed the period and periodic autocorrelation. In this paper, we first investigate linear complexity of the geometric sequences. In the case that the Chan-Games formula which describes linear complexity of geometric sequences does not hold, we show the new formula by considering the sequence of complement numbers, Hasse derivative and cyclotomic classes. Under some conditions, we can ensure that the geometric sequences have a large linear complexity from the results on linear complexity of Sidel'nikov sequences. The geometric sequences have a long period and large linear complexity under some conditions, however they do not have the balance property. In order to construct sequences that have the balance property, we propose interleaved sequences of the geometric sequence and its complement. Furthermore, we show the periodic autocorrelation and linear complexity of the proposed sequences. The proposed sequences have the balance property, and have a large linear complexity if the geometric sequences have a large one.
- Publication
- IEICE TRANSACTIONS on Fundamentals Vol.E101-A No.12 pp.2382-2391
- Publication Date
- 2018/12/01
- Publicized
- Online ISSN
- 1745-1337
- DOI
- 10.1587/transfun.E101.A.2382
- Type of Manuscript
- Special Section PAPER (Special Section on Signal Design and Its Applications in Communications)
- Category
- Cryptography and Information Security
Authors
Kazuyoshi TSUCHIYA
the Koden Electronics Co., Ltd.
Chiaki OGAWA
Okayama University
Yasuyuki NOGAMI
Okayama University
Satoshi UEHARA
the University of Kitakyushu
Keyword
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Kazuyoshi TSUCHIYA, Chiaki OGAWA, Yasuyuki NOGAMI, Satoshi UEHARA, "Linear Complexity of Geometric Sequences Defined by Cyclotomic Classes and Balanced Binary Sequences Constructed by the Geometric Sequences" in IEICE TRANSACTIONS on Fundamentals,
vol. E101-A, no. 12, pp. 2382-2391, December 2018, doi: 10.1587/transfun.E101.A.2382.
Abstract: Pseudorandom number generators are required to generate pseudorandom numbers which have good statistical properties as well as unpredictability in cryptography. An m-sequence is a linear feedback shift register sequence with maximal period over a finite field. M-sequences have good statistical properties, however we must nonlinearize m-sequences for cryptographic purposes. A geometric sequence is a sequence given by applying a nonlinear feedforward function to an m-sequence. Nogami, Tada and Uehara proposed a geometric sequence whose nonlinear feedforward function is given by the Legendre symbol, and showed the period, periodic autocorrelation and linear complexity of the sequence. Furthermore, Nogami et al. proposed a generalization of the sequence, and showed the period and periodic autocorrelation. In this paper, we first investigate linear complexity of the geometric sequences. In the case that the Chan-Games formula which describes linear complexity of geometric sequences does not hold, we show the new formula by considering the sequence of complement numbers, Hasse derivative and cyclotomic classes. Under some conditions, we can ensure that the geometric sequences have a large linear complexity from the results on linear complexity of Sidel'nikov sequences. The geometric sequences have a long period and large linear complexity under some conditions, however they do not have the balance property. In order to construct sequences that have the balance property, we propose interleaved sequences of the geometric sequence and its complement. Furthermore, we show the periodic autocorrelation and linear complexity of the proposed sequences. The proposed sequences have the balance property, and have a large linear complexity if the geometric sequences have a large one.
URL: https://global.ieice.org/en_transactions/fundamentals/10.1587/transfun.E101.A.2382/_p
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@ARTICLE{e101-a_12_2382,
author={Kazuyoshi TSUCHIYA, Chiaki OGAWA, Yasuyuki NOGAMI, Satoshi UEHARA, },
journal={IEICE TRANSACTIONS on Fundamentals},
title={Linear Complexity of Geometric Sequences Defined by Cyclotomic Classes and Balanced Binary Sequences Constructed by the Geometric Sequences},
year={2018},
volume={E101-A},
number={12},
pages={2382-2391},
abstract={Pseudorandom number generators are required to generate pseudorandom numbers which have good statistical properties as well as unpredictability in cryptography. An m-sequence is a linear feedback shift register sequence with maximal period over a finite field. M-sequences have good statistical properties, however we must nonlinearize m-sequences for cryptographic purposes. A geometric sequence is a sequence given by applying a nonlinear feedforward function to an m-sequence. Nogami, Tada and Uehara proposed a geometric sequence whose nonlinear feedforward function is given by the Legendre symbol, and showed the period, periodic autocorrelation and linear complexity of the sequence. Furthermore, Nogami et al. proposed a generalization of the sequence, and showed the period and periodic autocorrelation. In this paper, we first investigate linear complexity of the geometric sequences. In the case that the Chan-Games formula which describes linear complexity of geometric sequences does not hold, we show the new formula by considering the sequence of complement numbers, Hasse derivative and cyclotomic classes. Under some conditions, we can ensure that the geometric sequences have a large linear complexity from the results on linear complexity of Sidel'nikov sequences. The geometric sequences have a long period and large linear complexity under some conditions, however they do not have the balance property. In order to construct sequences that have the balance property, we propose interleaved sequences of the geometric sequence and its complement. Furthermore, we show the periodic autocorrelation and linear complexity of the proposed sequences. The proposed sequences have the balance property, and have a large linear complexity if the geometric sequences have a large one.},
keywords={},
doi={10.1587/transfun.E101.A.2382},
ISSN={1745-1337},
month={December},}
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TY - JOUR
TI - Linear Complexity of Geometric Sequences Defined by Cyclotomic Classes and Balanced Binary Sequences Constructed by the Geometric Sequences
T2 - IEICE TRANSACTIONS on Fundamentals
SP - 2382
EP - 2391
AU - Kazuyoshi TSUCHIYA
AU - Chiaki OGAWA
AU - Yasuyuki NOGAMI
AU - Satoshi UEHARA
PY - 2018
DO - 10.1587/transfun.E101.A.2382
JO - IEICE TRANSACTIONS on Fundamentals
SN - 1745-1337
VL - E101-A
IS - 12
JA - IEICE TRANSACTIONS on Fundamentals
Y1 - December 2018
AB - Pseudorandom number generators are required to generate pseudorandom numbers which have good statistical properties as well as unpredictability in cryptography. An m-sequence is a linear feedback shift register sequence with maximal period over a finite field. M-sequences have good statistical properties, however we must nonlinearize m-sequences for cryptographic purposes. A geometric sequence is a sequence given by applying a nonlinear feedforward function to an m-sequence. Nogami, Tada and Uehara proposed a geometric sequence whose nonlinear feedforward function is given by the Legendre symbol, and showed the period, periodic autocorrelation and linear complexity of the sequence. Furthermore, Nogami et al. proposed a generalization of the sequence, and showed the period and periodic autocorrelation. In this paper, we first investigate linear complexity of the geometric sequences. In the case that the Chan-Games formula which describes linear complexity of geometric sequences does not hold, we show the new formula by considering the sequence of complement numbers, Hasse derivative and cyclotomic classes. Under some conditions, we can ensure that the geometric sequences have a large linear complexity from the results on linear complexity of Sidel'nikov sequences. The geometric sequences have a long period and large linear complexity under some conditions, however they do not have the balance property. In order to construct sequences that have the balance property, we propose interleaved sequences of the geometric sequence and its complement. Furthermore, we show the periodic autocorrelation and linear complexity of the proposed sequences. The proposed sequences have the balance property, and have a large linear complexity if the geometric sequences have a large one.
ER -
English
