Trace representation and linear complexity of binary eth power residue sequences of period p

Zongduo Dai; Guang Gong; Hong Yeop Song; Dingfeng Ye

doi:10.1109/TIT.2010.2103757

Trace representation and linear complexity of binary eth power residue sequences of period p

Zongduo Dai, Guang Gong, Hong Yeop Song, Dingfeng Ye

Department of Electrical and Electronic Engineering

Research output: Contribution to journal › Article › peer-review

20 Citations (Scopus)

Abstract

Let p = ef+1 be an odd prime for some e and f, and let F_p be the finite field with pelements. In this paper, we explicitly describe the trace representations of the binary characteristic sequences (of period p) of all the cyclic difference sets D which are some union of cosets of eth powers H _e in F_p^* (Δ\= F_p \{0}) for e≤ 12. For this, we define eth power residue sequences of period p, which include all the binary characteristic sequences mentioned above as special cases, and reduce the problem of determining their trace representations to that of determining the values of the generating polynomials of cosets of H _e in F_p^* at some primitive pth root of unity, and some properties of these values are investigated. Based on these properties, the trace representation and linear complexity not only of the characteristic sequences of all the known eth residue difference sets, but of all the sixth power residue sequences are determined. Furthermore, we have determined the linear complexity of a nonconstant eth power residue sequence for any e to be either p-1 or p whenever (e,(p-1)/n)= 1, where n is the order of 2 mod p.

Original language	English
Article number	5714271
Pages (from-to)	1530-1547
Number of pages	18
Journal	IEEE Transactions on Information Theory
Volume	57
Issue number	3
DOIs	https://doi.org/10.1109/TIT.2010.2103757
Publication status	Published - 2011 Mar

Bibliographical note

Funding Information:
Manuscript received February 20, 2008; revised August 01, 2010; accepted August 21, 2010. Date of current version February 18, 2011. The work of G. Gong was supported by a NSERC Discovery Grant, Canada. The work of H.-Y. Song was supported by the Basic Science Research Program through the National Research Foundation of Korea (NRF) funded by the Ministry of Education, Science and Technology (2009-0083888). Part of the material in this paper was presented at the 2003 Workshop on Coding and Cryptography, INRIA, France, 2003.

All Science Journal Classification (ASJC) codes

Information Systems
Computer Science Applications
Library and Information Sciences

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title = "Trace representation and linear complexity of binary eth power residue sequences of period p",

abstract = "Let p = ef+1 be an odd prime for some e and f, and let Fp be the finite field with pelements. In this paper, we explicitly describe the trace representations of the binary characteristic sequences (of period p) of all the cyclic difference sets D which are some union of cosets of eth powers H e in Fp* (Δ\= Fp \{0}) for e≤ 12. For this, we define eth power residue sequences of period p, which include all the binary characteristic sequences mentioned above as special cases, and reduce the problem of determining their trace representations to that of determining the values of the generating polynomials of cosets of H e in Fp* at some primitive pth root of unity, and some properties of these values are investigated. Based on these properties, the trace representation and linear complexity not only of the characteristic sequences of all the known eth residue difference sets, but of all the sixth power residue sequences are determined. Furthermore, we have determined the linear complexity of a nonconstant eth power residue sequence for any e to be either p-1 or p whenever (e,(p-1)/n)= 1, where n is the order of 2 mod p.",

author = "Zongduo Dai and Guang Gong and Song, {Hong Yeop} and Dingfeng Ye",

note = "Funding Information: Manuscript received February 20, 2008; revised August 01, 2010; accepted August 21, 2010. Date of current version February 18, 2011. The work of G. Gong was supported by a NSERC Discovery Grant, Canada. The work of H.-Y. Song was supported by the Basic Science Research Program through the National Research Foundation of Korea (NRF) funded by the Ministry of Education, Science and Technology (2009-0083888). Part of the material in this paper was presented at the 2003 Workshop on Coding and Cryptography, INRIA, France, 2003.",

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AU - Ye, Dingfeng

N1 - Funding Information: Manuscript received February 20, 2008; revised August 01, 2010; accepted August 21, 2010. Date of current version February 18, 2011. The work of G. Gong was supported by a NSERC Discovery Grant, Canada. The work of H.-Y. Song was supported by the Basic Science Research Program through the National Research Foundation of Korea (NRF) funded by the Ministry of Education, Science and Technology (2009-0083888). Part of the material in this paper was presented at the 2003 Workshop on Coding and Cryptography, INRIA, France, 2003.

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N2 - Let p = ef+1 be an odd prime for some e and f, and let Fp be the finite field with pelements. In this paper, we explicitly describe the trace representations of the binary characteristic sequences (of period p) of all the cyclic difference sets D which are some union of cosets of eth powers H e in Fp* (Δ\= Fp \{0}) for e≤ 12. For this, we define eth power residue sequences of period p, which include all the binary characteristic sequences mentioned above as special cases, and reduce the problem of determining their trace representations to that of determining the values of the generating polynomials of cosets of H e in Fp* at some primitive pth root of unity, and some properties of these values are investigated. Based on these properties, the trace representation and linear complexity not only of the characteristic sequences of all the known eth residue difference sets, but of all the sixth power residue sequences are determined. Furthermore, we have determined the linear complexity of a nonconstant eth power residue sequence for any e to be either p-1 or p whenever (e,(p-1)/n)= 1, where n is the order of 2 mod p.

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