A Simpler Trace Representation of Legendre Sequences

Minglong QI; Shengwu XIONG; Jingling YUAN; Wenbi RAO; Luo ZHONG

doi:10.1587/transfun.E98.A.1026

A Simpler Trace Representation of Legendre Sequences

Minglong QI, Shengwu XIONG, Jingling YUAN, Wenbi RAO, Luo ZHONG

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Summary :

We found that the work of Kim et al. [1] on trace representation of the Legendre sequence with the periods p ≡ ±3 (mod 8) can be improved by restricting the selection of the periods p while maintaining the form p ≡ ±3 (mod 8) unchanged. Our method relies on forcing the multiplicative group of residue classes modulo p, Z_p^*, to take 2 as the least primitive root. On the other hand, by relaxing the very strong condition in the theorem of these authors and by using the product among powers of the primitive root and powers of any quadratic residue element to represent an element in Z_p^*, we could extend Kim's formula so that it becomes a special case of our formula more general.

Publication: IEICE TRANSACTIONS on Fundamentals Vol.E98-A No.4 pp.1026-1031

Publication Date: 2015/04/01

Publicized

Online ISSN: 1745-1337

DOI: 10.1587/transfun.E98.A.1026

Type of Manuscript: LETTER

Category: Cryptography and Information Security

Authors

Minglong QI
  Wuhan University of Technology
Shengwu XIONG
  Wuhan University of Technology
Jingling YUAN
  Wuhan University of Technology
Wenbi RAO
  Wuhan University of Technology
Luo ZHONG
  Wuhan University of Technology

Keyword

Legendre sequence, quadratic residue, quadratic non-residue, primitive root, pth root of unity

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Minglong QI, Shengwu XIONG, Jingling YUAN, Wenbi RAO, Luo ZHONG, "A Simpler Trace Representation of Legendre Sequences" in IEICE TRANSACTIONS on Fundamentals, vol. E98-A, no. 4, pp. 1026-1031, April 2015, doi: 10.1587/transfun.E98.A.1026.
Abstract: We found that the work of Kim et al. [1] on trace representation of the Legendre sequence with the periods p ≡ ±3 (mod 8) can be improved by restricting the selection of the periods p while maintaining the form p ≡ ±3 (mod 8) unchanged. Our method relies on forcing the multiplicative group of residue classes modulo p, Z_p^*, to take 2 as the least primitive root. On the other hand, by relaxing the very strong condition in the theorem of these authors and by using the product among powers of the primitive root and powers of any quadratic residue element to represent an element in Z_p^*, we could extend Kim's formula so that it becomes a special case of our formula more general.
URL: https://global.ieice.org/en_transactions/fundamentals/10.1587/transfun.E98.A.1026/_p

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@ARTICLE{e98-a_4_1026,
author={Minglong QI, Shengwu XIONG, Jingling YUAN, Wenbi RAO, Luo ZHONG, },
journal={IEICE TRANSACTIONS on Fundamentals},
title={A Simpler Trace Representation of Legendre Sequences},
year={2015},
volume={E98-A},
number={4},
pages={1026-1031},
abstract={We found that the work of Kim et al. [1] on trace representation of the Legendre sequence with the periods p ≡ ±3 (mod 8) can be improved by restricting the selection of the periods p while maintaining the form p ≡ ±3 (mod 8) unchanged. Our method relies on forcing the multiplicative group of residue classes modulo p, Z_p^*, to take 2 as the least primitive root. On the other hand, by relaxing the very strong condition in the theorem of these authors and by using the product among powers of the primitive root and powers of any quadratic residue element to represent an element in Z_p^*, we could extend Kim's formula so that it becomes a special case of our formula more general.},
keywords={},
doi={10.1587/transfun.E98.A.1026},
ISSN={1745-1337},
month={April},}

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TY - JOUR
TI - A Simpler Trace Representation of Legendre Sequences
T2 - IEICE TRANSACTIONS on Fundamentals
SP - 1026
EP - 1031
AU - Minglong QI
AU - Shengwu XIONG
AU - Jingling YUAN
AU - Wenbi RAO
AU - Luo ZHONG
PY - 2015
DO - 10.1587/transfun.E98.A.1026
JO - IEICE TRANSACTIONS on Fundamentals
SN - 1745-1337
VL - E98-A
IS - 4
JA - IEICE TRANSACTIONS on Fundamentals
Y1 - April 2015
AB - We found that the work of Kim et al. [1] on trace representation of the Legendre sequence with the periods p ≡ ±3 (mod 8) can be improved by restricting the selection of the periods p while maintaining the form p ≡ ±3 (mod 8) unchanged. Our method relies on forcing the multiplicative group of residue classes modulo p, Z_p^*, to take 2 as the least primitive root. On the other hand, by relaxing the very strong condition in the theorem of these authors and by using the product among powers of the primitive root and powers of any quadratic residue element to represent an element in Z_p^*, we could extend Kim's formula so that it becomes a special case of our formula more general.
ER -