Complexities of normal bases constructed from Gauss periods

We obtain a lower bound on the multiplicative order of Gauss periods which generate normal bases over finite fields. This bound improves the previous bound of von zur Gathen and Shparlinski.

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Gauss periods as constructions of low complexity normal bases

Designs Codes and Cryptography ◽

10.1007/s10623-011-9490-4 ◽

2011 ◽

Vol 62 (1) ◽

pp. 43-62 ◽

Cited By ~ 11

Author(s):

M. Christopoulou ◽

T. Garefalakis ◽

D. Panario ◽

D. Thomson

Keyword(s):

Low Complexity ◽

Normal Bases ◽

Gauss Periods

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Abelian Groups, Gauss Periods, and Normal Bases

Finite Fields and Their Applications ◽

10.1006/ffta.2000.0304 ◽

2001 ◽

Vol 7 (1) ◽

pp. 149-164 ◽

Cited By ~ 13

Author(s):

Shuhong Gao

Keyword(s):

Abelian Groups ◽

Normal Bases ◽

Gauss Periods

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Normal bases via general Gauss periods

Mathematics of Computation ◽

10.1090/s0025-5718-99-00988-6 ◽

1999 ◽

Vol 68 (225) ◽

pp. 271-291 ◽

Cited By ~ 32

Author(s):

Sandra Feisel ◽

Joachim von zur Gathen ◽

M. Amin Shokrollahi

Keyword(s):

Normal Bases ◽

Gauss Periods

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Normal Bases on Galois Ring Extensions

Symmetry ◽

10.3390/sym10120702 ◽

2018 ◽

Vol 10 (12) ◽

pp. 702

Author(s):

Aixian Zhang ◽

Keqin Feng

Keyword(s):

Signal Processing ◽

Finite Field ◽

Block Cipher ◽

Field Extension ◽

Galois Ring ◽

Normal Basis ◽

Galois Rings ◽

Ring Extension ◽

Galois Fields ◽

Normal Bases

Normal bases are widely used in applications of Galois fields and Galois rings in areas such as coding, encryption symmetric algorithms (block cipher), signal processing, and so on. In this paper, we study the normal bases for Galois ring extension R / Z p r , where R = GR ( p r , n ) . We present a criterion on the normal basis for R / Z p r and reduce this problem to one of finite field extension R ¯ / Z ¯ p r = F q / F p ( q = p n ) by Theorem 1. We determine all optimal normal bases for Galois ring extension.

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