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May  2012, 6(2): 249-258. doi: 10.3934/amc.2012.6.249

Bent functions on a Galois ring and systematic authentication codes

Claude Carlet 1, , Juan Carlos Ku-Cauich 2, and  Horacio Tapia-Recillas 2,

1. 

LAGA, Universities of Paris 8 and Paris 13, CNRS, Paris, France
2. 

Departamento de Matemáticas, Universidad Autónoma Metropolitana-Iztapalapa, 09349, México, D.F., Mexico, Mexico

Received  July 2011 Revised  December 2011 Published  April 2012

A class of bent functions on a Galois ring is introduced and based on these functions systematic authentication codes are presented. These codes generalize those appearing in [4] for finite fields.
Keywords: systematic authentication codes., Galois rings, bent functions.
Mathematics Subject Classification: Primary: 94A62; Secondary: 94C1.
Citation: Claude Carlet, Juan Carlos Ku-Cauich, Horacio Tapia-Recillas. Bent functions on a Galois ring and systematic authentication codes. Advances in Mathematics of Communications, 2012, 6 (2) : 249-258. doi: 10.3934/amc.2012.6.249
References:
[1]

C. Carlet, More correlation-immune and resilient functions over Galois fields and Galois rings, in "Advances in Cryptology-EUROCRYPT 97,'' 1233 (1997), 422-433.  Google Scholar

[2]

C. Carlet, C. Ding and H. Niederreiter, Authentication schemes from highly nonlinear functions, Des. Codes Cryptogr., 40 (2006), 71-79. doi: 10.1007/s10623-005-6407-0.  Google Scholar

[3]

C. Carlet and S. Dubuc, On generalized bent and $q$-ary perfect nonlinear functions, in "Proceedings of the 5th conference on Finite Fields and Applications,'' Springer-Verlag, (1999), 81-94.  Google Scholar

[4]

C. Ding, Systematic authentication codes from highly nonlinear functions, IEEE Trans. Inform. Theory, 50 (2004), 2421-2428. doi: 10.1109/TIT.2004.834788.  Google Scholar

[5]

E. N. Gilbert, F. J. Macwilliams and N. J. A. Sloane, Codes which detect deception, Bell Syst. Tech. J., 33 (1974), 405-424.  Google Scholar

[6]

A. R. Hammons, Jr., P. V. Kumar, A. R. Calderbank, N. J. A. Sloane and P. Solé, The $\mathbb Z_4$-linearity of Kerdock, Preparata, Goethals, and related codes, IEEE Trans. Inform. Theory, 40 (1994), 301-319. doi: 10.1109/18.312154.  Google Scholar

[7]

K. Ireland and M. Rosen, "A Classical Introduction to Modern Number Theory,'' 2nd edition, Springer-Verlag, New York, 1990.  Google Scholar

[8]

P. V. Kumar, R. A. Scholtz and L. R. Welch, Generalized bent functions and their properties, J. Comb. Theory, 40 (1985), 90-107. doi: 10.1016/0097-3165(85)90049-4.  Google Scholar

[9]

B. R. McDonald, "Finite Rings with Identity,'' Marcel Deckker, Inc., New York, 1974.  Google Scholar

[10]

F. Özbudak and Z. Saygi, Some constructions of systematic authentication codes using Galois rings, Des. Codes Cryptogr., 41 (2006), 343-357. doi: 10.1007/s10623-006-9021-x.  Google Scholar

[11]

G. J. Simmons, Authentication theory/coding theory, in "Advances in Cryptology-Crypto 84,'' Springer-Verlag, New York, (1984), 411-431. Google Scholar

[12]

G. J. Simmons, A survey of information authentication, in "Contemporary Cryptology: The Science of Information Integrity'' (ed. G.J. Simmons), IEEE Press, New York, (1992), 379-419.  Google Scholar

[13]

D. R. Stinson, "Cryptography: Theory and Practice,'' CRC Press, 1995.  Google Scholar

[14]

Z.-X. Wan, "Lectures on Finite Fields and Galois Rings,'' World Scientific, 2003.  Google Scholar

show all references

References:
[1]

C. Carlet, More correlation-immune and resilient functions over Galois fields and Galois rings, in "Advances in Cryptology-EUROCRYPT 97,'' 1233 (1997), 422-433.  Google Scholar

[2]

C. Carlet, C. Ding and H. Niederreiter, Authentication schemes from highly nonlinear functions, Des. Codes Cryptogr., 40 (2006), 71-79. doi: 10.1007/s10623-005-6407-0.  Google Scholar

[3]

C. Carlet and S. Dubuc, On generalized bent and $q$-ary perfect nonlinear functions, in "Proceedings of the 5th conference on Finite Fields and Applications,'' Springer-Verlag, (1999), 81-94.  Google Scholar

[4]

C. Ding, Systematic authentication codes from highly nonlinear functions, IEEE Trans. Inform. Theory, 50 (2004), 2421-2428. doi: 10.1109/TIT.2004.834788.  Google Scholar

[5]

E. N. Gilbert, F. J. Macwilliams and N. J. A. Sloane, Codes which detect deception, Bell Syst. Tech. J., 33 (1974), 405-424.  Google Scholar

[6]

A. R. Hammons, Jr., P. V. Kumar, A. R. Calderbank, N. J. A. Sloane and P. Solé, The $\mathbb Z_4$-linearity of Kerdock, Preparata, Goethals, and related codes, IEEE Trans. Inform. Theory, 40 (1994), 301-319. doi: 10.1109/18.312154.  Google Scholar

[7]

K. Ireland and M. Rosen, "A Classical Introduction to Modern Number Theory,'' 2nd edition, Springer-Verlag, New York, 1990.  Google Scholar

[8]

P. V. Kumar, R. A. Scholtz and L. R. Welch, Generalized bent functions and their properties, J. Comb. Theory, 40 (1985), 90-107. doi: 10.1016/0097-3165(85)90049-4.  Google Scholar

[9]

B. R. McDonald, "Finite Rings with Identity,'' Marcel Deckker, Inc., New York, 1974.  Google Scholar

[10]

F. Özbudak and Z. Saygi, Some constructions of systematic authentication codes using Galois rings, Des. Codes Cryptogr., 41 (2006), 343-357. doi: 10.1007/s10623-006-9021-x.  Google Scholar

[11]

G. J. Simmons, Authentication theory/coding theory, in "Advances in Cryptology-Crypto 84,'' Springer-Verlag, New York, (1984), 411-431. Google Scholar

[12]

G. J. Simmons, A survey of information authentication, in "Contemporary Cryptology: The Science of Information Integrity'' (ed. G.J. Simmons), IEEE Press, New York, (1992), 379-419.  Google Scholar

[13]

D. R. Stinson, "Cryptography: Theory and Practice,'' CRC Press, 1995.  Google Scholar

[14]

Z.-X. Wan, "Lectures on Finite Fields and Galois Rings,'' World Scientific, 2003.  Google Scholar

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