Bent functions on a Galois ring and systematic authentication codes

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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

Abstract
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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.
[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.
[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.
[4]	C. Ding, Systematic authentication codes from highly nonlinear functions, IEEE Trans. Inform. Theory, 50 (2004), 2421-2428. doi: 10.1109/TIT.2004.834788.
[5]	E. N. Gilbert, F. J. Macwilliams and N. J. A. Sloane, Codes which detect deception, Bell Syst. Tech. J., 33 (1974), 405-424.
[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.
[7]	K. Ireland and M. Rosen, "A Classical Introduction to Modern Number Theory,'' 2^nd edition, Springer-Verlag, New York, 1990.
[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.
[9]	B. R. McDonald, "Finite Rings with Identity,'' Marcel Deckker, Inc., New York, 1974.
[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.
[11]	G. J. Simmons, Authentication theory/coding theory, in "Advances in Cryptology-Crypto 84,'' Springer-Verlag, New York, (1984), 411-431.
[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.
[13]	D. R. Stinson, "Cryptography: Theory and Practice,'' CRC Press, 1995.
[14]	Z.-X. Wan, "Lectures on Finite Fields and Galois Rings,'' World Scientific, 2003.

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.
[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.
[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.
[4]	C. Ding, Systematic authentication codes from highly nonlinear functions, IEEE Trans. Inform. Theory, 50 (2004), 2421-2428. doi: 10.1109/TIT.2004.834788.
[5]	E. N. Gilbert, F. J. Macwilliams and N. J. A. Sloane, Codes which detect deception, Bell Syst. Tech. J., 33 (1974), 405-424.
[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.
[7]	K. Ireland and M. Rosen, "A Classical Introduction to Modern Number Theory,'' 2^nd edition, Springer-Verlag, New York, 1990.
[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.
[9]	B. R. McDonald, "Finite Rings with Identity,'' Marcel Deckker, Inc., New York, 1974.
[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.
[11]	G. J. Simmons, Authentication theory/coding theory, in "Advances in Cryptology-Crypto 84,'' Springer-Verlag, New York, (1984), 411-431.
[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.
[13]	D. R. Stinson, "Cryptography: Theory and Practice,'' CRC Press, 1995.
[14]	Z.-X. Wan, "Lectures on Finite Fields and Galois Rings,'' World Scientific, 2003.

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