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

Bent functions on a Galois ring and systematic authentication codes

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.
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, 1233 (1997), 422.   Google Scholar

[2]

C. Carlet, C. Ding and H. Niederreiter, Authentication schemes from highly nonlinear functions,, Des. Codes Cryptogr., 40 (2006), 71.  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, (1999), 81.   Google Scholar

[4]

C. Ding, Systematic authentication codes from highly nonlinear functions,, IEEE Trans. Inform. Theory, 50 (2004), 2421.  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.   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.  doi: 10.1109/18.312154.  Google Scholar

[7]

K. Ireland and M. Rosen, "A Classical Introduction to Modern Number Theory,'' 2nd edition,, Springer-Verlag, (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.  doi: 10.1016/0097-3165(85)90049-4.  Google Scholar

[9]

B. R. McDonald, "Finite Rings with Identity,'', Marcel Deckker, (1974).   Google Scholar

[10]

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

[11]

G. J. Simmons, Authentication theory/coding theory,, in, (1984), 411.   Google Scholar

[12]

G. J. Simmons, A survey of information authentication,, in, (1992), 379.   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, 1233 (1997), 422.   Google Scholar

[2]

C. Carlet, C. Ding and H. Niederreiter, Authentication schemes from highly nonlinear functions,, Des. Codes Cryptogr., 40 (2006), 71.  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, (1999), 81.   Google Scholar

[4]

C. Ding, Systematic authentication codes from highly nonlinear functions,, IEEE Trans. Inform. Theory, 50 (2004), 2421.  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.   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.  doi: 10.1109/18.312154.  Google Scholar

[7]

K. Ireland and M. Rosen, "A Classical Introduction to Modern Number Theory,'' 2nd edition,, Springer-Verlag, (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.  doi: 10.1016/0097-3165(85)90049-4.  Google Scholar

[9]

B. R. McDonald, "Finite Rings with Identity,'', Marcel Deckker, (1974).   Google Scholar

[10]

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

[11]

G. J. Simmons, Authentication theory/coding theory,, in, (1984), 411.   Google Scholar

[12]

G. J. Simmons, A survey of information authentication,, in, (1992), 379.   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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