# American Institute of Mathematical Sciences

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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