# American Institute of Mathematical Sciences

February  2018, 12(1): 1-16. doi: 10.3934/amc.2018001

## New constructions of systematic authentication codes from three classes of cyclic codes

 1 College of Liberal Arts and Sciences, National University of Defense Technology, Changsha, Hunan, China 2 imec-COSIC KU Leuven, Leuven, Belgium 3 College of Liberal Arts and Sciences and College of Computer, National University of Defense Technology, Changsha, Hunan, China

* Corresponding author: Yunwen Liu

Received  July 2015 Revised  July 2017 Published  March 2018

Recently, several classes of cyclic codes with three nonzero weights were constructed. With the generic construction presented by C. Ding, T. Helleseth, T. Kløve and X. Wang, we present new systematic authentication codes based on these cyclic codes. In this paper, we study three special classes of cyclic codes and their authentication codes. With the help of exponential sums, we calculate the maximum success probabilities of the impersonation and substitution attacks on the authentication codes. Our results show that these new authentication codes are better than some of the authentication codes in the literature. As a byproduct, the number of times that each element occurs as the coordinates in the codewords of the cyclic codes is settled, which is a difficult problem in general.

Citation: Yunwen Liu, Longjiang Qu, Chao Li. New constructions of systematic authentication codes from three classes of cyclic codes. Advances in Mathematics of Communications, 2018, 12 (1) : 1-16. doi: 10.3934/amc.2018001
##### References:
 [1] J. Bierbrauer, Universal hashing and geometric codes, Des. Codes Crypt., 11 (1997), 207-221.   Google Scholar [2] C. Carlet, C. Ding and J. Yuan, Linear codes from perfect nonlinear mappings and their secret sharing schemes, IEEE Trans. Inf. Theory, 51 (2005), 2089-2102.   Google Scholar [3] S. Chanson, C. Ding and A. Salomaa, Cartesian authentication codes from functions with optimal nonlinearity, Theor. Comp. Sci., 290 (2003), 1737-1752.   Google Scholar [4] S.-T. Choi, J.-Y. Kim, J.-S. No and H. Chung, Weight distribution of some cyclic codes, in 2012 IEEE Int. Symp. Inf. Theory Proc. (ISIT), 2012, 2901-2903. Google Scholar [5] P. Delsarte, On subfield subcodes of modified Reed-Solomon codes (corresp.), IEEE Trans. Inf. Theory, 21 (1975), 575-576.   Google Scholar [6] C. Ding, Y. Gao and Z. Zhou, Five families of three-weight ternary cyclic codes and their duals, IEEE Trans. Inf. Theory, 59 (2013), 7940-7946.   Google Scholar [7] C. Ding, T. Helleseth, T. Kløve and X. Wang, A generic construction of Cartesian authentication codes, IEEE Trans. Inf. Theory, 53 (2007), 2229-2235.   Google Scholar [8] C. Ding and H. Niederreiter, Systematic authentication codes from highly nonlinear functions, IEEE Trans. Inf. Theory, 50 (2004), 2421-2428.   Google Scholar [9] C. Ding and X. Wang, A coding theory construction of new systematic authentication codes, Theor. Comp. Sci., 330 (2005), 81-99.   Google Scholar [10] T. Helleseth and T. Johansson, Universal hash functions from exponential sums over finite fields and Galois rings, in Adv. Crypt. -CRYPTO'96, Springer, 1996, 31-44.  Google Scholar [11] G. A. Kabatianskii, B. Smeets and T. Johansson, On the cardinality of systematic authentication codes via error-correcting codes, IEEE Trans. Inf. Theory, 42 (1996), 566-578.   Google Scholar [12] C. Li, N. Li, T. Helleseth and C. Ding, The weight distributions of several classes of cyclic codes from APN monomials, IEEE Trans. Inf. Theory, 60 (2014), 4710-4721.   Google Scholar [13] R. Lidl and H. Niederreiter, Finite Fields, Cambridge Univ. Press, 1997.  Google Scholar [14] J. Luo and K. Feng, On the weight distributions of two classes of cyclic codes, IEEE Trans. Inf. Theory, 54 (2008), 5332-5344.   Google Scholar [15] F. Özbudak and Z. Saygi, Some constructions of systematic authentication codes using Galois rings, Des. Codes Crypt., 41 (2006), 343-357.   Google Scholar [16] R. S. Rees and D. R. Stinson, Combinatorial characterizations of authentication codes Ⅱ, Des. Codes Crypt., 7 (1996), 239-259.   Google Scholar [17] G. J. Simmons, Authentication theory/coding theory, in Adv. Crypt. -CRYPTO'84, Springer, 1984,411-431. Google Scholar [18] H. Wang, C. Xing and R. Safavi-Naini, Linear authentication codes: bounds and constructions, IEEE Trans. Inf. Theory, 49 (2003), 866-872.   Google Scholar [19] J. Yuan, C. Carlet and C. Ding, The weight distribution of a class of linear codes from perfect nonlinear functions, IEEE Trans. Inf. Theory, 52 (2006), 712-717.   Google Scholar [20] Z. Zhou and C. Ding, Seven classes of three-weight cyclic codes, IEEE Trans. Commun., 61 (2013), 4120-4126.   Google Scholar [21] Z. Zhou and C. Ding, A class of three-weight cyclic codes, Finite Fields Appl., 25 (2014), 79-93.   Google Scholar [22] Z. Zhou, C. Ding, J. Luo and A. Zhang, A family of five-weight cyclic codes and their weight enumerators, IEEE Trans. Inf. Theory, 59 (2013), 6674-6682.   Google Scholar

show all references

##### References:
 [1] J. Bierbrauer, Universal hashing and geometric codes, Des. Codes Crypt., 11 (1997), 207-221.   Google Scholar [2] C. Carlet, C. Ding and J. Yuan, Linear codes from perfect nonlinear mappings and their secret sharing schemes, IEEE Trans. Inf. Theory, 51 (2005), 2089-2102.   Google Scholar [3] S. Chanson, C. Ding and A. Salomaa, Cartesian authentication codes from functions with optimal nonlinearity, Theor. Comp. Sci., 290 (2003), 1737-1752.   Google Scholar [4] S.-T. Choi, J.-Y. Kim, J.-S. No and H. Chung, Weight distribution of some cyclic codes, in 2012 IEEE Int. Symp. Inf. Theory Proc. (ISIT), 2012, 2901-2903. Google Scholar [5] P. Delsarte, On subfield subcodes of modified Reed-Solomon codes (corresp.), IEEE Trans. Inf. Theory, 21 (1975), 575-576.   Google Scholar [6] C. Ding, Y. Gao and Z. Zhou, Five families of three-weight ternary cyclic codes and their duals, IEEE Trans. Inf. Theory, 59 (2013), 7940-7946.   Google Scholar [7] C. Ding, T. Helleseth, T. Kløve and X. Wang, A generic construction of Cartesian authentication codes, IEEE Trans. Inf. Theory, 53 (2007), 2229-2235.   Google Scholar [8] C. Ding and H. Niederreiter, Systematic authentication codes from highly nonlinear functions, IEEE Trans. Inf. Theory, 50 (2004), 2421-2428.   Google Scholar [9] C. Ding and X. Wang, A coding theory construction of new systematic authentication codes, Theor. Comp. Sci., 330 (2005), 81-99.   Google Scholar [10] T. Helleseth and T. Johansson, Universal hash functions from exponential sums over finite fields and Galois rings, in Adv. Crypt. -CRYPTO'96, Springer, 1996, 31-44.  Google Scholar [11] G. A. Kabatianskii, B. Smeets and T. Johansson, On the cardinality of systematic authentication codes via error-correcting codes, IEEE Trans. Inf. Theory, 42 (1996), 566-578.   Google Scholar [12] C. Li, N. Li, T. Helleseth and C. Ding, The weight distributions of several classes of cyclic codes from APN monomials, IEEE Trans. Inf. Theory, 60 (2014), 4710-4721.   Google Scholar [13] R. Lidl and H. Niederreiter, Finite Fields, Cambridge Univ. Press, 1997.  Google Scholar [14] J. Luo and K. Feng, On the weight distributions of two classes of cyclic codes, IEEE Trans. Inf. Theory, 54 (2008), 5332-5344.   Google Scholar [15] F. Özbudak and Z. Saygi, Some constructions of systematic authentication codes using Galois rings, Des. Codes Crypt., 41 (2006), 343-357.   Google Scholar [16] R. S. Rees and D. R. Stinson, Combinatorial characterizations of authentication codes Ⅱ, Des. Codes Crypt., 7 (1996), 239-259.   Google Scholar [17] G. J. Simmons, Authentication theory/coding theory, in Adv. Crypt. -CRYPTO'84, Springer, 1984,411-431. Google Scholar [18] H. Wang, C. Xing and R. Safavi-Naini, Linear authentication codes: bounds and constructions, IEEE Trans. Inf. Theory, 49 (2003), 866-872.   Google Scholar [19] J. Yuan, C. Carlet and C. Ding, The weight distribution of a class of linear codes from perfect nonlinear functions, IEEE Trans. Inf. Theory, 52 (2006), 712-717.   Google Scholar [20] Z. Zhou and C. Ding, Seven classes of three-weight cyclic codes, IEEE Trans. Commun., 61 (2013), 4120-4126.   Google Scholar [21] Z. Zhou and C. Ding, A class of three-weight cyclic codes, Finite Fields Appl., 25 (2014), 79-93.   Google Scholar [22] Z. Zhou, C. Ding, J. Luo and A. Zhang, A family of five-weight cyclic codes and their weight enumerators, IEEE Trans. Inf. Theory, 59 (2013), 6674-6682.   Google Scholar
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