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.

[2]

C. CarletC. Ding and J. Yuan, Linear codes from perfect nonlinear mappings and their secret sharing schemes, IEEE Trans. Inf. Theory, 51 (2005), 2089-2102.

[3]

S. ChansonC. Ding and A. Salomaa, Cartesian authentication codes from functions with optimal nonlinearity, Theor. Comp. Sci., 290 (2003), 1737-1752.

[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.

[5]

P. Delsarte, On subfield subcodes of modified Reed-Solomon codes (corresp.), IEEE Trans. Inf. Theory, 21 (1975), 575-576.

[6]

C. DingY. Gao and Z. Zhou, Five families of three-weight ternary cyclic codes and their duals, IEEE Trans. Inf. Theory, 59 (2013), 7940-7946.

[7]

C. DingT. HellesethT. Kløve and X. Wang, A generic construction of Cartesian authentication codes, IEEE Trans. Inf. Theory, 53 (2007), 2229-2235.

[8]

C. Ding and H. Niederreiter, Systematic authentication codes from highly nonlinear functions, IEEE Trans. Inf. Theory, 50 (2004), 2421-2428.

[9]

C. Ding and X. Wang, A coding theory construction of new systematic authentication codes, Theor. Comp. Sci., 330 (2005), 81-99.

[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.

[11]

G. A. KabatianskiiB. Smeets and T. Johansson, On the cardinality of systematic authentication codes via error-correcting codes, IEEE Trans. Inf. Theory, 42 (1996), 566-578.

[12]

C. LiN. LiT. Helleseth and C. Ding, The weight distributions of several classes of cyclic codes from APN monomials, IEEE Trans. Inf. Theory, 60 (2014), 4710-4721.

[13]

R. Lidl and H. Niederreiter, Finite Fields, Cambridge Univ. Press, 1997.

[14]

J. Luo and K. Feng, On the weight distributions of two classes of cyclic codes, IEEE Trans. Inf. Theory, 54 (2008), 5332-5344.

[15]

F. Özbudak and Z. Saygi, Some constructions of systematic authentication codes using Galois rings, Des. Codes Crypt., 41 (2006), 343-357.

[16]

R. S. Rees and D. R. Stinson, Combinatorial characterizations of authentication codes Ⅱ, Des. Codes Crypt., 7 (1996), 239-259.

[17]

G. J. Simmons, Authentication theory/coding theory, in Adv. Crypt. -CRYPTO'84, Springer, 1984,411-431.

[18]

H. WangC. Xing and R. Safavi-Naini, Linear authentication codes: bounds and constructions, IEEE Trans. Inf. Theory, 49 (2003), 866-872.

[19]

J. YuanC. 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.

[20]

Z. Zhou and C. Ding, Seven classes of three-weight cyclic codes, IEEE Trans. Commun., 61 (2013), 4120-4126.

[21]

Z. Zhou and C. Ding, A class of three-weight cyclic codes, Finite Fields Appl., 25 (2014), 79-93.

[22]

Z. ZhouC. DingJ. Luo and A. Zhang, A family of five-weight cyclic codes and their weight enumerators, IEEE Trans. Inf. Theory, 59 (2013), 6674-6682.

show all references

References:
[1]

J. Bierbrauer, Universal hashing and geometric codes, Des. Codes Crypt., 11 (1997), 207-221.

[2]

C. CarletC. Ding and J. Yuan, Linear codes from perfect nonlinear mappings and their secret sharing schemes, IEEE Trans. Inf. Theory, 51 (2005), 2089-2102.

[3]

S. ChansonC. Ding and A. Salomaa, Cartesian authentication codes from functions with optimal nonlinearity, Theor. Comp. Sci., 290 (2003), 1737-1752.

[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.

[5]

P. Delsarte, On subfield subcodes of modified Reed-Solomon codes (corresp.), IEEE Trans. Inf. Theory, 21 (1975), 575-576.

[6]

C. DingY. Gao and Z. Zhou, Five families of three-weight ternary cyclic codes and their duals, IEEE Trans. Inf. Theory, 59 (2013), 7940-7946.

[7]

C. DingT. HellesethT. Kløve and X. Wang, A generic construction of Cartesian authentication codes, IEEE Trans. Inf. Theory, 53 (2007), 2229-2235.

[8]

C. Ding and H. Niederreiter, Systematic authentication codes from highly nonlinear functions, IEEE Trans. Inf. Theory, 50 (2004), 2421-2428.

[9]

C. Ding and X. Wang, A coding theory construction of new systematic authentication codes, Theor. Comp. Sci., 330 (2005), 81-99.

[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.

[11]

G. A. KabatianskiiB. Smeets and T. Johansson, On the cardinality of systematic authentication codes via error-correcting codes, IEEE Trans. Inf. Theory, 42 (1996), 566-578.

[12]

C. LiN. LiT. Helleseth and C. Ding, The weight distributions of several classes of cyclic codes from APN monomials, IEEE Trans. Inf. Theory, 60 (2014), 4710-4721.

[13]

R. Lidl and H. Niederreiter, Finite Fields, Cambridge Univ. Press, 1997.

[14]

J. Luo and K. Feng, On the weight distributions of two classes of cyclic codes, IEEE Trans. Inf. Theory, 54 (2008), 5332-5344.

[15]

F. Özbudak and Z. Saygi, Some constructions of systematic authentication codes using Galois rings, Des. Codes Crypt., 41 (2006), 343-357.

[16]

R. S. Rees and D. R. Stinson, Combinatorial characterizations of authentication codes Ⅱ, Des. Codes Crypt., 7 (1996), 239-259.

[17]

G. J. Simmons, Authentication theory/coding theory, in Adv. Crypt. -CRYPTO'84, Springer, 1984,411-431.

[18]

H. WangC. Xing and R. Safavi-Naini, Linear authentication codes: bounds and constructions, IEEE Trans. Inf. Theory, 49 (2003), 866-872.

[19]

J. YuanC. 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.

[20]

Z. Zhou and C. Ding, Seven classes of three-weight cyclic codes, IEEE Trans. Commun., 61 (2013), 4120-4126.

[21]

Z. Zhou and C. Ding, A class of three-weight cyclic codes, Finite Fields Appl., 25 (2014), 79-93.

[22]

Z. ZhouC. DingJ. Luo and A. Zhang, A family of five-weight cyclic codes and their weight enumerators, IEEE Trans. Inf. Theory, 59 (2013), 6674-6682.

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