American Institute of Mathematical Sciences

February  2011, 5(1): 59-68. doi: 10.3934/amc.2011.5.59

Infinite families of optimal splitting authentication codes secure against spoofing attacks of higher order

 1 Division of Mathematical Sciences, School of Physical & Mathematical Sciences, Nanyang Technological University, 21 Nanyang Link, Singapore 637371, Singapore, Singapore 2 Department of Mathematics, Zhejiang University, Hangzhou, Zhejiang 310027, China

Received  May 2010 Revised  November 2010 Published  February 2011

We consider the problem of constructing optimal authentication codes with splitting. New infinite families of such codes are obtained. In particular, we establish the first known infinite family of optimal authentication codes with splitting that are secure against spoofing attacks of order two.
Citation: Yeow Meng Chee, Xiande Zhang, Hui Zhang. Infinite families of optimal splitting authentication codes secure against spoofing attacks of higher order. Advances in Mathematics of Communications, 2011, 5 (1) : 59-68. doi: 10.3934/amc.2011.5.59
References:
 [1] A. E. Brouwer, A. Schrijver and H. Hanani, Group divisible designs with block-size four, Discrete Math., 20 (1977), 1-10. doi: 10.1016/0012-365X(77)90037-1. [2] B. Du, Splitting balanced incomplete block designs with block size $3\times 2$, J. Combin. Des. 12 (2004), 404-420. doi: 10.1002/jcd.20025. [3] G. Ge, Y. Miao and L. Wang, Combinatorial constructions for optimal splitting authentication codes, SIAM J. Discrete Math., 18 (2005), 663-678. doi: 10.1137/S0895480103435469. [4] H. Hanani, Balanced incomplete block designs and related designs, Discrete Math., 11 (1975), 255-369. doi: 10.1016/0012-365X(75)90040-0. [5] A. Hartman, The fundamental construction for $3$-designs, Discrete Math., 124 (1994), 107-132. doi: 10.1016/0012-365X(92)00055-V. [6] Q. X. Huang, On the decomposition of $K\sb n$ into complete $m$-partite graphs, J. Graph Theory, 15 (1991), 1-6. doi: 10.1002/jgt.3190150102. [7] M. Huber, Combinatorial bounds and characterizations of splitting authentication codes, Crypt. Commun., 2 (2010), 173-185. doi: 10.1007/s12095-010-0020-4. [8] L. Ji, An improvement on $H$ design, J. Combin. Des., 17 (2009), 25-35. doi: 10.1002/jcd.20184. [9] K. Kurosawa and S. Obana, Combinatorial bounds on authentication codes with arbitration, Des. Codes Crypt., 22 (2001), 265-281. doi: 10.1023/A:1008398306907. [10] J. L. Massey, Cryptography, a selective survey, in "Digital Communications '85: Proceedings of the Second Tirrenia International Workshop on Digital Communications'' (eds. E. Biglieri and G. Prati), Elsevier, (1986), 3-25. [11] W. H. Mills, On the existence of $H$ designs, in "Proceedings of the Twenty-first Southeastern Conference on Combinatorics, Graph Theory, and Computing (Boca Raton, FL, 1990),'' 79 (1990), 129-141. [12] W. Ogata, K. Kurosawa, D. R. Stinson and H. Saido, New combinatorial designs and their applications to authentication codes and secret sharing schemes, Discrete Math., 279 (2004), 383-405. doi: 10.1016/S0012-365X(03)00283-8. [13] G. J. Simmons, A game theory model of digital message authentication, Congr. Numer., 34 (1982), 413-424. [14] G. J. Simmons, Message authentication: a game on hypergraphs, Congr. Numer., 45 (1984), 161-192. [15] G. J. Simmons, Authentication theory/coding theory, in "Advances in Cryptology - CRYPTO '84'' (eds. G.R. Blakely and D. Chaum), Springer-Verlag, (1985), 411-432. doi: 10.1007/3-540-39568-7_32. [16] G. J. Simmons, Message authentication with arbitration of transmitter/receiver disputes, in "Advances in Cryptology - {EUROCRYPT} '87,'' Springer-Verlag, (1987), 151-165. [17] G. J. Simmons, A Cartesian product construction for unconditionally secure authentication codes that permit arbitration, J. Cryptology, 2 (1990), 77-104. doi: 10.1007/BF00204449. [18] G. J. Simmons, A survey of information authentication, in "Contemporary Cryptology - The Science of Information Integrity ''(ed. G.J. Simmons), IEEE Press, (1992), 379-419. [19] J. Wang, A new class of optimal 3-splitting authentication codes, Des. Codes Crypt., 38 (2006), 373-381. doi: 10.1007/s10623-005-1501-x. [20] J. Wang and R. Su, Further results on the existence of splitting BIBDs and application to authentication codes, Acta Appl. Math., 109 (2010), 791-803. doi: 10.1007/s10440-008-9346-8. [21] R. M. Wilson, An existence theory for pairwise balanced designs. I. Composition theorems and morphisms, J. Combin. Theory Ser. A, 13 (1972), 220-245. doi: 10.1016/0097-3165(72)90028-3. [22] R. M. Wilson, An existence theory for pairwise balanced designs. II. The structure of PBD-closed sets and the existence conjectures, J. Combin. Theory Ser. A, 13 (1972), 246-273. doi: 10.1016/0097-3165(72)90029-5. [23] R. M. Wilson, Decompositions of complete graphs into subgraphs isomorphic to a given graph, in "Proceedings of the Fifth British Combinatorial Conference (Univ. Aberdeen, Aberdeen, 1975),'' Winnipeg, Man., (1976), 647-659.

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References:
 [1] A. E. Brouwer, A. Schrijver and H. Hanani, Group divisible designs with block-size four, Discrete Math., 20 (1977), 1-10. doi: 10.1016/0012-365X(77)90037-1. [2] B. Du, Splitting balanced incomplete block designs with block size $3\times 2$, J. Combin. Des. 12 (2004), 404-420. doi: 10.1002/jcd.20025. [3] G. Ge, Y. Miao and L. Wang, Combinatorial constructions for optimal splitting authentication codes, SIAM J. Discrete Math., 18 (2005), 663-678. doi: 10.1137/S0895480103435469. [4] H. Hanani, Balanced incomplete block designs and related designs, Discrete Math., 11 (1975), 255-369. doi: 10.1016/0012-365X(75)90040-0. [5] A. Hartman, The fundamental construction for $3$-designs, Discrete Math., 124 (1994), 107-132. doi: 10.1016/0012-365X(92)00055-V. [6] Q. X. Huang, On the decomposition of $K\sb n$ into complete $m$-partite graphs, J. Graph Theory, 15 (1991), 1-6. doi: 10.1002/jgt.3190150102. [7] M. Huber, Combinatorial bounds and characterizations of splitting authentication codes, Crypt. Commun., 2 (2010), 173-185. doi: 10.1007/s12095-010-0020-4. [8] L. Ji, An improvement on $H$ design, J. Combin. Des., 17 (2009), 25-35. doi: 10.1002/jcd.20184. [9] K. Kurosawa and S. Obana, Combinatorial bounds on authentication codes with arbitration, Des. Codes Crypt., 22 (2001), 265-281. doi: 10.1023/A:1008398306907. [10] J. L. Massey, Cryptography, a selective survey, in "Digital Communications '85: Proceedings of the Second Tirrenia International Workshop on Digital Communications'' (eds. E. Biglieri and G. Prati), Elsevier, (1986), 3-25. [11] W. H. Mills, On the existence of $H$ designs, in "Proceedings of the Twenty-first Southeastern Conference on Combinatorics, Graph Theory, and Computing (Boca Raton, FL, 1990),'' 79 (1990), 129-141. [12] W. Ogata, K. Kurosawa, D. R. Stinson and H. Saido, New combinatorial designs and their applications to authentication codes and secret sharing schemes, Discrete Math., 279 (2004), 383-405. doi: 10.1016/S0012-365X(03)00283-8. [13] G. J. Simmons, A game theory model of digital message authentication, Congr. Numer., 34 (1982), 413-424. [14] G. J. Simmons, Message authentication: a game on hypergraphs, Congr. Numer., 45 (1984), 161-192. [15] G. J. Simmons, Authentication theory/coding theory, in "Advances in Cryptology - CRYPTO '84'' (eds. G.R. Blakely and D. Chaum), Springer-Verlag, (1985), 411-432. doi: 10.1007/3-540-39568-7_32. [16] G. J. Simmons, Message authentication with arbitration of transmitter/receiver disputes, in "Advances in Cryptology - {EUROCRYPT} '87,'' Springer-Verlag, (1987), 151-165. [17] G. J. Simmons, A Cartesian product construction for unconditionally secure authentication codes that permit arbitration, J. Cryptology, 2 (1990), 77-104. doi: 10.1007/BF00204449. [18] G. J. Simmons, A survey of information authentication, in "Contemporary Cryptology - The Science of Information Integrity ''(ed. G.J. Simmons), IEEE Press, (1992), 379-419. [19] J. Wang, A new class of optimal 3-splitting authentication codes, Des. Codes Crypt., 38 (2006), 373-381. doi: 10.1007/s10623-005-1501-x. [20] J. Wang and R. Su, Further results on the existence of splitting BIBDs and application to authentication codes, Acta Appl. Math., 109 (2010), 791-803. doi: 10.1007/s10440-008-9346-8. [21] R. M. Wilson, An existence theory for pairwise balanced designs. I. Composition theorems and morphisms, J. Combin. Theory Ser. A, 13 (1972), 220-245. doi: 10.1016/0097-3165(72)90028-3. [22] R. M. Wilson, An existence theory for pairwise balanced designs. II. The structure of PBD-closed sets and the existence conjectures, J. Combin. Theory Ser. A, 13 (1972), 246-273. doi: 10.1016/0097-3165(72)90029-5. [23] R. M. Wilson, Decompositions of complete graphs into subgraphs isomorphic to a given graph, in "Proceedings of the Fifth British Combinatorial Conference (Univ. Aberdeen, Aberdeen, 1975),'' Winnipeg, Man., (1976), 647-659.
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