# 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:
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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. doi: 10.1016/0012-365X(77)90037-1. Google Scholar [2] B. Du, Splitting balanced incomplete block designs with block size $3\times 2$,, J. Combin. Des. \textbf{12} (2004), 12 (2004), 404. doi: 10.1002/jcd.20025. Google Scholar [3] G. Ge, Y. Miao and L. Wang, Combinatorial constructions for optimal splitting authentication codes,, SIAM J. Discrete Math., 18 (2005), 663. doi: 10.1137/S0895480103435469. Google Scholar [4] H. Hanani, Balanced incomplete block designs and related designs,, Discrete Math., 11 (1975), 255. doi: 10.1016/0012-365X(75)90040-0. Google Scholar [5] A. Hartman, The fundamental construction for $3$-designs,, Discrete Math., 124 (1994), 107. doi: 10.1016/0012-365X(92)00055-V. Google Scholar [6] Q. X. Huang, On the decomposition of $K\sb n$ into complete $m$-partite graphs,, J. Graph Theory, 15 (1991), 1. doi: 10.1002/jgt.3190150102. Google Scholar [7] M. Huber, Combinatorial bounds and characterizations of splitting authentication codes,, Crypt. Commun., 2 (2010), 173. doi: 10.1007/s12095-010-0020-4. Google Scholar [8] L. Ji, An improvement on $H$ design,, J. Combin. Des., 17 (2009), 25. doi: 10.1002/jcd.20184. Google Scholar [9] K. Kurosawa and S. Obana, Combinatorial bounds on authentication codes with arbitration,, Des. Codes Crypt., 22 (2001), 265. doi: 10.1023/A:1008398306907. Google Scholar [10] J. L. Massey, Cryptography, a selective survey,, in, (1986), 3. Google Scholar [11] W. H. Mills, On the existence of $H$ designs,, in, 79 (1990), 129. Google Scholar [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. doi: 10.1016/S0012-365X(03)00283-8. Google Scholar [13] G. J. Simmons, A game theory model of digital message authentication,, Congr. Numer., 34 (1982), 413. Google Scholar [14] G. J. Simmons, Message authentication: a game on hypergraphs,, Congr. Numer., 45 (1984), 161. Google Scholar [15] G. J. Simmons, Authentication theory/coding theory,, in, (1985), 411. doi: 10.1007/3-540-39568-7_32. Google Scholar [16] G. J. Simmons, Message authentication with arbitration of transmitter/receiver disputes,, in, (1987), 151. Google Scholar [17] G. J. Simmons, A Cartesian product construction for unconditionally secure authentication codes that permit arbitration,, J. Cryptology, 2 (1990), 77. doi: 10.1007/BF00204449. Google Scholar [18] G. J. Simmons, A survey of information authentication,, in, (1992), 379. Google Scholar [19] J. Wang, A new class of optimal 3-splitting authentication codes,, Des. Codes Crypt., 38 (2006), 373. doi: 10.1007/s10623-005-1501-x. Google Scholar [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. doi: 10.1007/s10440-008-9346-8. Google Scholar [21] R. M. Wilson, An existence theory for pairwise balanced designs. I. Composition theorems and morphisms,, J. Combin. Theory Ser. A, 13 (1972), 220. doi: 10.1016/0097-3165(72)90028-3. Google Scholar [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. doi: 10.1016/0097-3165(72)90029-5. Google Scholar [23] R. M. Wilson, Decompositions of complete graphs into subgraphs isomorphic to a given graph,, in, (1976), 647. Google Scholar
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