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Combinational constructions of splitting authentication codes with perfect secrecy

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Abstract

Splitting authentication codes were first introduced by Simmons in 1982. Ogata et al. introduced \((v,u\times c,1)\)-splitting balanced incomplete block designs in 2006 in order to construct twofold optimal c-splitting authentication codes. In 2020, Paterson and Stinson showed that there exists an authentication code with perfect secrecy for u uniformly distributed source states that is \(\epsilon \)-secure against message-substitution and key-substitution attacks if and only if there exists an \(\epsilon \)-secure robust (2, 2)-threshold scheme for u uniformly distributed secrets, and they used an equitably ordered \((v, u \times c, 1)\)-splitting balanced incomplete block design (briefly a \((v, u \times c, 1)\)-ESBIBD) to construct a (1/cu)-secure robust (2, 2)-threshold scheme for u equiprobable secrets. Note that \(v\equiv 1\ (\text{ mod }\ u(u-1)c)\) and \(v(v-1) \equiv 0\ (\text{ mod }\ u(u-1)c^2)\) if there is a \((v, u\times c, 1)\)-ESBIBD. In order to consider other orders v, we generalize the concept of a \((v, u\times c, 1)\)-ESBIBD to an equitably ordered \((v, u\times c, 1)\)-splitting packing design (briefly a \((v, u \times c, 1)\)-ESPD), which can also be used to construct a (1/cu)-secure robust (2, 2)-threshold scheme for u equiprobable secrets. In this paper, we study combinatorial constructions of \((v, u\times c, 1)\)-ESPDs and determine the existence of an optimal \((v,u\times c,1)\)-ESPD for \((u,c)\in \{ (2,k) :\ k\ \text{ is } \text{ a } \text{ positive } \text{ integer } \}\cup \{(3,1),(4,1),(3,2)\}\). Consequently, we obtain some new infinite classes of authentication codes with perfect secrecy and (1/cu)-secure robust (2, 2)-threshold schemes.

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Authors and Affiliations

  1. Department of Mathematics, Soochow University, Suzhou, 215006, People’s Republic of China

    Lijun Ji & Yanting Wang

  2. Department of Mathematics and Physics, Suzhou Vocational University, Suzhou, 215104, People’s Republic of China

    Miao Liang

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  1. Lijun Ji
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  2. Miao Liang
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  3. Yanting Wang
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Correspondence to Miao Liang.

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Communicated by M. Paterson.

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Research is supported by NSFC Grant 11871363 (L. Ji)

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Ji, L., Liang, M. & Wang, Y. Combinational constructions of splitting authentication codes with perfect secrecy. Des. Codes Cryptogr. 90, 2491–2515 (2022). https://doi.org/10.1007/s10623-022-01092-6

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  • DOI: https://doi.org/10.1007/s10623-022-01092-6

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