Combinational constructions of splitting authentication codes with perfect secrecy

Splitting authentication codes were first introduced by Simmons in 1982. Ogata et al. introduced (v, u x 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 x c, 1)-splitting balanced incomplete block design (briefly a (v, u x c, 1)-ESBIBD) to construct a (1/cu)-secure robust (2, 2)-threshold scheme for u equiprobable secrets. Note that v equivalent to 1 (modu(u-1)c) and v(v-1) equivalent to 0 (modu(u-1)c(2)) if there is a (v, u x c, 1)-ESBIBD. In order to consider other orders v, we generalize the concept of a (v, u xc, 1)-ESBIBD to an equitably ordered (v, u xc, 1)-splitting packing design (briefly a (v, u x 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 x c, 1)-ESPDs and determine the existence of an optimal (v, u x c, 1)-ESPD for (u, c) is an element of {(2, k) : k is a positive integer} boolean OR{(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.