Combinatorial interpretation of secret sharing schemes
In a perfect secret sharing scheme, it is known that log2 ¦V i ¦ ≥H(S), where S is a secret and V i is the share of user i. On the other hand, log2 ¦Ŝ¦ ≥H(S), where Ŝ is the domain of S. The equality holds if and only if S is uniformly distributed. Therefore, if S is uniformly distributed, we have ¦V i ¦≥¦Ŝ¦. However, if S is not uniformly distributed, log2 ¦Ŝ¦> H(S). In this case, we have log2¦V i ¦≥H(S) <log2¦Ŝ¦. Then, which is bigger, ¦Vi¦ or ¦Ŝ¦? The answer is not known.
In this paper, we first prove that ¦V i ¦ >-¦Ŝ¦ for any distribution of S by using a combinatorial argument. This is a more sharp lower bound on ¦V i ¦ for not uniformly distributed S. Our proof makes it intuitively clear why ¦V i ¦ must be so large, also. Further, we show an extension of our combinatorial technique for some access structures.
KeywordsSecret Sharing Access Structure Information Rate Secret Sharing Scheme Minimal Access
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