We consider secret sharing with binary shares. This model allows us to use the well developed theory of cryptographically strong Boolean functions. We prove that for given secret sharing, the average cheating probability over all cheating and original vectors, i.e., ρ{top}- = (1/n)·2{sup}(-n)∑{sub}(c=1){sup}n∑{sub}(α∈V{sub}n) ρ{sub}(c,α), satisfies ρ{top}-≥1/2, and the equality holds ←→ρ{sub}(c,α) satisfies ρ{sub}(c,α) = 1/2 for every cheating vector δ{sub}c and every original vector α. In this case the secret sharing is said to be cheating immune. We further establish a relationship between cheating-immune secret sharing and cryptographic criteria of Boolean functions. This enables us to construct cheating-immune secret sharing.
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