On the Size of a Share for Non-perfect Secret Sharing Schemes

Abstract

In a secret sharing scheme, a secret s is divided into multiple shares and distributed among n participants. As more shares are collected, the mutual information between the secret and the shares increases monotonically. In the perfect case, the best known explicit lower bound on the share size is given by $\frac{{{2^k} - 1}}{k} \cdot H(s)$, where H(s) is the entropy of the secret and k is the largest integer such that n ≥ 2k+k−2. This paper considers the non-perfect case, in which each share reveals at most a $\frac{1}{L}$ fraction of the secret's entropy, for an integer L > 1. A prior result in this setting provides a lower bound of $\frac{{{2^k} - 1}}{{kL}} \cdot H(s)$, under the condition n ≥ 2k+k+L−3. We provide a new lower bound that approaches the perfect-case bound, namely $\frac{{{2^k} - 2 + 1/L}}{k} \cdot H(s)$, which holds under the condition n ≥ 2k+k+2L−4. The proof generalizes perfect-case techniques by increasing the number of participants by 2(L−1). These results refine our understanding of the trade-off between share size and information leakage in non-perfect secret sharing. In particular, we construct non-perfect access structures that nearly attain the perfect lower bound with fewer participants than previous non-perfect bounds, thus complementing and extending prior work.