Almost Optimum Secret Sharing Schemes with Cheating Detection for Random Bit Strings

Abstract

Cheating detectable secret sharing is a secret sharing scheme with an extra property to detect forged shares in reconstructing a secret. Such a property is indispensable when we have to store shares in possibly malicious environment (e.g., cloud storage.) Because of its importance in the real world applications, cheating detectable secret sharing is actively studied, and many efficient schemes have been presented so far. However, interestingly, no optimum scheme is known when the secret is an element of finite field of characteristic two. Since $\mathbb {F}_{2^N}$ is the most natural representation of a bit string, an efficient scheme supporting $\mathbb {F}_{2^N}$ is highly desired.

In this paper, we present cheating detectable secret sharing schemes which is secure when the secret is an element of $\mathbb {F}_{2^N}$. The size of share of the proposed scheme is almost optimum in the sense that the bit length of the share meets the lower bound with equality. Moreover, the proposed schemes are applicable to any linear secret schemes.

This work was supported by JSPS KAKENHI Grant Number 15K00193.