A simple secret sharing scheme using a key and its security against substitution of shares

Secret sharing schemes which are secure under the presence of malicious participants who forge their shares have been investigated by many researchers so far. In this paper we consider a secret sharing scheme with a key at an encoder and a decoder and analyze its security against the substitution attack by malicious participants. We first give conditions that the secret sharing scheme with a key must satisfy. Next, we construct a simple secret sharing scheme with a key that generates n shares and detects forged shares with high probability. We can prove that, if at most n - 1 malicious participants collude and try to deceive the other participants by forged shares optimally, the success probability of such an attack is upper-bounded by |S|-1/|X|-1, where |S| and |X| denote the sizes of the secret and the n shares, respectively. In addition, we investigate basic properties of the scheme especially for the case of n = 2.