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 over |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.
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机译：到目前为止，许多研究人员已经研究了秘密共享机制，这种机制在恶意参与者伪造其股份的情况下是安全的。在本文中，我们考虑一种在编码器和解码器处具有密钥的秘密共享方案，并分析其针对恶意参与者的替代攻击的安全性。我们首先给出带有密钥的秘密共享方案必须满足的条件。接下来，我们使用密钥生成一个简单的秘密共享方案，该密钥生成n个份额并以高概率检测伪造的份额。我们可以证明，如果最多有n − 1个恶意参与者合谋并试图通过伪造份额来欺骗其他参与者，那么这种攻击的成功概率就比| X | −1高了| S | -1 | S |和| X |分别表示秘密和n份的大小。此外，我们研究了该方案的基本性质，尤其是在n = 2的情况下。
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