Detecting dealer cheating in secret sharing systems

The concept of secret sharing can be used in a wide range of business application. A secret sharing system can implement the policies of secret sharing, and control the distribution of the secrets to the participants under the secret sharing policies. But, it can be damaged when the dealer cheating occurs. If the secret sharing system is implemented by Shamir's (t, n)-threshold scheme, one of the dealer's cheatings is that the dealer uses incorrectly polynomials to generate shadows (or shares) and distributes these error shadows to the participants. How can we detect this cheating? In this paper we propose a method to handle the detection of that the dealer uses wrong degree of the polynomial which the dealer chooses to hide the key. The main idea of the proposed method is that we ask the dealer to generate a certificate polymomial and one-bit verifying keys to provide information when participants do the detection process.