Linear (k, n) Secret Sharing Scheme with Cheating Detection

Linear (k,n) secret sharing scheme is a class of (k,n) secret sharing, where all the n shares of a secret satisfy a linear relationship. It plays an important role in other cryptographic systems, such as multi-party computation and function sharing schemes. On the other hand, cheating problem in (k,n) secret sharing is an important issue, such that cheaters (dishonest players) submit forged shares during secret reconstruction to fool honest players. During decades of research on cheating prevention, vast (k,n) secret sharing schemes against cheating have been proposed. However, most of these schemes are not linear schemes since it contains redundant information in their shares to achieve cheating detection. Since linear (k,n) secret sharing is an important primitive in threshold cryptography, linear (k,n) secret sharing scheme with the capability of cheating detection is also worthwhile to be discussed. In this paper, we propose a linear (k,n) secret sharing scheme against cheating based on Shamir´s original scheme, which possesses the following merits: (1) Our scheme is just combination of two Shamir´s schemes. Therefore, our scheme can be used in other threshold cryptographic systems which are based on Shamir´s scheme. (2) The size of share in proposed scheme almost reaches its theoretic lower bound in (k,n) secret sharing with cheating detection. (3) In the phase of cheating detection, only one honest player can detect the cheating from other k-1 cheaters, which achieves a stronger detection effective than the previous linear secret sharing schemes against cheating.