Secret Sharing Schemes Based on Linear Codes Can Be Precisely Characterized by the Relative Generalized Hamming Weight

This paper precisely characterizes secret sharing schemes based on arbitrary linear codes by using the relative dimension/length profile (RDLP) and the relative generalized Hamming weight (RGHW). We first describe the equivocation Δm of the secret vector s = [s1, . . . ,sl ] given m shares in terms of the RDLP of linear codes. We also characterize two thresholds t1 and t2 in the secret sharing schemes by the RGHW of linear codes. One shows that any set of at most t1 shares leaks no information about s, and the other shows that any set of at least t2 shares uniquely determines s. It is clarified that both characterizations for t1 and t2 are better than Chen et al.’s ones derived by the regular minimum Hamming weight. Moreover, this paper characterizes the strong security in secret sharing schemes based on linear codes, by generalizing the definition of stronglysecure threshold ramp schemes. We define a secret sharing scheme achieving the α-strong security as the one such that the mutual information between any r elements of (s1, . . . ,sl) and any α−r+1 shares is always zero. Then, it is clarified that secret sharing schemes based on linear codes can always achieve the α-strong security where the value α is precisely characterized by the RGHW. key words: secret sharing scheme, linear code, relative generalized Hamming weight, relative dimension/length profile