Secret sharing schemes based on the dual of Golay codes

摘要

Linear codes are an important class of codes in coding theory and have been extensively studied due to their significant applications (including the design of secret sharing schemes) in practical systems. Interesting linear codes having several different real-world applications are the so-called Golay codes. Secret sharing schemes play a fundamental role in cryptography and have numerous applications in security systems. One approach to constructing secret sharing schemes is based on linear codes, especially for minimal linear codes and self-dual codes. Several minimal linear codes based on Boolean cryptographic functions and vectorial Boolean functions have been found. This paper proposes two secret sharing schemes based on the dual of the [23,12,7](2) and [11,6,5](3) Golay codes, respectively, where these two Golay codes are neither minimal nor self-dual. We determine the minimal access structures of our schemes by using the two Golay codes' combinatorial properties. To our surprise, our schemes are 3-democratic. This is interesting since our schemes are not threshold secret sharing schemes, and previous works propose some democratic secret sharing schemes that are based on the dual of minimal linear codes. On the other hand, we find that our schemes' minimal access structures contradict a result of Dougherty et al. (ITW 2008). We then revise the minimal access structures of the secret sharing schemes based on the [24,12,8](2) and [12,6,6](3) extended Golay codes, respectively, and we further discuss the democracies of these two schemes.