Secret sharing with a class of minimal linear codes
This work addresses a theoretical challenge in cryptography for researchers and practitioners, but it is incremental as it builds on existing coding theory methods.
The paper tackled the difficulty of determining access structures in secret sharing schemes based on linear codes by introducing minimal linear codes, which simplify this process for their duals, and proved that shortening codes preserve minimality while analyzing specific irreducible cyclic codes.
There are several methods for constructing secret sharing schemes, one of which is based on coding theory. Theoretically, every linear code can be used to construct secret sharing schemes. However, in general, determining the access structures of the schemes based on linear codes is very hard. This paper proposed the concept of minimal linear code, which makes the determination of the access structures of the schemes based on the duals of minimal linear codes easier. It is proved that the shortening codes of minimal linear codes are also minimal ones. Then the conditions whether several types of irreducible cyclic codes are minimal linear codes are presented. Furthermore, the access structures of secret sharing schemes based on the duals of minimal linear codes are studied, and these access structures in specific examples are obtained through programming.