| Abstrakt: |
Secret sharing scheme (SSS) allows to share some secret samong nparties in such a way that only certain subsets of them can recover the secret. The first SSSs were independently proposed by Shamir (Commun ACM 22:612–613, 1979) and Blakley (in: 1979 international workshop on managing requirements knowledge (MARK), 1979. https://doi.org/10.1109/MARK.1979.8817296) in 1976. Nowadays there are several general approaches for constructing secret sharing schemes. The most popular and well-studied are linear approachand modular approach. In a linear SSS the secret Scan be found by solving a system of linear equations. Both Shamir and Blakely schemes are linear. Modular SSS was first proposed by Mignotte (in: Beth (ed) Cryptography. EUROCRYPT 1982, Springer, Berlin, 1983. https://doi.org/10.1007/3-540-39466-4_27) and Asmuth and Bloom (IEEE Trans Inf Theory 29(2):208–210, 1983. https://doi.org/10.1109/TIT.1983.1056651) in 1983. It is based on the Chinese remainder theorem. In this paper we focus only on the modular approach. Such secret sharing schemes are most often called "CRT-based" in the literature. But we stress, that Asmuth and Bloom, who proposed this approach, called it modular. The aim of this paper is the following. First, we provide a comprehensive historical overview regarding modular secret sharing and discuss the main directions of its advancement. Second, we present our key research findings in this area, some of which have recently been partially rediscovered by other scientists. Finally, we give an exact formula for the best information rate of modular implementations of the general access structure obtained by the GM algorithm. |
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