Matroids And Secret Sharing Schemes,

The Wid theory has found its aPplication in the fields of combinatoryoPtindZation, integral programrning, network fiows and electronic network. In thelast l0 yeas, it is discovered that close connection between the theory of matIOidand the concept of secret shallng schemes. And a lot of excellent work is doneever since. In this paPer the author discusses the relationship betWeen the matroidand secret sharing schemes. He gets some secret sharing schemes with desiredProperties, by takng a matroid as a tool. And he presenis a method to conStructthem.There are six chapters in this paper. The sboe of it is as following.(l) 1n ChaPter 1, it is briefly intrOduced the origin of the matroid theory aswell as the current researching of secret sharing schemes.(2) In ChaPter 2, it is introduced some basic notations of the matroid, and thatof secret sharing schemes.(3) In the first section of ChaPter 3, it is presented some previous studies aboutthe matrOid and the schemes. In the second section, the authOr discussesthe ideal secret sharing schemes based on the network. It comes to aconclusion, that is, "For a given netWork, an ideal secret sharing schemecan be construted, such tha its minimal access sbefore is just the sst ofall the minimal cuts of the given netWork." The key of the constnJcting isthat an efficient represetalion of the cut matroid of an oriented graPh isround.(4)ChaPter 4 mainly discusses the existence of ideal muti-secret sharingschems, and gives a method tO constnJct them.lIIn the first section of it, the author presents two theorems. One of it is about the necessary conditions, the other is about the sufficient conditions. And the conditions in the theorems are related with the matroids concerned, so they are directly related to the access structures. It is practical.In the second section, the author presents a theorem of necessity and sufficient conditions for the existence of a binary ideal multi-secret sharing scheme.In the third section, it is proposed a theorem of necessary and sufficient conditions for the existence of an ideal multi-secret threshold scheme. The strong restriction between the access structures is proposed. The author shows the existence of a class of ideal multi-secret sharing schemes slightly different from the threshold schemes.(5)In Chapter 5, the secret sharing schemes, which are able to prevent some attacks, are discussed. A method is showed to construct such schemes.