Design And Analysis Of Some Cryptographic Protocols On Matrices

As a fundamental tool, matrix theory has found many applications in math-ematical and physical science, as well as fertile felds for research including statis-tical analysis, numerical calculation, optimization theory, diferential equations,probability and statistics, and system engineering. The rapid development ofcomputer network and information technology also ofers new opportunities forits further application. Among them, studies on the application of matrix the-ory in cryptography have been extensively carried out by scholars both at homeand abroad. For instance, secure and efcient cryptographic systems based onmatrix theory have been designed to solve the problems of information security.On the other hand, secure and efcient computing protocols utilizing diferentcryptographic techniques provide alternative access to multi-party sharing prob-lems in matrix theory. In this thesis, we focused our attention on solving severalopen questions in the feld of secure multi-party computation, secret sharing andidentity-based signcryption, thus providing many meaningful new results over theknown ones. These results further facilitate the developments on matrix theoryand information security to some extent. Accordingly, the main contributions ofthis dissertation are as follows,1.Researches on the design and analysis of a secure two-party protocol forsolving systems Ax=b of m linear equations in n variables over a fnite feld.Generalized inverses play an important role in the solution of such linear sys-tems of equations. Given an implementation of oblivious transfer, we can solveprivacy-preserving cooperative linear system of equations of the form Ax=bby computing the generalized inverse. Based on this probabilistic algorithm, wepresent a secure two-party protocol to enrich the research contents on securelysolving linear system of equations.2.Researches on the design and analysis of a secure multi-party protocol of matrix addition. Matrix addition represents a basic operation in matrix theory,hence we propose a privacy-preserving cooperative matrix addition protocol inthe k-party model via oblivious transfer protocol and recursive method. In suchcase, the security is proven through constructing simulator. This basic protocolcan serve as a new tool to protect privacy in practical applications.3.Researches on the attack analysis of a secret sharing scheme using matrixprojection. Matrix projection and its invariance property provide a novel ideafor constructing threshold secret sharing system. In this thesis, the securitydefciency against cheating, which lies in a threshold secret sharing scheme usingprojection matrix, is presented. We exemplify that there exists only single cheaterpassing the check that makes other participants reconstruct an invalid secretwithout being detected. Most importantly, we also give a strict proof according tomatrix projection and matrix theory over fnite feld, which show that the cheaterhas non-negligible advantage in above deception. Moreover, the probability ofsuccessfully cheating is given.4.Researches on the design and analysis of ID-based signcryption from lattice.Lattice is an important algebraic structure with matrix as its basic tool. Owing toits strong resistance against quantum attack and security proofs based on average-case hardness, lattice-based cryptographic constructions hold a great promisefor post-quantum cryptography. In view of the average-case hard problems inq modular lattice such as LWE and SIS, we propose an ID-based signcryptionscheme, which security can be proved in the random oracle model of IND-CCA2and EUF-CMA security formally. Since linear operations in lattice reduce thecomputational cost, the presented scheme becomes more efcient.