68 research outputs found

    Application of Quasigroups in Cryptography and Data Communications

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    In the past decade, quasigroup theory has proven to be a fruitfull field for production of new cryptographic primitives and error-corecting codes. Examples include several finalists in the flagship competitions for new symmetric ciphers, as well as several assimetric proposals and cryptcodes. Since the importance of cryptography and coding theory for secure and reliable data communication can only grow within our modern society, investigating further the power of quasigroups in these fields is highly promising research direction. Our team of researchers has defined several research objectives, which can be devided into four main groups: 1. Design of new cryptosystems or their building blocks based on quasigroups - we plan to make a classification of small quasigroups based on new criteria, as well as to identify new optimal 8–bit S-boxes produced by small quasigroups. The results will be used to design new stream and block ciphers. 2. Cryptanalysis of some cryptosystems based on quasigroups - we will modify and improve the existing automated tools for differential cryptanalysis, so that they can be used for prove the resistance to differential cryptanalysis of several existing ciphers based on quasigroups. This will increase the confidence in these ciphers. 3. Codes based on quasigroups - we will designs new and improve the existing error correcting codes based on combinatorial structures and quasigroups. 4. Algebraic curves over finite fields with their cryptographic applications - using some known and new tools, we will investigate the rational points on algebraic curves over finite fields, and explore the possibilities of applying the results in cryptography

    On the structure of non-full-rank perfect codes

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    The Krotov combining construction of perfect 1-error-correcting binary codes from 2000 and a theorem of Heden saying that every non-full-rank perfect 1-error-correcting binary code can be constructed by this combining construction is generalized to the qq-ary case. Simply, every non-full-rank perfect code CC is the union of a well-defined family of μ\mu-components KμK_\mu, where μ\mu belongs to an "outer" perfect code C∗C^*, and these components are at distance three from each other. Components from distinct codes can thus freely be combined to obtain new perfect codes. The Phelps general product construction of perfect binary code from 1984 is generalized to obtain μ\mu-components, and new lower bounds on the number of perfect 1-error-correcting qq-ary codes are presented.Comment: 8 page

    On the number of 1-perfect binary codes: a lower bound

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    We present a construction of 1-perfect binary codes, which gives a new lower bound on the number of such codes. We conjecture that this lower bound is asymptotically tight.Comment: 5pp(Eng)+7pp(Rus) V2: revised V3: + Russian version, + reference

    On decomposability of 4-ary distance 2 MDS codes, double-codes, and n-quasigroups of order 4

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    A subset SS of {0,1,...,2t−1}n\{0,1,...,2t-1\}^n is called a tt-fold MDS code if every line in each of nn base directions contains exactly tt elements of SS. The adjacency graph of a tt-fold MDS code is not connected if and only if the characteristic function of the code is the repetition-free sum of the characteristic functions of tt-fold MDS codes of smaller lengths. In the case t=2t=2, the theory has the following application. The union of two disjoint (n,4n−1,2)(n,4^{n-1},2) MDS codes in {0,1,2,3}n\{0,1,2,3\}^n is a double-MDS-code. If the adjacency graph of the double-MDS-code is not connected, then the double-code can be decomposed into double-MDS-codes of smaller lengths. If the graph has more than two connected components, then the MDS codes are also decomposable. The result has an interpretation as a test for reducibility of nn-quasigroups of order 4. Keywords: MDS codes, n-quasigroups, decomposability, reducibility, frequency hypercubes, latin hypercubesComment: 19 pages. V2: revised, general case q=2t is added. Submitted to Discr. Mat
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