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Let $${{\mathcal S}}$$ be one of the two multiplicative semigroups: M × M Boolean matrices, or the semigroup of M × M matrices over the field GF(2). Then for any matrix $${A\in {\mathcal S}}$$ there exist two unique smallest numbers, namely the index and period k, d, such that Ak = Ak+d. This fact allows us to form a new statistical test for randomness which we call the Semigroup Matrix...
The design of stream ciphers based on the synthesis of linear feedback shift registers (LFSRs) has been the research target of many cryptographers for more than 50 years.There is no general way for such a synthesis that results in design of a ''secure'' stream cipher. One of the design approaches is the use of basic design principles followed by properties testing procedures (system-theoretic approach)...
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