Abstract
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 A k = A k+d. This fact allows us to form a new statistical test for randomness which we call the Semigroup Matrix Test. In this paper, we present details and results of our experiments for this test. We use Boolean matrices for M = 2, . . . , 5, and matrices over GF(2) of the size M = 2, . . . , 6. We also compare the results with the results obtained by the well-known Binary Matrix Rank Test.
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This material is based upon work supported under grant No. VEGA 1/3115/06 and the grant NIL-I-004 from Iceland, Lichtenstein and Norway through the EEA Financial Mechanism and the Norwegian Financial Mechanism.
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Grošek, O., Vojvoda, M. & Krchnavý, R. A new matrix test for randomness. Computing 85, 21–36 (2009). https://doi.org/10.1007/s00607-009-0033-z
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DOI: https://doi.org/10.1007/s00607-009-0033-z