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We introduce a procedure in which two trusted individuals, Alice and Bob, may share a secret matrix K from the non-abelian general linear group of matrices. In this procedure, the matrix K is concealed from an eavesdropper, Eve, by a sequence of conjugations by elements from a pre-determined abelian subgroup of the general linear group. We demonstrate that the group of invertible circulant matrices is one abelian subgroup that may be able to withstand a brute force attack. To analyze this we need a technique to determine the order of this group, and to do this we make use of a well-known isomorphism between the ring of circulants over a finite field and a quotient of the ring of polynomials over a finite field. After we show empirically that the order of the group of invertible circulants increases exponentially in dimension n with degree q of the field fixed, we will give a universal lower bound on the order of this group to prove this mathematically as well.
Frederick, Hannah B., "Conjugation by Circulant Matrices in Non-commutative Cryptography" (2020). Student Research Submissions. 321.