In its most general form, the Hill cipher's keyspace consists of all matrices of a given dimension that are invertible over . Working from known results over finite fields, we assemble and prove a formula for the number of such matrices. We also compare this result with the total number of matrices and the number of involutory matrices for a given dimension and modulus, identifying the effects of change in dimension and modulus on the order of the keyspace.
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Overbey et al. (2005) studied this question.