The linear complexity is a measure for the unpredictability of a sequence over a finite field and thus for its suitability in cryptography. In 2012, Diem introduced a new figure of merit for cryptographic sequences called expansion complexity. We study the relationship between linear complexity and expansion complexity. In particular, we show that for purely periodic sequences both figures of merit provide essentially the same quality test for a sufficiently long part of the sequence. However, if we study shorter parts of the period or nonperiodic sequences, then we can show, roughly speaking, that the expansion complexity provides a stronger test. We demonstrate this by analyzing a sequence of binomial coefficients modulo p. Finally, we establish a probabilistic result on the behavior of the expansion complexity of random sequences over a finite field.2000 Mathematics Subject Classification: 11T71, 11Y16, 94A60, 94A55, 68Q25

The equivalence problem for a group G is the problem of deciding which equations hold in G. It is known that for finite nilpotent groups and certain other solvable groups, the equivalence problem has polynomial-time complexity. We prove that the equivalence problem for a finite nonsolvable group G is co-NP-complete by reducing the k-coloring problem for graphs to the equivalence problem, where k is the cardinality of G.

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