Factorization of determinants over finite fields and application in stream ciphers

Binary sequences being generated by nonlinearly filtering maximal length sequences with period 2n - 1 are studied in this paper. In particular, we focus on two well-known classes of nonlinear filters, namely the equidistant and normal filters, and provide new improved lower bounds on the linear complexity of the generated keystreams. In order to achieve this, properties of certain determinants over finite fields, i. e. generalized Vandermonde and linearized determinants, are first analyzed in terms of their factorization. The value of the derived methodology is demonstrated by the simplification that occurs in the generalized version of the root presence test, which has been commonly used to obtain lower bounds on the linear complexity. Moreover, it is shown how these results can be applied to reason about the properties of more complex nonlinear filters. © 2008 Springer Science + Business Media, LLC