@MISC{Lauder98polynomialswith, author = {Alan G. B. Lauder}, title = {Polynomials with Odd Orthogonal Multiplicity}, year = {1998} }

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Abstract

Let the orthogonal multiplicity of a monic polynomial g over a field F be the number of polynomials f over F, coprime to g and of degree less than that of g, such that all the partial quotients of the continued fraction expansion of f=g are of degree 1. Polynomials with positive orthogonal multiplicity arise in stream cipher theory, part of cryptography, as the minimal polynomials of the initial segments of sequences which have perfect linear complexity profiles. This paper focuses on polynomials which have odd orthogonal multiplicity; such polynomials are characterized and a lower bound on their orthogonal multiplicity is given. A special case of a conjecture on rational functions over the finite field of two elements with partial quotients of degree 1 or 2 in their continued fraction expansion is also proved. 1 Introduction If f and g are polynomials over an arbitrary field F with gcd(f; g) = 1 and deg g 1, then the rational function f=g has a unique continued fraction expansion...