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This subject is covered in another paper under preparation by the authors. In this article, we report on a successful implementation of Giesbrecht's algorithm, making some specific comments concerning the probabilistic part of the algorithm. We also recall the work of Oystein Ore, showing how Giesbrecht's algorithm is equivalent to an algorithm described by Ore sixty years earlier. We also consider implications of Ore's work to the security of the Hidden Field Equations (HFE) cryptosystem. 2 Giesbrecht's algorithm and the work of Ore Giesbrecht 13 introduces a probabilistic polynomial-time algorithm for obtaining a complete (essentially unique) factorisation of a polynomial in some classes of skew-polynomial ring defined over a finite field.

J. Abbott, M. Bronstein, and T. Mulders. Fast deterministic computation of determinants of dense matrices. In S. Dooley, editor, ISSAC 99 25 2. 3. 4. 5. 6. 7. 8. 9. 10. 11. 12. 13. 14. 15. 16. Proc. 1999 Internat. Symp. , pages 181-188, New York, N. , 1999. ACM Press. A. Aho, J. Hopcroft, and J. Ullman. The Design and Analysis of Algorithms. Addison and Wesley, Reading, MA, 1974. W. Baur and V. Strassen. The complexity of partial derivatives. Theoretical Comp. Sci, 22:317-330, 1983. B. Beckermann and G.

WILF, H. S. A Global Bisection algorithm for computing the zeros of late polynomials in the complex plane. Journal of the ACM, 25(3), 1978. GIESBRECHT'S ALGORITHM, THE HFE CRYPTOSYSTEM A N D ORE'S ^-POLYNOMIALS R O B E R T S. au We report on a recent implementation of Giesbrecht's algorithm for factoring polynomials in a skew-polynomial ring. We also discuss the equivalence between factoring polynomials in a skew-polynomial ring and decomposing p s -polynomials over a finite field, and how Giesbrecht's algorithm is outlined in some detail by Ore in the 1930's.

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