By Greenberg H.J.
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20. O. Ore, Theory of non-commutative polynomials, Annals of Math. 34 (1933), 480-508. 45 21. O. Ore, Contributions to the theory of finite fields, Trans. Amer. Math. Soc. 36 (1934), 243-274. 22. J. Patarin, Hidden fields equations (HFE) and isomorphisms of polynomials (IP): two new families of asymmetric algorithms, Advances in Cryptology - Eurocrypt '96 (U. ), Lecture Notes in Computer Science, vol. 1070, 1996, pp. 33-48. 23. F. Ritt, Prime and composite polynomials, Trans. Amer. Math. Soc. 23 (1922), 51-66.
1 Introduction Algorithms that manipulate recursive sequences, as it occurs with greatest common divisors, division of polynomials, Sturm sequences, and others have a problem with growth of the intermediate expressions. Such problem have been addressed by 1 - 5 . Also it is known that for Chebyshev polynomials the length of the coefficients decreases in a Sturm sequence2 . Such property was used by6 who presents an algebraic algorithm to enumerate zeros inside a circle, based on Chebyschev polynomials.
9. F. Dorey and G. Whaples, Prime and composite polynomials, J. Algebra 28 (1974), 88-101. 10. T. Engstrom, Polynomial substitutions, Amer. J. Math. 63 (1941), 249-255. 11. D. E. MacRae, On the invariance of chains of fields, Illinois J. Math. 13 (1969), 165-171. 12. J. von zur Gathen, Functional decomposition of polynomials: the tame case, J. Symbolic Computation 9 (1990), 281-299. 13. M. Giesbrecht, Factoring in skew-polynomial rings over finite fields, J. Symbolic Comput. 26 (1998), 463-486.