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On the efficiency of algorithms for polynomial factoring

Author: Robert T. Moenck
Journal: Math. Comp. 31 (1977), 235-250
MSC: Primary 12-04; Secondary 68A10, 68A20
MathSciNet review: 0422193
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Abstract: Algorithms for factoring polynomials over finite fields are discussed. A construction is shown which reduces the final step of Berlekamp's algorithm to the problem of finding the roots of a polynomial in a finite field $ {Z_p}$.

It is shown that if the characteristic of the field is of the form $ p = L \cdot {2^l} + 1$, where $ l \simeq L$, then the roots of a polynomial of degree n may be found in $ O({n^2}\log p + n{\log ^2}p)$ steps.

As a result, a modification of Berlekamp's method can be performed in $ O({n^3} + {n^2}\log p + n{\log ^2}p)$ steps. If n is very large then an alternative method finds the factors of the polynomial in $ O({n^2}{\log ^2}n + {n^2}\log n\log p)$. Some consequences and empirical evidence are discussed.

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Keywords: Algebraic manipulation, polynomial factoring, roots in finite fields, analysis of algorithms
Article copyright: © Copyright 1977 American Mathematical Society