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:The algorithm focuses on polynomials which satisfy the congruence:
:These polynomials form a subalgebra of R (which can be considered as an -dimensional vector space over ), called the Berlekamp subalgebra. The Berlekamp subalgebra is of interest because the polynomials it contains satisfy

:

In general, not every GCD in the above product will be a non-trivial factor of , but some are, providing the factors we seek.

Berlekamp's algorithm finds polynomials suitable for use with

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