In general terms, the invention provides a finite field engine and methods for operating on elements in a finite field. The finite field engine provides finite field sub-engines suitable for any finite field size requiring a fixed number of machine words. The engine reuses these engines, along with some general purpose component or specific component providing modular reduction associated with the exact reduction (polynomial or prime) of a specific finite field. The engine has wordsized suitable code capable of adding, subtracting, multiplying, squaring, or inverting finite field elements, as long as the elements are representable in no more than the given number of words. The wordsized code produces unreduced values. Specific reduction is then applied to the unreduced value, as is suitable for the specific finite field. In this way, fast engines can be produced for many specific finite fields, without duplicating the bulk of the engine instructions (program).