Comparing the combinational complexities of arithmetic functions

Methods are presented for finding reductions between the computations of certain arithmetic functions that preserve asymptotic Boolean complexities (circuit depth or size). They can be used to show, for example, that all nonlinear algebraic functions are as difficult as integer multiplication with respect to circuit size. As a consequence, any lower or upper bound (e.g., <italic>O</italic>(<italic>n</italic> log <italic>n</italic> log log <italic>n</italic>)) for one of them applies to the whole class. It is also shown that, with respect to depth and size simultaneously, multiplication is reducible to any nonlinear and division to any nonpolynomial algebraic function.

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