Evaluation of polynomials using the structure of the coefficients

A new algorithm to evaluate polynomials that exploits the structure of their coefficients is proposed. This algorithm is an extension of one due to Savage with the advantage that it is not restricted to coefficients from a set whose cardinality is small compared to the degree of the polynomial. The new algorithm is shown to be asymptotically optimum for evaluating arbitrary functions in finite fields and for evaluating polynomials with real coefficients in a binary fixed-point representation. To illustrate that it is nonasymptotically useful as well, the new algorithm is shown to reduce the time for syndrome calculation for binary Bose-Chaudhuri-Hocquenghem (BCH) codes of practical interest.