Efficient evaluation of polynomials over finite fields
  A method is described which allows to evaluate efficiently a polynomial in a (possibly trivial) extension of the finite field of its coefficients. Its complexity is shown to be lower than that of standard techniques when the degree of the polynomial is large with respect to the base field. Applications to the syndrome computation in the decoding of cyclic codes, Reed-Solomon codes in particular, are highlighted. 
Davide Schipani, Michele Elia, Joachim Rosenthal
cs.IT math.IT math.NT