Prime Numbers

514 Chapter 9 FAST ALGORITHMS FOR LARGE-INTEGER ARITHMETIC
Whatever method used for polynomial gcd, the fast polynomial remaindering
scheme of this section can be applied as desired for the internal polynomial
mod operations.
9.6.3 Polynomial evaluation
We next discuss polynomial evaluation techniques. The essential problem is
to evaluate a polynomial x(t) = D−1 j=0 xjtj at, say, each of n field values
t0,...,tn−1. It turns out that the entire sequence (x(t0),x(t1),...,x(tn−1))
canbeevaluatedin
O n ln 2 min{n, D}
field operations. We shall split the problem into three basic cases:
(1) The arguments t0,...,tn−1 lie in arithmetic progression.
(2) The arguments t0,...,tn−1 lie in geometric progression.
(3) The arguments t0,...,tn−1 are arbitrary.
Of course, case (3) covers the other two, but in (1), (2) it can happen that
special enhancements apply.
Algorithm 9.6.5 (Evaluation of polynomial on arithmetic progression).
Let x(t) = D−1 j=0 xjtj . This algorithm returns the n evaluations x(a),x(a +
d),x(a +2d),...,x(a +(n− 1)d). (The method attains its best efficiency when
n is much greater than D.)
1. [Evaluate at first D points]
for(0 ≤ j