Prime Numbers

512 Chapter 9 FAST ALGORITHMS FOR LARGE-INTEGER ARITHMETIC
multiplication loops, one need not handle terms of degree higher than
indicated. In convolution-theory language, we are therefore doing “half-cyclic”
convolutions, so when transform methods are used, there is also gain to be
realized because of the truncation.
As is typical of Newton methods, the dynamical precision degree n
essentially doubles on each pass of the Newton loop. Let us give an example
of the workings of the algorithm. Take
x(t) =1+t + t 2 +4t 3
and call the algorithm to output R[x, 8]. Then the values of g(t) attheendof
each pass of the Newton loop come out as
1 − t,
1 − t − 3t 3 ,
1 − t − 3t 3 +7t 4 − 4t 5 +9t 6 − 33t 7 ,
1 − t − 3t 3 +7t 4 − 4t 5 +9t 6 − 33t 7 +40t 8 ,
and indeed, this last output of g(t) multiplied by the original x(t) is
1+43t 9 − 92t 10 + 160t 11 , showing that the last output g(t) is correct through
O(t 8 ).
Polynomial remaindering (polynomial mod operation) can be performed
in much the same way as some of our mod algorithms for integers used a
“reciprocal.” However, it is not always possible to divide one polynomial by
another and get a unique and legitimate remainder: This can depend on the
ring of coefficients for the polynomials. However, if the divisor polynomial has
its high coefficient invertible in the ring, then there is no problem with divide
and remainder; see the discussion in Section 2.2.1. For simplicity, we shall
restrict to the case that the divisor polynomial is monic, that is, the high
coefficient is 1, since generalizing is straightforward. Assume that x(t),y(t)
are polynomials and that y(t) is monic. Then there are unique polynomials
q(t),r(t) such that
x(t) =q(t)y(t)+r(t),
and r =0ordeg(r) < deg(x). We shall write
r(t) =x(t) mody(t),
and view q(t) asthequotientandr(t) as the remainder. Incidentally, for some
polynomial operations one demands that coefficients lie in a field, for example
in the evaluation of polynomial gcd’s, but many polynomial operations do not
require field coefficients. Before exhibiting a fast polynomial remaindering
algorithm, we establish some nomenclature:
Definition 9.6.3 (Polynomial operations). Let x(t) = D−1
j=0 xjt j be a
polynomial. We define the reversal of x by degree d as the polynomial
rev(x, d) =
d
j=0
xd−jt j ,