Prime Numbers

9.6 Polynomial arithmetic 511
Incidentally, if polynomial multiplication in rings is done via fast integer
convolution (recall that acyclic convolution is sufficient, and so zero-padded
cyclic will do), then one may obtain a different expression for the complexity
bound. For the Nussbaumer Algorithm 9.5.25 one requires O(M(ln m)D ln D)
bit operations, where M is the usual integer-multiplication complexity. It is
interesting to compare these various estimates for polynomial multiplication
(see Exercise 9.70).
9.6.2 Fast polynomial inversion and remaindering
Let x(t) = D−1
j=0 xjt j be a polynomial. If x0 = 0,thereisaformalinversion
1/x(t) =1/x0 − (x1/x 2 0)t +(x 2 1/x 3 0 − x2/x 2 0)t 2 + ···
that admits of rapid evaluation, by way of a scheme we have already invoked
for reciprocation, the celebrated Newton method. We describe the scheme in
the case that x0 = 1, from which case generalizations are easily inferred. In
what follows, the notation
z(t) modt k
is a polynomial remainder (which we cover later), but in this setting it is
simple truncation: The result of the mod operation is a polynomial consisting
of the terms of polynomial z(t) through order t k−1 inclusive. Let us define,
then, a truncated reciprocal,
R[x, N] =x(t) −1 mod t N+1
as the series of 1/x(t) through degree t N ,inclusive.
Algorithm 9.6.2 (Fast polynomial inversion). Let x(t) be a polynomial
with first coefficient x0 = 1. This algorithm returns the truncated reciprocal
R[x, N] through a desired degree N.
1. [Initialize]
g(t) =1; // Degree-zero polynomial.
n =1; // Working degree precision.
2. [Newton loop]
while(n N+1) n = N +1;
h(t) =x(t) modt n ; // Simple truncation.
h(t) =h(t)g(t) modt n ;
g(t) =g(t)(2 − h(t)) mod t n ; // Newton iteration.
}
return g(t);
One point that should be stressed right off is that in principle, an operation
f(t)g(t) modt n is simple truncation of a product (the operands usually
themselves being approximately of degree n). This means that within