Prime Numbers

9.6 Polynomial arithmetic 509
can itself be performed with a fast divide-and-conquer algorithm of the type
discussed in Chapter 8.8 (for example, Exercise 9.74). As an example of the
operation of Algorithm 9.5.26, let us take r = 8 = 23 and eight moduli:
(m1,...,m8) =(3, 5, 7, 11, 13, 17, 19, 23). Then we use these moduli along with
the product M = 8 i=1 mi = 111546435, to obtain at the [Precomputation]
phase M1,...,M8, which are, respectively,
37182145, 22309287, 15935205, 10140585, 8580495, 6561555, 5870865, 4849845,
and the tableau
(v1,...,v8) =(1, 3, 6, 3, 1, 11, 9, 17),
(q00,...,q70) =(3, 5, 7, 11, 13, 17, 19, 23),
(q01,...,q61) = (15, 35, 77, 143, 221, 323, 437),
(q02,...,q42) = (1155, 5005, 17017, 46189, 96577),
q03 = 111546435,
where we recall that for fixed j there exist qij for i ∈ [0,r−2 j ]. It is important
to note that all of the computation up through the establishment of the q
tableau can be done just once—as long as the CRT moduli mi are not going
to change in future runs of the algorithm. Now, when specific residues ni of
some mystery n are to be processed, let us say
(n1,...,n8) =(1, 1, 1, 1, 3, 3, 3, 3),
we have after the [Reconstruction loop] step, the value
n0k = 878271241,
whichwhenreducedmodq03 is the correct result n = 97446196. Indeed, a
quick check shows that
97446196 mod (3, 5, 7, 11, 13, 17, 19, 23) = (1, 1, 1, 1, 3, 3, 3, 3).
The computational complexity of Algorithm 9.5.26 is known in the
following form [Aho et al. 1974, pp. 294–298], assuming that fast multiplication
isused.Ifeachofther moduli mi has b bits, then the complexity is
O(br ln r ln(br) ln ln(br))
bit operations, on the assumption that all of the precomputation for the
algorithm is in hand.
9.6 Polynomial arithmetic
It is an important observation that polynomial multiplication/division is
not quite the same as large-integer multiplication/division. However, ideas
discussed in the previous sections can be applied, in a somewhat different
manner, in the domain of arithmetic of univariate polynomials.