Prime Numbers

532 Chapter 9 FAST ALGORITHMS FOR LARGE-INTEGER ARITHMETIC
9.63. If the Mersenne prime p = 2 89 − 1 is used in the DGT integer
convolution Algorithm 9.5.22 for zero-padded, large-integer multiply, and
the elements of signals x, y are interpreted as digits in base B =2 16 ,how
large can x, y be? What if balanced digit representation (with each digit in
[−2 15 , 2 15 − 1]) is used?
9.64. Describe how to use Algorithm 9.5.22 with a set of Mersenne primes
to effect integer convolution via CRT reconstruction, including the precise
manner of reconstruction. (Incidentally, CRT reconstruction for a Mersenne
prime set is especially straightforward.)
9.65. Analyze the complexity of Algorithm 9.5.22, with a view to the type
of recursion seen in the Schönhage Algorithm 9.5.23, and explain how this
compares to the entries of Table 9.1.
9.66. Describe how DWT ideas can be used to obviate the need for zeropadding
in Algorithm 9.5.25. Specifically, show how to use not a length-(2m)
cyclic, rather a length-m cyclic and a length-m negacyclic. This is possible
becausewehaveaprimitivem-th root of −1, so a DWT can be used for the
negacyclic. Note that this does not significantly change the complexity, but
in practice it reduces memory requirements.
9.67. Prove the complexity claim following the Nussbaumer Algorithm
9.5.25 for the O(D ln D) operation bound. Then analyze the somewhat
intricate problem of bit-complexity for the algorithm. One way to start on such
bit-complexity analysis is to decide upon the optimal base B, as intimated in
the complexity table of Section 9.5.8.
9.68. For odd primes p, the Nussbaumer Algorithm 9.5.25 will serve to
evaluate cyclic or negacyclic convolutions (mod p); that is, for ring R identified
with Fp. All that is required is to perform all R-element operations (mod p),
so the structure of the algorithm as given does not change. Use such a
Nussbaumer implementation to establish Fermat’s last theorem for some large
exponents p, by invoking a convolution to effect the Shokrollahi DFT. There
are various means for converting DFTs into convolutions. One method is to
invoke the Bluestein reindexing trick, another is to consider the DFT to be
a polynomial evaluation problem, and yet another is Rader’s trick (in the
case that signal length is a prime power). Furthermore, convolutions of notpower-of-two
length can be embedded in larger, more convenient convolutions
(see [Crandall 1996a] for a discussion of such interplay between transforms
and convolutions). You would use Theorem 9.5.14, noting first that the DFT
length can be brought down to (p − 1)/2. Then evaluate the DFT via a
cyclic convolution of power-of-two length by invoking the Nussbaumer method
(mod p). Aside from the recent and spectacular theoretical success of A. Wiles
in proving the “last theorem,” numerical studies have settled all exponents
p