Prime Numbers

524 Chapter 9 FAST ALGORITHMS FOR LARGE-INTEGER ARITHMETIC
algorithm that inputs x, y and requires only four internal variables to calculate
the inverse of x.
9.32. Can Algorithm 9.4.4 be generalized to composite p?
9.33. Prove that Algorithms 9.4.4 and 9.4.5 work. For the latter algorithm,
it may help to observe how one inverts a pure power of two modulo a Mersenne
prime.
9.34. In the spirit of the special-case mod Algorithm 9.2.13, which relied
heavily on bit shifting, recast Algorithm 9.4.5 to indicate the actual shifts
required in the various steps. In particular, not only the mod operation but
multiplication by a power of two is especially simple for Mersenne prime
moduli, so use these simplifications to rewrite the algorithm.
9.35. CanoneperformagcdontwonumberseachofsizeN in polynomial
time (i.e., time proportional to some power lg α N), using a polynomial number
of parallelized processors (i.e., lg β N of them)? An interesting reference is
[Cesari 1998], where it is explained that it is currently unknown whether such
a scheme is possible.
9.36. Write out a clear algorithm for a full integer multiplication using the
Karatsuba method. Make sure to show the recursive nature of the method,
and also to handle properly the problem of carry, which must be addressed
when any final digits overflow the base size.
9.37. Show that a Karatsuba-like recursion on the (D =3)Toom–Cook
method (i.e., recursion on Algorithm 9.5.2) yields integer multiplication of
two size-N numbers in what is claimed in the text, namely, O((ln N) ln 5/ ln 3 )
word multiplies. (All of this assumes that we count neither additions nor the
constant multiplies as they would arise in every recursive [Reconstruction]
step of Algorithm 9.5.2.)
9.38. Recast the [Initialize] step of Algorithm 9.5.2 so that the ri,si can be
most efficiently calculated.
9.39. We have seen that an acyclic convolution of length N can be effected
in 2N −1 multiplies (aside from multiplications by constants; e.g., a term such
as 4x can be done with left-shift alone, no explicit multiply). It turns out that
a cyclic convolution can be effected in 2N − d(N) multiplies, where d is the
standard divisor function (the number of divisors of n), while a negacyclic can
be effected in 2N − 1 multiplies. (These wonderful results are due chiefly to
S. Winograd; see the older but superb collection [McClellan and Rader 1979].)
Here are some explicit related problems:
(1) Show that two complex numbers a + bi, c + di may be multiplied via only
three real multiplies.
(2) Work out an algorithm that performs a length-4 negacyclic in nine
multiplies, but with all constant mul or div operations being by powers of