Prime Numbers

9.8 Research problems 539
9.82. A certain prime suggested in [Craig-Wood 1998], namely
p =2 64 − 2 32 +1,
has advantageous properties in regard to CRT-based convolution. Investigate
some of these advantages, for example by stating the possible signal lengths
for number-theoretical transforms modulo p, exhibiting a small-magnitude
element of order 64 (such elements might figure well into certain FFT
structures), and so on.
9.83. Here is a surprising result: Length-8 cyclic convolution modulo a
Mersenne prime can be done via only eleven multiplies. It is surprising because
the Winograd bound would be 2 · 8 − 4 = 12 multiplies, as in Exercise 9.39.
Of course, the resolution of this paradox is that the Mersenne mod changes
the problem slightly.
To reveal the phenomenon, first establish the existence of an 8-th
root of unity in F p 2,withp being a Mersenne prime and the root being
symbolically simple enough that DGTs can be performed without explicit
integer multiplications. Then consider the length-8 DGT, used to cyclically
convolve two integer signals x, y. Next, argue that the transforms X, Y have
sufficient symmetry that the dyadic product X ∗Y requires two real multiplies
and three complex multiplies. This is the requisite count of 11 muls.
An open question is: Are there similar “violations” of the Winograd bound
for lengths greater than eight?
9.84. Study the interesting observations of [Yagle 1995], who notes that
matrix multiplication involving n×n matrices can be effected via a convolution
of length n 3 . This is not especially surprising, since we cannot do an
arbitrary length-n convolution faster than O(n ln n) operations. However,
Yagle saw that the indicated convolution is sparse, and this leads to interesting
developments, touching, even, on number-theoretical transforms.