Prime Numbers

9.5 Large-integer multiplication 493
There is an elegant relation due to Shokrollahi in 1994; see [Buhler et al. 2000],
that gives a congruence for precisely these Bernoulli numbers:
Theorem 9.5.14. Let g be a primitive root of the odd prime p, and set:
−1 j (g mod p)(g mod p)
cj =
g
p
−j
for j ∈ [0,p− 2]. Then for k ∈ [1, (p − 3)/2] we have
p−2
cjg 2kj ≡ 1 − g 2k B2k
2kg
j=0
(mod p). (9.28)
We see that Shokrollahi’s relation involves a length-(p − 1) DFT, with the
operant field being Fp. One could proceed with an FFT algorithm, except
that there are two problems with that approach. First, the best lengths for
standard FFTs are powers of two; and second, one cannot use floating-point
arithmetic, especially when the prime p is large, unless the precision is extreme
(and somehow guaranteed). But we have the option of performing a DFT
itself via convolution (see Algorithm 9.6.6), so the Shokrollahi procedure for
determining regular primes; indeed, for finding precise irregularity indices of
any prime, can be effected via power-of-two length convolutions. As we shall
see later, there are “symbolic FFT” means to do this, notably in Nussbaumer
convolution, which avoids floating-point arithmetic and so is suitable for pureinteger
convolution. These approaches—Shokrollahi identity and Nussbaumer
convolution—have been used together to determine all regular primes p<
12000000 [Buhler et al. 2000].
9.5.4 Discrete weighted transform (DWT) methods
One variant of DFT-based convolution that has proved important for modern
primality and factorization studies (and when the relevant integers are large,
say in the region of 2 1000000 and beyond) is the discrete weighted transform
(DWT). This transform is defined as follows:
Definition 9.5.15 (Discrete weighted transform (DWT)). Let x be a signal
of length D, and let a be a signal (called the weight signal) of the
same length, with the property that every aj is invertible. Then the discrete
weighted transform X = DWT(x, a) is the signal of elements
D−1
Xk = (a ∗ x)jg −jk , (9.29)
j=0
with the inverse DWT −1 (X, a) =x given by
xj = 1
D−1
Xkg
Daj
jk . (9.30)
k=0