Prime Numbers

9.7 Exercises 533
9.69. Implement Algorithm 9.6.1 for multiplication of polynomials with
coefficients (mod p). Such an implementation is useful in, say, the Schoof
algorithm for counting points on elliptic curves, for in that method, one has
not only to multiply large polynomials, but create powering ladders that rely
on the large-degree-polynomial multiplies.
9.70. Prove both complexity claims in the text following Algorithm 9.6.1.
Describe under what conditions, e.g., what D, p ranges, or what memory
constraints, and so on, which of the methods indicated—Nussbaumer
convolution or binary-segmentation method—would be the more practical.
For further analysis, you might consider the Shoup method for polynomial
multiplication [Shoup 1995], which is a CRT-convolution-based method, which
will have its own complexity formula. To which of the two above methods does
the Shoup method compare most closely, in complexity terms?
9.71. Say that polynomials x(t),y(t) havecoefficients(modp) and degrees
≈ N. For Algorithm 9.6.4, which calls Algorithm 9.6.2, what is the asymptotic
bit complexity of the polynomial mod operation x mod y, in terms of p
and N? (You need to make an assumption about the complexity of the
integer multiplication for products of coefficients.) What if one is, as in many
integer mod scenarios, doing many polynomial mods with the same modulus
polynomial y(t), so that one has only to evaluate the truncated inverse R[y, ]
once?
9.72. Here we explore another relation for Bernoulli numbers (mod p).
Prove the theorem that if p ≥ 5isprime,a is coprime to p, and we define
d = −p −1 mod a, then for even m in [2,p− 3],
Bm
m (am p−1
− 1) ≡ j m−1 (dj mod a) (modp).
Then establish the corollary that
j=0
Bm
m (2−m − 1) ≡ 1
(p−1)/2
2
j=1
j m−1 (mod p).
Now achieve the interesting conclusion that if p ≡ 3 (mod 4), then B (p+1)/2
cannot vanish (mod p).
Such summation formulae have some practical value, but more computationally
efficient forms exist, in which summation indices need cover only a
fraction of the integers in the interval [0,p− 1], see [Wagstaff 1978], [Tanner
and Wagstaff 1987].
9.73. Prove that Algorithm 9.6.5 works. Then modify the algorithm for
a somewhat different problem, which is to evaluate a polynomial given in
product form
x(t) =t(t + d)(t +2d) ···(t +(n − 1)d),