Prime Numbers

6.5 Exercises 311
of the rational polynomial G(x) have the same sign. Deduce from this that
1 ≤|g(1)| = |G(−0.49)| < |G(0.51)| = |g(2)|, and similarly |h(2)| > 1, so that
the factorization n = g(2)h(2) is nontrivial.
6.11. Use the method of Exercise 6.9 to factor n = 187 using the base
m = 10. Do the same with n = 4189,m= 29.
6.12. Generalize the x(u, v),y(u, v) construction in Section 6.1.7 to arbitrary
numbers n satisfying (6.4).
6.13. Give a heuristic argument for the complexity bound
exp (c + o(1))(ln n) 1/3 (ln ln n) 2/3
operations, with c = (32/9) 1/3 , for the special number field sieve (SNFS).
6.14. Here we sketch some practical QS examples that can serve as guidance
for the creation of truly powerful QS implementations. In particular, the
reader who chooses to implement QS can use the following examples for
program checking. Incidentally, each one of the examples below—except
the last—can be effected on a typical symbolic processor possessed of
multiprecision operations. So the exercise shows that numbers in the 30digit
region and beyond can be handled even without fast, compiled
implementations.
(1) In Algorithm 6.1.1 let us take the very small example n = 10807 and,
because this n is well below typical ranges of applicability of practical
QS, let us force at the start of the algorithm the smoothness limit
B = 200. Then you should find k = 21 appropriate primes, You then get a
21 × 21 binary matrix, and can Gaussian-reduce said matrix. Incidentally,
packages exist for such matrix algebra, e.g., in the Mathematica language
a matrix m can be reduced for such purpose with the single statement
r = NullSpace[Transpose[m], Modulus->2];
(although, as pointed out to us by D. Lichtblau one may optimize the
overall operation by intervention at a lower level, using bit operations
rather than (mod 2) reduction, say). With such a command, there is a
row of the reduced matrix r that has just three 1’s, and this leads to the
relation:
3 4 · 11 4 · 13 4 ≡ 106 2 · 128 2 · 158 2 (mod n),
and thus a factorization of n.
(2) Now for a somewhat larger composite, namely n = 7001 · 70001, try using
the B assignment of Algorithm 6.1.1 as is, in which case you should have
B = 2305, k = 164. The resulting 164 × 164 matrix is not too unwieldy
in this day and age, so you should be able to factor n using the same
approach as in the previous item.
(3) Now try to factor the Mersenne number n =2 67 −1 but using smoothness
bound B = 80000, leading to k = 3962. Not only will this example start