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5.8. MULTIVARIATE FACTORING 163
Open Problem 12 (Better Choice of ‘Best’ Prime) In algorithm 30, we
picked the ‘best’ prime and the corresponding factorization. In principle, it is
possible to do better, as in the following example, but the author has seen no
systematic treatment of this.
Example 10 Suppose f p factors as polynomials of degree 2, 5 and 5, and f q
factors as polynomials of degree 3, 4 and 5. The factorization modulo p implies
that the allowable degrees of (proper) factors are 2, 5, 7 and 10. The factorization
modulo q implies that the allowable degrees of (proper) factors are 3, 4, 5,
7, 8 and 9. Hence the only possible degrees of proper factors over the integers
are 7 and 12. The factorization modulo p can yield this in two ways, but the
fatcorization modulo q can only yield this by combining the factors of degrees
3 and 4. Hence we should do this, and lift the factorization of f q .
Open Problem 13 (Low-degree Factorization) Although no system to the
author’s knowledge implements it, it would be possible to implement an efficient
procedure to find all factors of limited total degree d. This would be efficient for
three reasons:
• the Cantor–Zassenhaus algorithm (section 5.3.3) need only run up to maximum
degree d;
• it should be possible to use smaller bounds on the lifting
• The potentially-exponential recombination process need only run up to total
degree d. In particular, if d = 1, no recombination is needed.
At the moment, the user who only wants low-degree factors has to either program
such a search, or rely on the system’s factor, which may waste large amounts
of time looking for high-degree factors, which the user does not want.
5.8 Multivariate Factoring
5.9 Other Applications
5.10 Conclusions