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5.7. UNIVARIATE FACTORING SOLVED 161
1. [Col79] pointed out that there were two obvious ways to code the “for
subsets
∏
T of S” loop: increasing cardinality of T and increasing degree of
g∈T
g. He showed that, subject to two very plausible conjectures, the
average number of products actually formed with the cardinality ordering
was O(n 2 ), thus the average running time would be polynomial in n.
2. [LLJL82] had a completely different approach to algorithm 29. They
asked, for each d < n, “given f 1 ∈ S, what is the polynomial g of degree
d which divides f over the integers and is divisible by f 1 modulo
p k ?”. Unfortunately, answering this question needed a k far larger than
that implied by the Landau–Mignotte bound, and the complexity, while
polynomial in n, was O(n 12 ), at least while using classical arithmetic. This
paper introduced the ‘LLL’ lattice reduction algorithm, which has many
applications in computer algebra and far beyond.
3. [ABD85] showed that, by a combination of simple divisibility tests and
“early abort” trial division (Proposition 44) it was possible to make dramatic
reductions, at the time up to four orders of magnitude, in the constant
implied in the statement “exponential in r”.
4. [ASZ00] much improved this, and the authors were able to eliminate whole
swathes of possible T at one go.
5. [vH02] reduces the problem of finding T to a ‘knapsack’ problem, which, as
in method 2, is solved by LLL, but the lattices involved are much smaller
— of dimension r rather than n. At the time of writing, this seems to
be the best known method. His paper quoted a polynomial of degree
n = 180, with r = 36 factors of degree 5 modulo p = 19, but factoring as
two polynomials of degree 90 over the integers. This took 152 seconds to
factor.
Open Problem 11 (Evaluate [vH02] against [ASZ00]) How does the factorization
algorithm of [vH02] perform on the large factorizations successfully
solved by [ASZ00]? Clearly the algorithm of [vH02] is asymptotically faster, but
where is the cut-off point? Note also that [vH02]’s example of x 128 − x 112 +
x 80 − x 64 + x 48 − x 16 + 1 is in fact a disguised cyclotomic polynomial (as shown
by the methods of [BD89]), being
(
x 240 − 1 ) ( x 16 − 1 )
(x 80 − 1) (x 48 − 1) = ∏
1 ≤ k < 15
gcd(k, 15) = 1
5.7 Univariate Factoring Solved
(
x 16 − e 2πik/15) .
We can put together the components we have seen to deduce an algorithm
(Figure 5.8) for factoring square-free polynomials over Z[x].