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152 CHAPTER 5. P -ADIC METHODS
Figure 5.2: Algorithm23: Distinct Degree Factorization
Algorithm 23 (Distinct Degree Factorization)
Input: f(x) a square-free polynomial modulo p, not divisible by x; a prime p
Output: A decomposition f = ∏ f i , where each f i is the product of irreducibles
of degree i
i:=1
while 2i ≤ deg(f)
g := x pi −1
(mod f) (*)
f i := gcd(g − 1, f)
f := f/f i
i := i + 1
if f ≠ 1
then f deg(f) := f
Note that the computation in line (*) should be done by the repeated squaring
method, reducing modulo f at each stage. We can save time in practice by
re-using the previous g.
Corollary 15 Half of the irreducible polynomials of degree d (except for x itself
in the case d = 1) with coefficients modulo p divide (x − a) (pd −1)/2 − 1, for any
a (but a different 50%,depending on a).
If f i has degree i, then it is clearly irreducible: otherwise we have to split
it. This is the purpose of Algorithm 24, which relies on a generalization of
Corollary 15.
Proposition 50 ([CZ81, p. 589]) Let f be a product of r > 1 irreducible
polynomials of degree d modulo p, and g a random (non-constant) polynomial
of degree < d. Then the probability that gcd(g (pd −1)/2 − 1, f) is either 1 or f is
at most 2 1−r .
Proposition 51 The running time of the Cantor–Zassenhaus Algorithm (i.e.
Algorithm 23 followed by Algorithm 24) is O(d 3 log p), where d is the degree of
the polynomial being factored.
We do O(d) operations on polynomials of degree d, and the factor log p comes
from the x pi (mod p) computations.
5.4 From Z p to Z?
Now that we know that factoring over the integers modulo p is possible, the
obvious strategy for factoring polynomials over the integers would seem to be
to follow one of algorithms 13 or 14. This would depend on having ‘good’
reduction, which one would naturally define as follows.