Contents - Student subdomain for University of Bath

5.4. FROM Z P TO Z? 153
Figure 5.3: Algorithm24: Split a Distinct Degree Factorization
Algorithm 24 (Split a Distinct Degree Factorization)
Input: A prime p, a degree d and a polynomial f(x) (mod p) known to be the
product of irreducibles of degree d (and not divisible by x)
Output: The factorization of f (modulo p) into irreducibles of degree d.
if d = deg(f)
then return f # degree d so is irreducible
W := {f}
ans := ∅
while W ≠ ∅ # factors of degree d found
h :=RandomPoly(p,d)
h := h (pi −1)/2
(mod g) (*)
V := ∅ # list of polynomials to be processed
for g ∈ W do
h 1 := gcd(h − 1, g)
if deg(h 1 ) = 0 ∨ deg(h 1 ) = deg(g)
then V := V ∪ {g} # we made no progress
else process(h 1 )
process(g/h 1 )
W := V
return ans
Here RandomPoly(p,d) returns a random non-constant polynomial modulo p of
degree less than d, and the sub-function process(g 1 ) takes a polynomial g 1 which
is a result of splitting g by h 1 , and adds it to ans if it has degree d, otherwise
adds it to V .