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5.5. HENSEL LIFTING 159
Figure 5.7: Algorithm 28
Algorithm 28 (Hensel Lifting (Quadratic version))
Input: f, g (1)
1 , . . . , g(1) n , p, k with f primitive and ≡ ∏ g (1)
i (mod p)
Output: g 1 , . . . , g n with f ≡ ∏ g i (mod p 2k )
for j := 1 . . . n
g (1)
j := lc(f)
lc(g (1) ) g(1) j
j
F := lc(f) n−1 f #leading coefficients imposed
for i := 1 . . . k #f = ∏ g j (mod p 2i−1 )
∆ := F − ∏ j g j (mod p 2i )
p 2i−1
for j := 1 . . . n
g (r)
j , h (r)
j := Algorithm 4(g j , ∏ l≠j g l) in Z p 2
g (c)
j
:= ∆ ∗ h (r)
j (mod (p 2i−1 , g j ))
g j := g j + p 2i−1 g (c)
j
#f = ∏ g j (mod p 2i )
i−1 [x]
for j := 1 . . . n
g j := pp(g j ) #undo the imposed leading coefficients
return (g 1 , . . . , g n )