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5.5. HENSEL LIFTING 157
Figure 5.4: Algorithm 25
Algorithm 25 (Hensel Lifting (Linear Two Factor version))
Input: f, g (1) , h (1) , p, k with f monic and ≡ g (1) h (1) (mod p)
Output: g (k) , h (k) with f ≡ g (k) h (k) (mod p k )
g := g (1)
h := h (1)
g (r) , h (r) := Algorithm 4(g (1) , h (1) ) in Z p [x]
for i := 2 . . . k
∆ := f − gh (mod pi )
p i−1
g (c) := ∆ ∗ h (r) (mod (p, g (1) ))
h (c) := ∆ ∗ g (r) (mod (p, h (1) ))
g := g + p i−1 g (c)
h := h + p i−1 h (c)
return (g, h)
Figure 5.5: Algorithm 26
Algorithm 26 (Hensel Lifting (Linear version))
Input: f, g (1)
1 , . . . , g(1) n , p, k with f primitive and ≡ ∏ g (1)
Output: g (k)
1 , . . . , g(k) n
for j := 1 . . . n
g (1)
j := lc(f)
lc(g (1) ) g(1) j
j
with f ≡ ∏ g (k)
i (mod p k )
F := lc(f) n−1 f #leading coefficients imposed
for j := 1 . . . n
g (r)
j , h (r)
j := Algorithm 4(g (1)
j , ∏ i≠j g(1) i ) in Z p [x]
for i := 2 . . . k
∆ := F − ∏ j g j (mod p i )
p i−1
for j := 1 . . . n
g (c)
j
:= ∆ ∗ h (r)
j (mod (p, g (1)
j ))
g j := g j + p i−1 g (c)
j
for j := 1 . . . n
g j := pp(g j ) #undo the imposed leading coefficients
return (g 1 , . . . , g n )
i (mod p)