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-- Methods related to the Berlekamp factoring algorithm.
-- Square-free factorization in the modular field Zp.
function PolynomialRing:modularsquarefreefactorization()
local monic = self / self:lc()
local terms = {}
terms[0] = PolynomialRing.gcd(monic, monic:derivative())
local b = monic // terms[0]
local c = monic:derivative() // terms[0]
local d = c - b:derivative()
local i = 1
while b ~= Integer.one() do
terms[i] = PolynomialRing.gcd(b, d)
b, c = b // terms[i], d // terms[i]
i = i + 1
d = c - b:derivative()
end
if not (terms[i-1]:derivative().degree == Integer.zero() and terms[i-1]:derivative().coefficients[0] == Integer.zero()) then
return terms
end
local recursiveterms = terms[i-1]:collapseterms(self.ring.modulus):modularsquarefreefactorization()
for k, poly in ipairs(recursiveterms) do
recursiveterms[k] = poly:expandterms(self.ring.modulus)
end
return JoinArrays(terms, recursiveterms)
end
-- Returns a new polnomial consisting of every nth term of the old one - helper method for square-free factorization
function PolynomialRing:collapseterms(n)
local new = {}
local loc = 0
local i = 0
local nn = n:asnumber()
while loc <= self.degree:asnumber() do
new[i] = self.coefficients[loc]
loc = loc + nn
i = i + 1
end
return PolynomialRing(new, self.symbol, self.degree // n)
end
-- Returns a new polnomial consisting of every nth term of the old one - helper method for square-free factorization
function PolynomialRing:expandterms(n)
local new = {}
local loc = 0
local i = 0
local nn = n:asnumber()
while i <= self.degree:asnumber() do
new[loc] = self.coefficients[i]
for j = 1, nn do
new[loc + j] = IntegerModN(Integer.zero(), n)
end
loc = loc + nn
i = i + 1
end
return PolynomialRing(new, self.symbol, self.degree * n)
end
-- Uses Berlekamp's Algorithm to factor polynomials in mod p
function PolynomialRing:berlekampfactor()
if self.degree == 0 or self.degree == 1 then
return {self}
end
local R = self:RMatrix()
local S = self:auxillarybasis(R)
if #S == 1 then
return {self}
end
return self:findfactors(S)
end
-- Gets the R Matrix for Berlekamp factorization
function PolynomialRing:RMatrix()
local R = {}
for i = 1, self.degree:asnumber() do
R[i] = {}
end
for i = 0, self.degree:asnumber()-1 do
local remainder = PolynomialRing({IntegerModN(Integer.one(), self.ring.modulus)}, self.symbol):multiplyDegree(self.ring.modulus:asnumber()*i) % self
for j = 0, self.degree:asnumber()-1 do
R[j + 1][i + 1] = remainder.coefficients[j]
if j == i then
R[j + 1][i + 1] = R[j + 1][i + 1] - IntegerModN(Integer.one(), self.ring.modulus)
end
end
end
return R
end
-- Creates an auxillary basis using the R matrix
function PolynomialRing:auxillarybasis(R)
local P = {}
local n = self.degree:asnumber()
for i = 1, n do
P[i] = 0
end
S = {}
local q = 1
for j = 1, n do
local i = 1
local pivotfound = false
while not pivotfound and i <= n do
if R[i][j] ~= self:zeroc() and P[i] == 0 then
pivotfound = true
else
i = i + 1
end
end
if pivotfound then
P[i] = j
local a = R[i][j]:inv()
for l = 1, n do
R[i][l] = a * R[i][l]
end
for k = 1, n do
if k ~= i then
local f = R[k][j]
for l = 1, n do
R[k][l] = R[k][l] - f*R[i][l]
end
end
end
else
local s = {}
s[j] = self:onec()
for l = 1, j - 1 do
local e = 0
i = 1
while e == 0 and i <= n do
if l == P[i] then
e = i
else
i = i + 1
end
end
if e > 0 then
local c = -R[e][j]
s[l] = c
else
s[l] = self:zeroc()
end
end
S[#S+1] = PolynomialRing(s, self.symbol)
end
end
return S
end
-- Uses the auxilary basis to find the irreirrducible factors of the polynomial.
function PolynomialRing:findfactors(S)
local r = #S
local p = self.ring.modulus
local factors = {self}
for k = 2,r do
local b = S[k]
local old_factors = Copy(factors)
for i = 1,#old_factors do
local w = old_factors[i]
local j = 0
while j <= p:asnumber() - 1 do
local g = PolynomialRing.gcd(b-IntegerModN(Integer(j), p), w)
if g == Integer.one() then
j = j + 1
elseif g == w then
j = p:asnumber()
else
factors = Remove(factors, w)
local q = w // g
factors[#factors+1] = g
factors[#factors+1] = q
if #factors == r then
return factors
else
j = j + 1
w = q
end
end
end
end
end
end
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