Berlekamp’s algorithm
gap> x:=X(GF(3),"x");; f:=x^9+x^8-x^6-x^5-x^4-x^3+Z(3)^0;;
Calculate the powers of xq and store the coefficients in a matrix.
gap> xq:=PowerMod(x,q,f);;
gap> pows:=List([0..8],i->PowerMod(xq,i,f));
[ Z(3)^0, x^3, x^6, -x^8+x^6+x^5+x^4+x^3-Z(3)^0, x^7+x^6+x^4-x^3+x^2-x,
-x^7-x^5-x, x^8-x^7-x^4+x^3+x-Z(3)^0, -x^8+x^7-x^6+x^5+x^4+x^3-x^2-x+Z(3)^0,
-x^7-x^6+x^4+x^3+x^2+x ]
gap> qmat:=List(pows,CoefficientsOfUnivariatePolynomial);
[ [ Z(3)^0 ], [ 0*Z(3), 0*Z(3), 0*Z(3), Z(3)^0 ],
...
The rows are not equal lengths, we have to extend the shorter ones. As they are immutable, we have to
make a copy first
gap>
[ 1,
gap>
gap>
List(qmat,Length);
4, 7, 9, 8, 8, 9, 9, 8 ]
qmat:=List(qmat,ShallowCopy);;
for i in qmat do while Length(i)<9 do Add(i,0*Z(3));od;od;
Q now is quadratic, as we could check with Display
gap> Display(qmat);
1 . . . . . . . .
...
Now create an identity matrix and consider Q − I.
gap> id:=IdentityMat(9,Z(3)^0);;
gap> RankMat(qmat-id);
6
Rank 6 = 9 − 3 implies: 3 factors. Calculate a basis of the nullspace. (GAPs linear algebra works on row
vectors, the nullspace thus is the left null space. From the coefficient lists, we go back to polynomials.
gap> bas:=NullspaceMat(qmat-id);
[ [ Z(3)^0, 0*Z(3), 0*Z(3), 0*Z(3), 0*Z(3), 0*Z(3), 0*Z(3), 0*Z(3), 0*Z(3) ],
. <3 vectors total >..
gap> bpols:=List(bas,i->UnivariatePolynomial(GF(3),i,1));
[ Z(3)^0, x^7+x^6-x^5-x^2+x, x^8-x^6-x^5-x^4+x^3-x ]
We now create a random nullspace element. It happens to have nontrivial gcd with f , so we got a factor.
gap> a:=Sum([1,2,3],i->Random(GF(3))*bpols[i]);
x^8-x^7+x^6-x^4+x^3+x^2+x+Z(3)^0
gap> g:=Gcd(a,f);
x^3+x^2-Z(3)^0
gap> f:=f/g;
x^6-x^2-Z(3)^0
The next a has trivial gcd. We thus calculate b = a(q−1)/2 mod f and form the gcd with b − 1. This turns
out to give a factor.
gap> a:=Sum([1,2,3],i->Random(GF(3))*bpols[i]);;
gap> Gcd(a,f);
Z(3)^0
gap> b:=PowerMod(a,(q-1)/2,f);;
gap> Gcd(b-1,f);
x^3+x^2-x+Z(3)^0