n: X^12 + 12;
r: 7*X^11 + 4*X^10 + X^9 + 12*X^8 + 2*X^7 + 12*X^6;

q0: 2*X+10;
q1: 2*X+2;
q2: 2*X+10;

a0: n;
b0: r;
r0: expand(a0 - q0 * b0);

a1: b0;
b1: r0;
r1: expand(a1 - q1 * b1);

a2: b1;
b2: r1;
r2: expand(a2 - q2 * b2);

K: matrix([1,0],[0,1]);

K: expand(K . matrix([q0,1],[1,0]));
K: expand(K . matrix([q1,1],[1,0]));
K: expand(K . matrix([q2,1],[1,0]));

u: 8*X^3+10*X^2+11*X+12;
v: 4*X^2+11*X+8;

factor(u), modulus:13;