# how to compute x^ip (mod P) in F2

hello

i want to compute x^iq (mod P) for q=2,0<=i<=3. By using the Euclidean algorithm; but i don

```
p = 3
X = ZZ['X'].gen()
P = 1+X+X^2+X^3
K.<X> = GF(p)['X'].quotient(P)
K.<X> = PolynomialRing(GF(2),'X')
X = K.gen()
S = K.quotient(P, 'X')
X = S.gen()
M = matrix([(X^(p*i).list() for i in range(p)]).transpose()
```

but my matrix M does not correspond to what i have when i do it by hand. thanks

Please describe the situation mathematically first, since the provided code cannot be a substitute. Do we want to work in characteristic two or in characteristic three? The ring $$\Bbb F_3[X]\ /\ (1+X+X^2+X^3)$$ is immediately overwritten. Why do we do this? The prime $p=3$ is finally used also in the matrix $M$. Which should be this matrix?