Hello,I'm stalled try to calculate a division in a polynomial ring defined over a field that is a finite field.sage: _.<z> = GF(2)[]sage: K.<z> = GF(2^8, modulus=z^8+z^4+z^3+z+1)
sage: R = PolynomialRing(K,'x')sage: l = x^4 + 1sage: c = (z+1)*x^3+x^2+x+(z)sage: l/c(x^4 + 1)/((z + 1)*x^3 + x^2 + x + z)But I like to find the quotient and the reminder of this division. How should I write it?
sage: _.<z0> = GF(2)[]
sage: K.<z> = GF(2^8, modulus=z0^8+z0^4+z0^3+z0+1)
sage: R.<x> = K[]
sage: l = x^4 + 1
sage: c = (z+1)*x^3+x^2+x+(z)
sage: l.quo_rem(c)
((z^7 + z^6 + z^5 + z^4 + z^2 + z)*x + z^6 + z^4 + z,
(z^7 + z^5 + z^2)*x^2 + (z^7 + z^5 + z^2 + 1)*x + z^7 + z^5 + z^2 + 1)
sage: l // c
(z^7 + z^6 + z^5 + z^4 + z^2 + z)*x + z^6 + z^4 + z
sage: l % c
Can you help for thatt:
sage: P2.<x> = GF(2)[];sage: print (x^8 + x^6 + x^3 + x^2 + 1).is_irreducible()sage: p = x^8 + x^4 + x^3 + x^2 + 1; #twofish irreducible polsage: GF256 = GF(2^8, 'X', modulus=p)sage: GF256([1,0,1,1,0,1,0,1]);