galois.is_primitive

galois.is_primitive(poly)

Checks whether the polynomial \(f(x)\) over \(\mathrm{GF}(p)\) is primitive.

A degree-\(n\) polynomial \(f(x)\) over \(\mathrm{GF}(p)\) is primitive if \(f(x)\ |\ (x^k - 1)\) for \(k = p^n - 1\) and no \(k\) less than \(p^n - 1\).

Parameters

poly (galois.Poly) – A polynomial \(f(x)\) over \(\mathrm{GF}(p)\).

Returns

True if the polynomial is primitive.

Return type

bool

Examples

All Conway polynomials are primitive.

In [383]: f = galois.conway_poly(2, 8); f
Out[383]: Poly(x^8 + x^4 + x^3 + x^2 + 1, GF(2))

In [384]: galois.is_primitive(f)
Out[384]: True

In [385]: f = galois.conway_poly(3, 5); f
Out[385]: Poly(x^5 + 2x + 1, GF(3))

In [386]: galois.is_primitive(f)
Out[386]: True

The irreducible polynomial of \(\mathrm{GF}(2^8)\) for AES is not primitive.

In [387]: f = galois.Poly.Degrees([8,4,3,1,0]); f
Out[387]: Poly(x^8 + x^4 + x^3 + x + 1, GF(2))

In [388]: galois.is_primitive(f)
Out[388]: False