galois.conway_poly

class galois.conway_poly(p, n)

Returns the degree-\(n\) Conway polynomial \(C_{p,n}\) over \(\mathrm{GF}(p)\).

A Conway polynomial is a an irreducible and primitive polynomial over \(\mathrm{GF}(p)\) that provides a standard representation of \(\mathrm{GF}(p^n)\) as a splitting field of \(C_{p,n}\). Conway polynomials provide compatability between fields and their subfields, and hence are the common way to represent extension fields.

The Conway polynomial \(C_{p,n}\) is defined as the lexicographically-minimal monic irreducible polynomial of degree \(n\) over \(\mathrm{GF}(p)\) that is compatible with all \(C_{p,m}\) for \(m\) dividing \(n\).

This function uses Frank Luebeck’s Conway polynomial database for fast lookup, not construction.

Parameters

  • p (int) – The prime characteristic of the field \(\mathrm{GF}(p)\).

  • n (int) – The degree \(n\) of the Conway polynomial.

Returns

The degree-\(n\) Conway polynomial \(C_{p,n}\) over \(\mathrm{GF}(p)\).

Return type

galois.Poly

Raises

LookupError – If the Conway polynomial \(C_{p,n}\) is not found in Frank Luebeck’s database.

Warning

If the \(\mathrm{GF}(p)\) field hasn’t previously been created, it will be created in this function since it’s needed for the construction of the return polynomial.

Examples

In [470]: galois.conway_poly(2, 100)
Out[470]: Poly(x^100 + x^57 + x^56 + x^55 + x^52 + x^48 + x^47 + x^46 + x^45 + x^44 + x^43 + x^41 + x^37 + x^36 + x^35 + x^34 + x^31 + x^30 + x^27 + x^25 + x^24 + x^22 + x^20 + x^19 + x^16 + x^15 + x^11 + x^9 + x^8 + x^6 + x^5 + x^3 + 1, GF(2))

In [471]: galois.conway_poly(7, 13)
Out[471]: Poly(x^13 + 6x^2 + 4, GF(7))