galois.conway_poly¶
- galois.conway_poly(characteristic: int, degree: int) Poly¶
Returns the Conway polynomial \(C_{p,m}(x)\) over \(\mathrm{GF}(p)\) with degree \(m\).
- Parameters
- characteristic
The prime characteristic \(p\) of the field \(\mathrm{GF}(p)\) that the polynomial is over.
- degree
The degree \(m\) of the Conway polynomial.
- Returns
The degree-\(m\) Conway polynomial \(C_{p,m}(x)\) over \(\mathrm{GF}(p)\).
See also
- Raises
LookupError – If the Conway polynomial \(C_{p,m}(x)\) is not found in Frank Luebeck’s database.
Notes
A Conway polynomial is an irreducible and primitive polynomial over \(\mathrm{GF}(p)\) that provides a standard representation of \(\mathrm{GF}(p^m)\) as a splitting field of \(C_{p,m}(x)\). Conway polynomials provide compatability between fields and their subfields and, hence, are the common way to represent extension fields.
The Conway polynomial \(C_{p,m}(x)\) is defined as the lexicographically-minimal monic primitive polynomial of degree \(m\) over \(\mathrm{GF}(p)\) that is compatible with all \(C_{p,n}(x)\) for \(n\) dividing \(m\).
This function uses Frank Luebeck’s Conway polynomial database for fast lookup, not construction.
Examples
Notice
primitive_poly()returns the lexicographically-minimal primitive polynomial butconway_poly()returns the lexicographically-minimal primitive polynomial that is consistent with smaller Conway polynomials.This is sometimes the same polynomial.
In [1]: galois.primitive_poly(2, 4) Out[1]: Poly(x^4 + x + 1, GF(2)) In [2]: galois.conway_poly(2, 4) Out[2]: Poly(x^4 + x + 1, GF(2))However, it is not always.
In [3]: galois.primitive_poly(7, 10) Out[3]: Poly(x^10 + 5x^2 + x + 5, GF(7)) In [4]: galois.conway_poly(7, 10) Out[4]: Poly(x^10 + x^6 + x^5 + 4x^4 + x^3 + 2x^2 + 3x + 3, GF(7))