galois.Poly.is_conway_consistent(search: bool = False) bool

Determines whether the degree-$$m$$ polynomial $$f(x)$$ over $$\mathrm{GF}(p)$$ is consistent with smaller Conway polynomials $$C_{p,n}(x)$$ for all $n\ |m$.

Why is this a method and not a property?

This is a method to indicate it is a computationally-expensive task.

Parameters:

Manually search for Conway polynomials if they are not included in Frank Luebeck’s database. The default is False.

Slower performance

Manually searching for a Conway polynomial is very computationally expensive.

Returns:

True if the polynomial $$f(x)$$ is primitive and consistent with smaller Conway polynomials $$C_{p,n}(x)$$ for all $n\ |m$.

Raises:

LookupError – If search=False and a smaller Conway polynomial $$C_{p,n}$$ is not found in Frank Luebeck’s database.

Notes

A degree-$$m$$ polynomial $$f(x)$$ over $$\mathrm{GF}(p)$$ is compatible with Conway polynomials $$C_{p,n}(x)$$ for $n\ |\ m$ if $$C_{p,n}(x^r)$$ divides $$f(x)$$, where $$r = \frac{p^m - 1}{p^n - 1}$$.

A Conway-consistent polynomial has all the properties of a Conway polynomial except that it is not necessarily lexicographically first (according to a special ordering).

References

Examples

All Conway polynomials are primitive.

In [1]: GF = galois.GF(7)

In [2]: f = galois.Poly([1, 1, 2, 4], field=GF); f
Out[2]: Poly(x^3 + x^2 + 2x + 4, GF(7))

In [3]: g = galois.Poly([1, 6, 0, 4], field=GF); g
Out[3]: Poly(x^3 + 6x^2 + 4, GF(7))

In [4]: f.is_primitive()
Out[4]: True

In [5]: g.is_primitive()
Out[5]: True


They are also consistent with all smaller Conway polynomials.

In [6]: f.is_conway_consistent()
Out[6]: True

In [7]: g.is_conway_consistent()
Out[7]: True


Among the multiple candidate Conway polynomials, the lexicographically-first (accordingly to a special lexicographical order) is the Conway polynomial.

In [8]: f.is_conway()
Out[8]: False

In [9]: g.is_conway()
Out[9]: True

In [10]: galois.conway_poly(7, 3)
Out[10]: Poly(x^3 + 6x^2 + 4, GF(7))