galois.Poly.is_conway

galois.Poly.is_conway(search: bool = False) → bool

Checks whether the degree- $m$ polynomial $f (x)$ over $GF (p)$ is the Conway polynomial $C_{p, m} (x)$ .

Why is this a method and not a property?

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

Parameters:¶

search: bool = False¶

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 the Conway polynomial $C_{p, m} (x)$ .

Raises:¶

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

Notes

A degree- $m$ polynomial $f (x)$ over $GF (p)$ is the Conway polynomial $C_{p, m} (x)$ if it is monic, primitive, compatible with Conway polynomials $C_{p, n} (x)$ for all $n ∣ m$ , and is lexicographically first according to a special ordering.

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

The Conway lexicographic ordering is defined as follows. Given two degree- $m$ polynomials $g (x) = \sum_{i = 0}^{m} g_{i} x^{i}$ and $h (x) = \sum_{i = 0}^{m} h_{i} x^{i}$ , then $g < h$ if and only if there exists $i$ such that $g_{j} = h_{j}$ for all $j > i$ and $(- 1)^{m - i} g_{i} < (- 1)^{m - i} h_{i}$ .

References

http://www.math.rwth-aachen.de/~Frank.Luebeck/data/ConwayPol/CP7.html
Lenwood S. Heath, Nicholas A. Loehr, New algorithms for generating Conway polynomials over finite fields, Journal of Symbolic Computation, Volume 38, Issue 2, 2004, Pages 1003-1024, https://www.sciencedirect.com/science/article/pii/S0747717104000331.

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))