-
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 \(n \mid m\).
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.
- search: bool =
- Returns:¶
True
if the polynomial \(f(x)\) is primitive and consistent with smaller Conway polynomials \(C_{p,n}(x)\) for all \(n \mid m\).- Raises:¶
LookupError – If
search=False
and a smaller Conway polynomial \(C_{p,n}\) is not found in Frank Luebeck’s database.
See also
Notes¶
A degree-\(m\) polynomial \(f(x)\) over \(\mathrm{GF}(p)\) is compatible with Conway polynomials \(C_{p,n}(x)\) for \(n \mid m\) 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¶
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))