Constructs a random polynomial over $\mathrm{GF}(p^m)$ with degree $d$.

Parameters:¶

degree: int¶

The degree of the polynomial.

seed: int | numpy.integer | numpy.random._generator.Generator | None = None¶

Non-negative integer used to initialize the PRNG. The default is None which means that unpredictable entropy will be pulled from the OS to be used as the seed. A numpy.random.Generator can also be passed.

field: type[Array] | None = None¶

The Galois field $\mathrm{GF}(p^m)$ the polynomial is over. The default is None which corresponds to GF2.

Returns:¶

The polynomial $f(x)$.

Examples¶

Construct a random degree-5 polynomial over $\mathrm{GF}(2)$.

In [1]: galois.Poly.Random(5)
Out[1]: Poly(x^5 + x^3 + x^2 + 1, GF(2))

Construct a random degree-5 polynomial over $\mathrm{GF}(3^5)$ with a given seed. This produces repeatable results.

In [2]: GF = galois.GF(3**5)

In [3]: galois.Poly.Random(5, seed=123456789, field=GF)
Out[3]: Poly(56x^5 + 228x^4 + 157x^3 + 218x^2 + 148x + 43, GF(3^5))

In [4]: galois.Poly.Random(5, seed=123456789, field=GF)
Out[4]: Poly(56x^5 + 228x^4 + 157x^3 + 218x^2 + 148x + 43, GF(3^5))

Construct multiple polynomials with one global seed.

In [5]: rng = np.random.default_rng(123456789)

In [6]: galois.Poly.Random(5, seed=rng, field=GF)
Out[6]: Poly(56x^5 + 228x^4 + 157x^3 + 218x^2 + 148x + 43, GF(3^5))

In [7]: galois.Poly.Random(5, seed=rng, field=GF)
Out[7]: Poly(194x^5 + 195x^4 + 200x^3 + 141x^2 + 164x + 119, GF(3^5))