Routines for Conway and pseudo-Conway polynomials¶

AUTHORS:

• David Roe

• Jean-Pierre Flori

• Peter Bruin

class sage.rings.finite_rings.conway_polynomials.PseudoConwayLattice(p, use_database=True)

A pseudo-Conway lattice over a given finite prime field.

The Conway polynomial $$f_n$$ of degree $$n$$ over $$\Bold{F}_p$$ is defined by the following four conditions:

• $$f_n$$ is irreducible.

• In the quotient field $$\Bold{F}_p[x]/(f_n)$$, the element $$x\bmod f_n$$ generates the multiplicative group.

• The minimal polynomial of $$(x\bmod f_n)^{\frac{p^n-1}{p^m-1}}$$ equals the Conway polynomial $$f_m$$, for every divisor $$m$$ of $$n$$.

• $$f_n$$ is lexicographically least among all such polynomials, under a certain ordering.

The final condition is needed only in order to make the Conway polynomial unique. We define a pseudo-Conway lattice to be any family of polynomials, indexed by the positive integers, satisfying the first three conditions.

INPUT:

• p – prime number

• use_database – boolean. If True, use actual Conway polynomials whenever they are available in the database. If False, always compute pseudo-Conway polynomials.

EXAMPLES:

sage: from sage.rings.finite_rings.conway_polynomials import PseudoConwayLattice
sage: PCL = PseudoConwayLattice(2, use_database=False)
sage: PCL.polynomial(3)
x^3 + x + 1
check_consistency(n)

Check that the pseudo-Conway polynomials of degree dividing $$n$$ in this lattice satisfy the required compatibility conditions.

EXAMPLES:

sage: from sage.rings.finite_rings.conway_polynomials import PseudoConwayLattice
sage: PCL = PseudoConwayLattice(2, use_database=False)
sage: PCL.check_consistency(6)
sage: PCL.check_consistency(60)  # long time
polynomial(n)

Return the pseudo-Conway polynomial of degree $$n$$ in this lattice.

INPUT:

• n – positive integer

OUTPUT:

• a pseudo-Conway polynomial of degree $$n$$ for the prime $$p$$.

ALGORITHM:

Uses an algorithm described in [HL1999], modified to find pseudo-Conway polynomials rather than Conway polynomials. The major difference is that we stop as soon as we find a primitive polynomial.

EXAMPLES:

sage: from sage.rings.finite_rings.conway_polynomials import PseudoConwayLattice
sage: PCL = PseudoConwayLattice(2, use_database=False)
sage: PCL.polynomial(3)
x^3 + x + 1
sage: PCL.polynomial(4)
x^4 + x^3 + 1
sage: PCL.polynomial(60)
x^60 + x^59 + x^58 + x^55 + x^54 + x^53 + x^52 + x^51 + x^48 + x^46 + x^45 + x^42 + x^41 + x^39 + x^38 + x^37 + x^35 + x^32 + x^31 + x^30 + x^28 + x^24 + x^22 + x^21 + x^18 + x^17 + x^16 + x^15 + x^14 + x^10 + x^8 + x^7 + x^5 + x^3 + x^2 + x + 1
sage.rings.finite_rings.conway_polynomials.conway_polynomial(p, n)

Return the Conway polynomial of degree $$n$$ over GF(p).

If the requested polynomial is not known, this function raises a RuntimeError exception.

INPUT:

• p – prime number

• n – positive integer

OUTPUT:

• the Conway polynomial of degree $$n$$ over the finite field GF(p), loaded from a table.

Note

The first time this function is called a table is read from disk, which takes a fraction of a second. Subsequent calls do not require reloading the table.

See also the ConwayPolynomials() object, which is the table of Conway polynomials used by this function.

EXAMPLES:

sage: conway_polynomial(2,5)
x^5 + x^2 + 1
sage: conway_polynomial(101,5)
x^5 + 2*x + 99
sage: conway_polynomial(97,101)
Traceback (most recent call last):
...
RuntimeError: requested Conway polynomial not in database.
sage.rings.finite_rings.conway_polynomials.exists_conway_polynomial(p, n)

Check whether the Conway polynomial of degree $$n$$ over GF(p) is known.

INPUT:

• p – prime number

• n – positive integer

OUTPUT:

• boolean: True if the Conway polynomial of degree $$n$$ over GF(p) is in the database, False otherwise.

If the Conway polynomial is in the database, it can be obtained using the command conway_polynomial(p,n).

EXAMPLES:

sage: exists_conway_polynomial(2,3)
True
sage: exists_conway_polynomial(2,-1)
False
sage: exists_conway_polynomial(97,200)
False
sage: exists_conway_polynomial(6,6)
False