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)[source]¶
- Bases: - WithEqualityById,- SageObject- 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: # needs sage.rings.finite_rings sage: from sage.rings.finite_rings.conway_polynomials import PseudoConwayLattice sage: PCL = PseudoConwayLattice(2, use_database=False) sage: PCL.polynomial(3) # random x^3 + x + 1 - >>> from sage.all import * >>> # needs sage.rings.finite_rings >>> from sage.rings.finite_rings.conway_polynomials import PseudoConwayLattice >>> PCL = PseudoConwayLattice(Integer(2), use_database=False) >>> PCL.polynomial(Integer(3)) # random x^3 + x + 1 - check_consistency(n)[source]¶
- Check that the pseudo-Conway polynomials of degree dividing \(n\) in this lattice satisfy the required compatibility conditions. - EXAMPLES: - sage: # needs sage.rings.finite_rings 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 - >>> from sage.all import * >>> # needs sage.rings.finite_rings >>> from sage.rings.finite_rings.conway_polynomials import PseudoConwayLattice >>> PCL = PseudoConwayLattice(Integer(2), use_database=False) >>> PCL.check_consistency(Integer(6)) >>> PCL.check_consistency(Integer(60)) # long time 
 - polynomial(n)[source]¶
- 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: # needs sage.rings.finite_rings sage: from sage.rings.finite_rings.conway_polynomials import PseudoConwayLattice sage: PCL = PseudoConwayLattice(2, use_database=False) sage: PCL.polynomial(3) # random x^3 + x + 1 sage: PCL.polynomial(4) # random x^4 + x^3 + 1 sage: PCL.polynomial(60) # random 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 - >>> from sage.all import * >>> # needs sage.rings.finite_rings >>> from sage.rings.finite_rings.conway_polynomials import PseudoConwayLattice >>> PCL = PseudoConwayLattice(Integer(2), use_database=False) >>> PCL.polynomial(Integer(3)) # random x^3 + x + 1 >>> PCL.polynomial(Integer(4)) # random x^4 + x^3 + 1 >>> PCL.polynomial(Integer(60)) # random 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)[source]¶
- Return the Conway polynomial of degree \(n\) over - GF(p).- If the requested polynomial is not known, this function raises a - RuntimeErrorexception.- 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) # needs conway_polynomials x^5 + x^2 + 1 sage: conway_polynomial(101,5) # needs conway_polynomials x^5 + 2*x + 99 sage: conway_polynomial(97,101) # needs conway_polynomials Traceback (most recent call last): ... RuntimeError: requested Conway polynomial not in database. - >>> from sage.all import * >>> conway_polynomial(Integer(2),Integer(5)) # needs conway_polynomials x^5 + x^2 + 1 >>> conway_polynomial(Integer(101),Integer(5)) # needs conway_polynomials x^5 + 2*x + 99 >>> conway_polynomial(Integer(97),Integer(101)) # needs conway_polynomials Traceback (most recent call last): ... RuntimeError: requested Conway polynomial not in database. 
- sage.rings.finite_rings.conway_polynomials.exists_conway_polynomial(p, n)[source]¶
- Check whether the Conway polynomial of degree \(n\) over - GF(p)is known.- INPUT: - p– prime number
- n– positive integer
 - OUTPUT: - boolean: - Trueif the Conway polynomial of degree \(n\) over- GF(p)is in the database,- Falseotherwise.
 - 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) # needs conway_polynomials True sage: exists_conway_polynomial(2,-1) False sage: exists_conway_polynomial(97,200) False sage: exists_conway_polynomial(6,6) False - >>> from sage.all import * >>> exists_conway_polynomial(Integer(2),Integer(3)) # needs conway_polynomials True >>> exists_conway_polynomial(Integer(2),-Integer(1)) False >>> exists_conway_polynomial(Integer(97),Integer(200)) False >>> exists_conway_polynomial(Integer(6),Integer(6)) False