Univariate Polynomials over GF(p^e) via NTL’s ZZ_pEX

AUTHOR:

• Yann Laigle-Chapuy (2010-01) initial implementation

• Lorenz Panny (2023-01): minpoly_mod()

• Giacomo Pope (2023-08): reverse(), inverse_series_trunc()

class sage.rings.polynomial.polynomial_zz_pex.Polynomial_ZZ_pEX

Bases: Polynomial_template

Univariate Polynomials over \(\GF{p^n}\) via NTL’s ZZ_pEX.

EXAMPLES:

sage: K.<a> = GF(next_prime(2**60)**3)
sage: R.<x> = PolynomialRing(K, implementation='NTL')
sage: (x^3 + a*x^2 + 1) * (x + a)
x^4 + 2*a*x^3 + a^2*x^2 + x + a

inverse_series_trunc(prec)

Compute and return the inverse of self modulo \(x^{prec}\).

The constant term of self must be invertible.

EXAMPLES:

sage: R.<x> = GF(101^2)[]
sage: z2 =  R.base_ring().gen()
sage: f = (3*z2 + 57)*x^3 + (13*z2 + 94)*x^2 + (7*z2 + 2)*x + 66*z2 + 15
sage: f.inverse_series_trunc(1)
51*z2 + 92
sage: f.inverse_series_trunc(2)
(30*z2 + 30)*x + 51*z2 + 92
sage: f.inverse_series_trunc(3)
(42*z2 + 94)*x^2 + (30*z2 + 30)*x + 51*z2 + 92
sage: f.inverse_series_trunc(4)
(99*z2 + 96)*x^3 + (42*z2 + 94)*x^2 + (30*z2 + 30)*x + 51*z2 + 92

is_irreducible(algorithm='fast_when_false', iter=1)

Return True precisely when self is irreducible over its base ring.

INPUT:

• algorithm – a string (default "fast_when_false"), there are 3 available algorithms: "fast_when_true", "fast_when_false", and "probabilistic".

• iter – (default: 1) if the algorithm is "probabilistic", defines the number of iterations. The error probability is bounded by \(q^{\text{-iter}}\) for polynomials in \(\GF{q}[x]\).

EXAMPLES:

sage: K.<a> = GF(next_prime(2**60)**3)
sage: R.<x> = PolynomialRing(K, implementation='NTL')
sage: P = x^3 + (2-a)*x + 1
sage: P.is_irreducible(algorithm="fast_when_false")
True
sage: P.is_irreducible(algorithm="fast_when_true")
True
sage: P.is_irreducible(algorithm="probabilistic")
True
sage: Q = (x^2+a)*(x+a^3)
sage: Q.is_irreducible(algorithm="fast_when_false")
False
sage: Q.is_irreducible(algorithm="fast_when_true")
False
sage: Q.is_irreducible(algorithm="probabilistic")
False

list(copy=True)

Return the list of coefficients.

EXAMPLES:

sage: K.<a> = GF(5^3)
sage: P = PolynomialRing(K, 'x')
sage: f = P.random_element(100)
sage: f.list() == [f[i] for i in range(f.degree()+1)]
True
sage: P.0.list()
[0, 1]

minpoly_mod(other)

Compute the minimal polynomial of this polynomial modulo another polynomial in the same ring.

ALGORITHM:

NTL’s MinPolyMod(), which uses Shoup’s algorithm [Sho1999].

EXAMPLES:

sage: R.<x> = GF(101^2)[]
sage: f = x^17 + x^2 - 1
sage: (x^2).minpoly_mod(f)
x^17 + 100*x^2 + 2*x + 100

resultant(other)

Return the resultant of self and other, which must lie in the same polynomial ring.

INPUT:

• other – a polynomial

OUTPUT: an element of the base ring of the polynomial ring

EXAMPLES:

sage: K.<a> = GF(next_prime(2**60)**3)
sage: R.<x> = PolynomialRing(K, implementation='NTL')
sage: f = (x-a)*(x-a**2)*(x+1)
sage: g = (x-a**3)*(x-a**4)*(x+a)
sage: r = f.resultant(g)
sage: r == prod(u - v for (u,eu) in f.roots() for (v,ev) in g.roots())
True

reverse(degree=None)

Return the polynomial obtained by reversing the coefficients of this polynomial. If degree is set then this function behaves as if this polynomial has degree degree.

EXAMPLES:

sage: R.<x> = GF(101^2)[]
sage: f = x^13 + 11*x^10 + 32*x^6 + 4
sage: f.reverse()
4*x^13 + 32*x^7 + 11*x^3 + 1
sage: f.reverse(degree=15)
4*x^15 + 32*x^9 + 11*x^5 + x^2
sage: f.reverse(degree=2)
4*x^2

shift(n)

EXAMPLES:

sage: K.<a> = GF(next_prime(2**60)**3)
sage: R.<x> = PolynomialRing(K, implementation='NTL')
sage: f = x^3 + x^2 + 1
sage: f.shift(1)
x^4 + x^3 + x
sage: f.shift(-1)
x^2 + x

class sage.rings.polynomial.polynomial_zz_pex.Polynomial_template

Bases: Polynomial

Template for interfacing to external C / C++ libraries for implementations of polynomials.

AUTHORS:

• Robert Bradshaw (2008-10): original idea for templating

• Martin Albrecht (2008-10): initial implementation

This file implements a simple templating engine for linking univariate polynomials to their C/C++ library implementations. It requires a ‘linkage’ file which implements the celement_ functions (see sage.libs.ntl.ntl_GF2X_linkage for an example). Both parts are then plugged together by inclusion of the linkage file when inheriting from this class. See sage.rings.polynomial.polynomial_gf2x for an example.

We illustrate the generic glueing using univariate polynomials over \(\mathop{\mathrm{GF}}(2)\).

Note

Implementations using this template MUST implement coercion from base ring elements and get_unsafe(). See Polynomial_GF2X for an example.

degree()

EXAMPLES:

sage: P.<x> = GF(2)[]
sage: x.degree()
1
sage: P(1).degree()
0
sage: P(0).degree()
-1

gcd(other)

Return the greatest common divisor of self and other.

EXAMPLES:

sage: P.<x> = GF(2)[]
sage: f = x*(x+1)
sage: f.gcd(x+1)
x + 1
sage: f.gcd(x^2)
x

get_cparent()

is_gen()

EXAMPLES:

sage: P.<x> = GF(2)[]
sage: x.is_gen()
True
sage: (x+1).is_gen()
False

is_one()

EXAMPLES:

sage: P.<x> = GF(2)[]
sage: P(1).is_one()
True

is_zero()

EXAMPLES:

sage: P.<x> = GF(2)[]
sage: x.is_zero()
False

list(copy=True)

EXAMPLES:

sage: P.<x> = GF(2)[]
sage: x.list()
[0, 1]
sage: list(x)
[0, 1]

quo_rem(right)

EXAMPLES:

sage: P.<x> = GF(2)[]
sage: f = x^2 + x + 1
sage: f.quo_rem(x + 1)
(x, 1)

shift(n)

EXAMPLES:

sage: P.<x> = GF(2)[]
sage: f = x^3 + x^2 + 1
sage: f.shift(1)
x^4 + x^3 + x
sage: f.shift(-1)
x^2 + x

truncate(n)

Returns this polynomial mod \(x^n\).

EXAMPLES:

sage: R.<x> =GF(2)[]
sage: f = sum(x^n for n in range(10)); f
x^9 + x^8 + x^7 + x^6 + x^5 + x^4 + x^3 + x^2 + x + 1
sage: f.truncate(6)
x^5 + x^4 + x^3 + x^2 + x + 1

If the precision is higher than the degree of the polynomial then the polynomial itself is returned:

sage: f.truncate(10) is f
True

If the precision is negative, the zero polynomial is returned:

sage: f.truncate(-1)
0

xgcd(other)

Computes extended gcd of self and other.

EXAMPLES:

sage: P.<x> = GF(7)[]
sage: f = x*(x+1)
sage: f.xgcd(x+1)
(x + 1, 0, 1)
sage: f.xgcd(x^2)
(x, 1, 6)

sage.rings.polynomial.polynomial_zz_pex.make_element(parent, args)