Univariate Polynomials over GF(2) via NTL’s GF2X.

AUTHOR: - Martin Albrecht (2008-10) initial implementation

class sage.rings.polynomial.polynomial_gf2x.Polynomial_GF2X

Univariate Polynomials over GF(2) via NTL’s GF2X.

EXAMPLE:

sage: P.<x> = GF(2)[]
sage: x^3 + x^2 + 1
x^3 + x^2 + 1
__getitem__()

EXAMPLE:

sage: P.<x> = GF(2)[]
sage: f = x^3 + x^2 + 1; f
x^3 + x^2 + 1
sage: f[0]
1
sage: f[1]
0
__init__()
x.__init__(...) initializes x; see x.__class__.__doc__ for signature
static __new__()
T.__new__(S, ...) -> a new object with type S, a subtype of T
_pari_()

EXAMPLE:

sage: P.<x> = GF(2)[]
sage: f = x^3 + x^2 + 1
sage: pari(f)
Mod(1, 2)*x^3 + Mod(1, 2)*x^2 + Mod(1, 2)
is_irreducible()

Return True precisely if this polynomial is irreducible over GF(2).

EXAMPLES:

sage: R.<x> = GF(2)[]
sage: (x^2 + 1).is_irreducible()
False
sage: (x^3 + x + 1).is_irreducible()
True
modular_composition()

Compute f(g) \pmod h.

Both implementations use Brent-Kung’s Algorithm 2.1 (Fast Algorithms for Manipulation of Formal Power Series, JACM 1978).

INPUT:

  • g – a polynomial
  • h – a polynomial
  • algorithm – either ‘native’ or ‘ntl’ (default: ‘native’)

EXAMPLE:

sage: P.<x> = GF(2)[]
sage: r = 279
sage: f = x^r + x +1
sage: g = x^r
sage: g.modular_composition(g, f) == g(g) % f
True

sage: P.<x> = GF(2)[]
sage: f = x^29 + x^24 + x^22 + x^21 + x^20 + x^16 + x^15 + x^14 + x^10 + x^9 + x^8 + x^7 + x^6 + x^5 + x^2
sage: g = x^31 + x^30 + x^28 + x^26 + x^24 + x^21 + x^19 + x^18 + x^11 + x^10 + x^9 + x^8 + x^5 + x^2 + 1
sage: h = x^30 + x^28 + x^26 + x^25 + x^24 + x^22 + x^21 + x^18 + x^17 + x^15 + x^13 + x^12 + x^11 + x^10 + x^9 + x^4
sage: f.modular_composition(g,h) == f(g) % h
True

AUTHORS:

  • Paul Zimmermann (2008-10) initial implementation
  • Martin Albrecht (2008-10) performance improvements
class sage.rings.polynomial.polynomial_gf2x.Polynomial_template

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 __getitem__. See Polynomial_GF2X for an example.

__copy__()

EXAMPLE:

sage: P.<x> = GF(2)[]
sage: copy(x) is x
False
sage: copy(x) == x
True
__eq__()
x.__eq__(y) <==> x==y
__floordiv__()

EXAMPLE:

sage: P.<x> = GF(2)[]
sage: x//(x + 1)
1
sage: (x + 1)//x
1
__ge__()
x.__ge__(y) <==> x>=y
__getslice__()

x.__getslice__(i, j) <==> x[i:j]

Use of negative indices is not supported.

__gt__()
x.__gt__(y) <==> x>y
__hash__()
x.__hash__() <==> hash(x)
__init__()
x.__init__(...) initializes x; see x.__class__.__doc__ for signature
__int__()
x.__int__() <==> int(x)
__le__()
x.__le__(y) <==> x<=y
__long__()

EXAMPLE:

sage: P.<x> = GF(2)[]
sage: int(x)
...
ValueError: Cannot coerce polynomial with degree 1 to integer.

sage: int(P(1))
1
__lshift__()

EXAMPLE:

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

EXAMPLE:

sage: P.<x> = GF(2)[]
sage: (x^2 + 1) % x^2
1
__ne__()
x.__ne__(y) <==> x!=y
__neg__()

EXAMPLE:

sage: P.<x> = GF(2)[]
sage: -x
x
static __new__()
T.__new__(S, ...) -> a new object with type S, a subtype of T
__nonzero__()
x.__nonzero__() <==> x != 0
__pow__()
x.__pow__(y[, z]) <==> pow(x, y[, z])
__reduce__()

EXAMPLE:

sage: P.<x> = GF(2)[]
sage: loads(dumps(x)) == x
True
__rfloordiv__()
x.__rfloordiv__(y) <==> y//x
__rlshift__()
x.__rlshift__(y) <==> y<<x
__rmod__()
x.__rmod__(y) <==> y%x
__rpow__()
y.__rpow__(x[, z]) <==> pow(x, y[, z])
__rrshift__()
x.__rrshift__(y) <==> y>>x
__rshift__()

EXAMPLE:

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

EXAMPLE:

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

Returns the formal derivative of self with respect to var.

var must be either the generator of the polynomial ring to which this polynomial belongs, or None (either way the behaviour is the same).

See also

derivative()

EXAMPLES:

sage: R.<x> = Integers(77)[]
sage: f = x^4 - x - 1
sage: f._derivative()
4*x^3 + 76
sage: f._derivative(None)
4*x^3 + 76

sage: f._derivative(2*x)
...
ValueError: cannot differentiate with respect to 2*x

sage: y = var("y")
sage: f._derivative(y)
...
ValueError: cannot differentiate with respect to y
_mul_()

EXAMPLE:

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

Return Singular representation of this polynomial

INPUT:

  • singular – Singular interpreter (default: default interpreter)
  • have_ring – set to True if the ring was already set in Singular

EXAMPLE:

sage: P.<x> = PolynomialRing(GF(7))
sage: f = 3*x^2 + 2*x + 5
sage: singular(f)
3*x^2+2*x-2
_sub_()

EXAMPLE:

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

EXAMPLE:

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

Return the greatest common divisor of self and other.

EXAMPLE:

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

EXAMPLE:

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

EXAMPLE:

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

EXAMPLE:

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

EXAMPLE:

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

EXAMPLE:

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

EXAMPLE:

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()

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
xgcd()

Computes extended gcd of self and other.

EXAMPLE:

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_gf2x.make_element()

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