def lfsr_connection_polynomial(s):
"""
INPUT:
- ``s`` -- a sequence of elements of a finite field of even length
OUTPUT:
- ``C(x)`` -- the connection polynomial of the minimal LFSR.
This implements the algorithm in section 3 of J. L. Massey's article
[Mas1969]_.
EXAMPLES::
sage: F = GF(2)
sage: F
Finite Field of size 2
sage: o = F(0); l = F(1)
sage: key = [l,o,o,l]; fill = [l,l,o,l]; n = 20
sage: s = lfsr_sequence(key,fill,n); s
[1, 1, 0, 1, 0, 1, 1, 0, 0, 1, 0, 0, 0, 1, 1, 1, 1, 0, 1, 0]
sage: lfsr_connection_polynomial(s)
x^4 + x + 1
sage: from sage.matrix.berlekamp_massey import berlekamp_massey
sage: berlekamp_massey(s)
x^4 + x^3 + 1
Notice that ``berlekamp_massey`` returns the reverse of the connection
polynomial (and is potentially must faster than this implementation).
"""
# Initialization:
FF = s[0].base_ring()
R = PolynomialRing(FF, "x")
x = R.gen()
C = R.one()
B = R.one()
m = 1
b = FF.one()
L = 0
N = 0
while N < len(s):
if L > 0:
r = min(L + 1, C.degree() + 1)
d = s[N] + sum([(C.list())[i] * s[N - i] for i in range(1, r)])
if L == 0:
d = s[N]
if d == 0:
m += 1
N += 1
if d > 0:
if 2 * L > N:
C = C - d * b**(-1) * x**m * B
m += 1
N += 1
else:
T = C
C = C - d * b**(-1) * x**m * B
L = N + 1 - L
m = 1
b = d
B = T
N += 1
return C
def __call__(self, P):
r"""
Return a rational point P in the abstract Homset J(K), given:
0. A point P in J = Jac(C), returning P;
1. A point P on the curve C such that J = Jac(C), where C is
an odd degree model, returning [P - oo];
2. A pair of points (P, Q) on the curve C such that J = Jac(C),
returning [P-Q];
3. A list of polynomials (a,b) such that `b^2 + h*b - f = 0 mod a`,
returning [(a(x),y-b(x))].
EXAMPLES::
sage: P.<x> = PolynomialRing(QQ)
sage: f = x^5 - x + 1; h = x
sage: C = HyperellipticCurve(f,h,'u,v')
sage: P = C(0,1,1)
sage: J = C.jacobian()
sage: Q = J(QQ)(P)
sage: for i in range(6): i*Q
(1)
(u, v - 1)
(u^2, v + u - 1)
(u^2, v + 1)
(u, v + 1)
(1)
::
sage: F.<a> = GF(3)
sage: R.<x> = F[]
sage: f = x^5-1
sage: C = HyperellipticCurve(f)
sage: J = C.jacobian()
sage: X = J(F)
sage: a = x^2-x+1
sage: b = -x +1
sage: c = x-1
sage: d = 0
sage: D1 = X([a,b])
sage: D1
(x^2 + 2*x + 1, y + x + 2)
sage: D2 = X([c,d])
sage: D2
(x + 2, y)
sage: D1+D2
(x^2 + 2*x + 2, y + 2*x + 1)
"""
if isinstance(P, (Integer, int)) and P == 0:
R = PolynomialRing(self.value_ring(), 'x')
return JacobianMorphism_divisor_class_field(
self, (R.one(), R.zero()))
elif isinstance(P, (list, tuple)):
if len(P) == 1 and P[0] == 0:
R = PolynomialRing(self.value_ring(), 'x')
return JacobianMorphism_divisor_class_field(
self, (R.one(), R.zero()))
elif len(P) == 2:
P1 = P[0]
P2 = P[1]
if is_Integer(P1) and is_Integer(P2):
R = PolynomialRing(self.value_ring(), 'x')
P1 = R(P1)
P2 = R(P2)
return JacobianMorphism_divisor_class_field(self, (P1, P2))
if is_Integer(P1) and is_Polynomial(P2):
R = PolynomialRing(self.value_ring(), 'x')
P1 = R(P1)
return JacobianMorphism_divisor_class_field(self, (P1, P2))
if is_Integer(P2) and is_Polynomial(P1):
R = PolynomialRing(self.value_ring(), 'x')
P2 = R(P2)
return JacobianMorphism_divisor_class_field(self, (P1, P2))
if is_Polynomial(P1) and is_Polynomial(P2):
return JacobianMorphism_divisor_class_field(self, tuple(P))
if is_SchemeMorphism(P1) and is_SchemeMorphism(P2):
return self(P1) - self(P2)
raise TypeError("argument P (= %s) must have length 2" % P)
elif isinstance(
P,
JacobianMorphism_divisor_class_field) and self == P.parent():
return P
elif is_SchemeMorphism(P):
x0 = P[0]
y0 = P[1]
R, x = PolynomialRing(self.value_ring(), 'x').objgen()
return self((x - x0, R(y0)))
raise TypeError(
"argument P (= %s) does not determine a divisor class" % P)