def computeR(P, Q):
m = P.degree()
n = Q.degree()
t = trace_from_minpoly(P, 2 * m * n + 4)
u = trace_from_minpoly(Q, 2 * m * n + 4)
v = [t[i] * u[i] for i in range(2 * m * n + 4)]
from sage.matrix.berlekamp_massey import berlekamp_massey
BM = berlekamp_massey(v)
return BM
def _composed_product(P, Q):
'composed product of P and Q'
m = P.degree()
n = Q.degree()
t = _trace_from_minpoly(P, 2*m*n+4)
u = _trace_from_minpoly(Q, 2*m*n+4)
v = [t[i]*u[i] for i in range(2*m*n+4)]
BM = berlekamp_massey(v)
return BM
def computeR(P, Q):
m = P.degree()
n = Q.degree()
t = trace_from_minpoly(P, 2*m*n+4)
u = trace_from_minpoly(Q, 2*m*n+4)
v = [t[i]*u[i] for i in range(2*m*n+4)]
from sage.matrix.berlekamp_massey import berlekamp_massey
BM = berlekamp_massey(v)
return BM
def guess(self, sequence, algorithm='sage'):
"""
Return the minimal CFiniteSequence that generates the sequence.
Assume the first value has index 0.
INPUT:
- ``sequence`` -- list of integers
- ``algorithm`` -- string
- 'sage' - the default is to use Sage's matrix kernel function
- 'pari' - use Pari's implementation of LLL
- 'bm' - use Sage's Berlekamp-Massey algorithm
OUTPUT:
- a CFiniteSequence, or 0 if none could be found
With the default kernel method, trailing zeroes are chopped
off before a guessing attempt. This may reduce the data
below the accepted length of six values.
EXAMPLES::
sage: C.<x> = CFiniteSequences(QQ)
sage: C.guess([1,2,4,8,16,32])
C-finite sequence, generated by -1/2/(x - 1/2)
sage: r = C.guess([1,2,3,4,5])
Traceback (most recent call last):
...
ValueError: Sequence too short for guessing.
With Berlekamp-Massey, if an odd number of values is given, the last one is dropped.
So with an odd number of values the result may not generate the last value::
sage: r = C.guess([1,2,4,8,9], algorithm='bm'); r
C-finite sequence, generated by -1/2/(x - 1/2)
sage: r[0:5]
[1, 2, 4, 8, 16]
"""
S = self.polynomial_ring()
if algorithm == 'bm':
from sage.matrix.berlekamp_massey import berlekamp_massey
if len(sequence) < 2:
raise ValueError('Sequence too short for guessing.')
R = PowerSeriesRing(QQ, 'x')
if len(sequence) % 2:
sequence.pop()
l = len(sequence) - 1
denominator = S(berlekamp_massey(sequence).reverse())
numerator = R(S(sequence) * denominator, prec=l).truncate()
return CFiniteSequence(numerator / denominator)
elif algorithm == 'pari':
global _gp
if len(sequence) < 6:
raise ValueError('Sequence too short for guessing.')
if _gp is None:
_gp = Gp()
_gp("ggf(v)=local(l,m,p,q,B);l=length(v);B=floor(l/2);\
if(B<3,return(0));m=matrix(B,B,x,y,v[x-y+B+1]);\
q=qflll(m,4)[1];if(length(q)==0,return(0));\
p=sum(k=1,B,x^(k-1)*q[k,1]);\
q=Pol(Pol(vector(l,n,v[l-n+1]))*p+O(x^(B+1)));\
if(polcoeff(p,0)<0,q=-q;p=-p);q=q/p;p=Ser(q+O(x^(l+1)));\
for(m=1,l,if(polcoeff(p,m-1)!=v[m],return(0)));q")
_gp.set('gf', sequence)
_gp("gf=ggf(gf)")
num = S(sage_eval(_gp.eval("Vec(numerator(gf))"))[::-1])
den = S(sage_eval(_gp.eval("Vec(denominator(gf))"))[::-1])
if num == 0:
return 0
else:
return CFiniteSequence(num / den)
else:
from sage.matrix.constructor import matrix
from sage.functions.other import ceil
from numpy import trim_zeros
seq = sequence[:]
while seq and sequence[-1] == 0:
seq.pop()
l = len(seq)
if l == 0:
return 0
if l < 6:
raise ValueError('Sequence too short for guessing.')
hl = ceil(ZZ(l) / 2)
A = matrix([sequence[k:k + hl] for k in range(hl)])
K = A.kernel()
if K.dimension() == 0:
return 0
R = PolynomialRing(QQ, 'x')
den = R(trim_zeros(K.basis()[-1].list()[::-1]))
if den == 1:
return 0
offset = next((i for i, x in enumerate(sequence) if x), None)
S = PowerSeriesRing(QQ, 'x', default_prec=l - offset)
num = S(R(sequence) * den).truncate(ZZ(l) // 2 + 1)
if num == 0 or sequence != S(num / den).list():
return 0
else:
return CFiniteSequence(num / den)
def guess(self, sequence, algorithm="sage"):
"""
Return the minimal CFiniteSequence that generates the sequence.
Assume the first value has index 0.
INPUT:
- ``sequence`` -- list of integers
- ``algorithm`` -- string
- 'sage' - the default is to use Sage's matrix kernel function
- 'pari' - use Pari's implementation of LLL
- 'bm' - use Sage's Berlekamp-Massey algorithm
OUTPUT:
- a CFiniteSequence, or 0 if none could be found
With the default kernel method, trailing zeroes are chopped
off before a guessing attempt. This may reduce the data
below the accepted length of six values.
EXAMPLES::
sage: C.<x> = CFiniteSequences(QQ)
sage: C.guess([1,2,4,8,16,32])
C-finite sequence, generated by 1/(-2*x + 1)
sage: r = C.guess([1,2,3,4,5])
Traceback (most recent call last):
...
ValueError: Sequence too short for guessing.
With Berlekamp-Massey, if an odd number of values is given, the last one is dropped.
So with an odd number of values the result may not generate the last value::
sage: r = C.guess([1,2,4,8,9], algorithm='bm'); r
C-finite sequence, generated by 1/(-2*x + 1)
sage: r[0:5]
[1, 2, 4, 8, 16]
"""
S = self.polynomial_ring()
if algorithm == "bm":
from sage.matrix.berlekamp_massey import berlekamp_massey
if len(sequence) < 2:
raise ValueError("Sequence too short for guessing.")
R = PowerSeriesRing(QQ, "x")
if len(sequence) % 2 == 1:
sequence = sequence[:-1]
l = len(sequence) - 1
denominator = S(berlekamp_massey(sequence).list()[::-1])
numerator = R(S(sequence) * denominator, prec=l).truncate()
return CFiniteSequence(numerator / denominator)
elif algorithm == "pari":
global _gp
if len(sequence) < 6:
raise ValueError("Sequence too short for guessing.")
if _gp is None:
_gp = Gp()
_gp(
"ggf(v)=local(l,m,p,q,B);l=length(v);B=floor(l/2);\
if(B<3,return(0));m=matrix(B,B,x,y,v[x-y+B+1]);\
q=qflll(m,4)[1];if(length(q)==0,return(0));\
p=sum(k=1,B,x^(k-1)*q[k,1]);\
q=Pol(Pol(vector(l,n,v[l-n+1]))*p+O(x^(B+1)));\
if(polcoeff(p,0)<0,q=-q;p=-p);q=q/p;p=Ser(q+O(x^(l+1)));\
for(m=1,l,if(polcoeff(p,m-1)!=v[m],return(0)));q"
)
_gp.set("gf", sequence)
_gp("gf=ggf(gf)")
num = S(sage_eval(_gp.eval("Vec(numerator(gf))"))[::-1])
den = S(sage_eval(_gp.eval("Vec(denominator(gf))"))[::-1])
if num == 0:
return 0
else:
return CFiniteSequence(num / den)
else:
from sage.matrix.constructor import matrix
from sage.functions.other import floor, ceil
from numpy import trim_zeros
l = len(sequence)
while l > 0 and sequence[l - 1] == 0:
l -= 1
sequence = sequence[:l]
if l == 0:
return 0
if l < 6:
raise ValueError("Sequence too short for guessing.")
hl = ceil(ZZ(l) / 2)
A = matrix([sequence[k : k + hl] for k in range(hl)])
K = A.kernel()
if K.dimension() == 0:
return 0
R = PolynomialRing(QQ, "x")
den = R(trim_zeros(K.basis()[-1].list()[::-1]))
if den == 1:
return 0
offset = next((i for i, x in enumerate(sequence) if x != 0), None)
S = PowerSeriesRing(QQ, "x", default_prec=l - offset)
num = S(R(sequence) * den).add_bigoh(floor(ZZ(l) / 2 + 1)).truncate()
if num == 0 or sequence != S(num / den).list():
return 0
else:
return CFiniteSequence(num / den)
N = 128
m = 8
F2 = GF(2)
XOR = lambda s1,s2: bytes([x^y for x,y in zip(s1,s2)])
BIT2BYTE = lambda l: bytes(ZZ(l[::-1], base=2).digits(base=256))[::-1]
BYTE2BIT = lambda b: ZZ(list(b[::-1]), base=256).digits(base=2, padto=8*len(b))[::-1]
cipher = open('msg.enc', 'rb').read()
cipherstream = BYTE2BIT(cipher)
BITS_LENGTH = len(cipherstream)
c = [F2(cipherstream[m*i]) for i in range(2*N)]
Pol = berlekamp_massey(c)
per = 2**Pol.degree() - 1
m_inv = inverse_mod(m, per)
T = companion_matrix(Pol, format='right')
TT = T**m_inv
cc = Matrix(F2, c[:N])
bb = []
for _ in range(2*N):
bb.append( cc[0, 0] )
cc = cc*TT
fill = bb[:N]
gg = berlekamp_massey(bb)
key = gg.coefficients(sparse=False)[:-1]
def guess(sequence, algorithm='pari'):
"""
Return the minimal CFiniteSequence that generates the sequence. Assume the first
value has index 0.
INPUT:
- ``sequence`` -- list of integers
- ``algorithm`` -- string
- 'pari' - use Pari's implementation of LLL (fast)
- 'bm' - use Sage's Berlekamp-Massey algorithm
OUTPUT:
- a CFiniteSequence, or 0 if none could be found
EXAMPLES::
sage: CFiniteSequence.guess([1,2,4,8,16,32])
C-finite sequence, generated by 1/(-2*x + 1)
sage: r = CFiniteSequence.guess([1,2,3,4,5])
Traceback (most recent call last):
...
ValueError: Sequence too short for guessing.
sage: CFiniteSequence.guess([1,0,0,0,0,0])
Finite sequence [1], offset = 0
With Pari LLL, all values are taken into account, and if no o.g.f.
can be found, `0` is returned::
sage: CFiniteSequence.guess([1,0,0,0,0,1])
0
With Berlekamp-Massey, if an odd number of values is given, the last one is dropped.
So with an odd number of values the result may not generate the last value::
sage: r = CFiniteSequence.guess([1,2,4,8,9], algorithm='bm'); r
C-finite sequence, generated by 1/(-2*x + 1)
sage: r[0:5]
[1, 2, 4, 8, 16]
"""
S = PolynomialRing(QQ, 'x')
if algorithm == 'bm':
if len(sequence) < 2:
raise ValueError('Sequence too short for guessing.')
R = PowerSeriesRing(QQ, 'x')
if len(sequence) % 2 == 1: sequence = sequence[:-1]
l = len(sequence) - 1
denominator = S(berlekamp_massey(sequence).list()[::-1])
numerator = R(S(sequence) * denominator, prec=l).truncate()
return CFiniteSequence(numerator / denominator)
else:
global _gp
if len(sequence) < 6:
raise ValueError('Sequence too short for guessing.')
if _gp is None:
_gp = Gp()
_gp("ggf(v)=local(l,m,p,q,B);l=length(v);B=floor(l/2);\
if(B<3,return(0));m=matrix(B,B,x,y,v[x-y+B+1]);\
q=qflll(m,4)[1];if(length(q)==0,return(0));\
p=sum(k=1,B,x^(k-1)*q[k,1]);\
q=Pol(Pol(vector(l,n,v[l-n+1]))*p+O(x^(B+1)));\
if(polcoeff(p,0)<0,q=-q;p=-p);q=q/p;p=Ser(q+O(x^(l+1)));\
for(m=1,l,if(polcoeff(p,m-1)!=v[m],return(0)));q")
_gp.set('gf', sequence)
_gp("gf=ggf(gf)")
num = S(sage_eval(_gp.eval("Vec(numerator(gf))"))[::-1])
den = S(sage_eval(_gp.eval("Vec(denominator(gf))"))[::-1])
if num == 0:
return 0
else:
return CFiniteSequence(num / den)
"""