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| import time
debug = True
strict = False
helpful_only = True
dimension_min = 7
def helpful_vectors(BB, modulus):
nothelpful = 0
for ii in range(BB.dimensions()[0]):
if BB[ii,ii] >= modulus:
nothelpful += 1
print( nothelpful, "/", BB.dimensions()[0], " vectors are not helpful")
def matrix_overview(BB, bound):
for ii in range(BB.dimensions()[0]):
a = ('%02d ' % ii)
for jj in range(BB.dimensions()[1]):
a += '0' if BB[ii,jj] == 0 else 'X'
if BB.dimensions()[0] < 60:
a += ' '
if BB[ii, ii] >= bound:
a += '~'
print( a)
def remove_unhelpful(BB, monomials, bound, current):
if current == -1 or BB.dimensions()[0] <= dimension_min:
return BB
for ii in range(current, -1, -1):
if BB[ii, ii] >= bound:
affected_vectors = 0
affected_vector_index = 0
for jj in range(ii + 1, BB.dimensions()[0]):
if BB[jj, ii] != 0:
affected_vectors += 1
affected_vector_index = jj
if affected_vectors == 0:
print( "* removing unhelpful vector", ii)
BB = BB.delete_columns([ii])
BB = BB.delete_rows([ii])
monomials.pop(ii)
BB = remove_unhelpful(BB, monomials, bound, ii-1)
return BB
elif affected_vectors == 1:
affected_deeper = True
for kk in range(affected_vector_index + 1, BB.dimensions()[0]):
if BB[kk, affected_vector_index] != 0:
affected_deeper = False
if affected_deeper and abs(bound - BB[affected_vector_index, affected_vector_index]) < abs(bound - BB[ii, ii]):
print( "* removing unhelpful vectors", ii, "and", affected_vector_index)
BB = BB.delete_columns([affected_vector_index, ii])
BB = BB.delete_rows([affected_vector_index, ii])
monomials.pop(affected_vector_index)
monomials.pop(ii)
BB = remove_unhelpful(BB, monomials, bound, ii-1)
return BB
return BB
def boneh_durfee(pol, modulus, mm, tt, XX, YY):
PR.<u, x, y> = PolynomialRing(ZZ)
Q = PR.quotient(x*y + 1 - u)
polZ = Q(pol).lift()
UU = XX*YY + 1
gg = []
for kk in range(mm + 1):
for ii in range(mm - kk + 1):
xshift = x^ii * modulus^(mm - kk) * polZ(u, x, y)^kk
gg.append(xshift)
gg.sort()
monomials = []
for polynomial in gg:
for monomial in polynomial.monomials():
if monomial not in monomials:
monomials.append(monomial)
monomials.sort()
for jj in range(1, tt + 1):
for kk in range(floor(mm/tt) * jj, mm + 1):
yshift = y^jj * polZ(u, x, y)^kk * modulus^(mm - kk)
yshift = Q(yshift).lift()
gg.append(yshift)
for jj in range(1, tt + 1):
for kk in range(floor(mm/tt) * jj, mm + 1):
monomials.append(u^kk * y^jj)
nn = len(monomials)
BB = Matrix(ZZ, nn)
for ii in range(nn):
BB[ii, 0] = gg[ii](0, 0, 0)
for jj in range(1, ii + 1):
if monomials[jj] in gg[ii].monomials():
BB[ii, jj] = gg[ii].monomial_coefficient(monomials[jj]) * monomials[jj](UU,XX,YY)
if helpful_only:
BB = remove_unhelpful(BB, monomials, modulus^mm, nn-1)
nn = BB.dimensions()[0]
if nn == 0:
print( "failure")
return 0,0
if debug:
helpful_vectors(BB, modulus^mm)
det = BB.det()
bound = modulus^(mm*nn)
if det >= bound:
print( "We do not have det < bound. Solutions might not be found.")
print( "Try with highers m and t.")
if debug:
diff = (log(det) - log(bound)) / log(2)
print( "size det(L) - size e^(m*n) = ", floor(diff))
if strict:
return -1, -1
else:
print( "det(L) < e^(m*n) (good! If a solution exists < N^delta, it will be found)")
if debug:
matrix_overview(BB, modulus^mm)
if debug:
print( "optimizing basis of the lattice via LLL, this can take a long time")
BB = BB.LLL()
if debug:
print( "LLL is done!")
if debug:
print( "looking for independent vectors in the lattice")
found_polynomials = False
for pol1_idx in range(nn - 1):
for pol2_idx in range(pol1_idx + 1, nn):
PR.<w,z> = PolynomialRing(ZZ)
pol1 = pol2 = 0
for jj in range(nn):
pol1 += monomials[jj](w*z+1,w,z) * BB[pol1_idx, jj] / monomials[jj](UU,XX,YY)
pol2 += monomials[jj](w*z+1,w,z) * BB[pol2_idx, jj] / monomials[jj](UU,XX,YY)
PR.<q> = PolynomialRing(ZZ)
rr = pol1.resultant(pol2)
if rr.is_zero() or rr.monomials() == [1]:
continue
else:
print( "found them, using vectors", pol1_idx, "and", pol2_idx)
found_polynomials = True
break
if found_polynomials:
break
if not found_polynomials:
print( "no independant vectors could be found. This should very rarely happen...")
return 0, 0
rr = rr(q, q)
soly = rr.roots()
if len(soly) == 0:
print( "Your prediction (delta) is too small")
return 0, 0
soly = soly[0][0]
ss = pol1(q, soly)
solx = ss.roots()[0][0]
return solx, soly
def example():
N =
e =
c =
delta = .291
m = 4
t = int((1-2*delta) * m)
X = 2*floor(N^delta)
Y = floor(N^(1/2))
P.<x,y> = PolynomialRing(ZZ)
A = int((N+1)/2)
pol = 1 + x * (A + y)
if debug:
print( "=== checking values ===")
print( "* delta:", delta)
print( "* delta < 0.292", delta < 0.292)
print( "* size of e:", int(log(e)/log(2)))
print( "* size of N:", int(log(N)/log(2)))
print( "* m:", m, ", t:", t)
if debug:
print( "=== running algorithm ===")
start_time = time.time()
solx, soly = boneh_durfee(pol, e, m, t, X, Y)
if solx > 0:
print( "=== solution found ===")
if False:
print( "x:", solx)
print( "y:", soly)
d = int(pol(solx, soly) / e)
m = pow(c,d,N)
print( '[-]d is ' + str(d))
print( '[-]m is: ' + str(m))
print( '[-]hex(m) is: ' + '{:x}'.format(int(m)))
else:
print( "[!]no solution was found!")
print( '[!]All Done!')
if debug:
print(("[!]Timer: %s s" % (time.time() - start_time)))
print( '[!]All Done!')
if __name__ == "__main__":
example()
|
