def orthogonalityDefect(basis):
n = len(basis)
gsBasis = gramSchmidtOrthogonalization(basis)[0]
defect = 0.0
for i in range(n):
defect += math.log10(op.norm(basis[i]) / op.norm(gsBasis[i]))
return defect
def isLLLReduced(basis, delta=BKZ_constants.delta):
n = len(basis)
gsBasis, coeff = GSO.gramSchmidtOrthogonalization(basis)
if not GSO.isSizeReduced(coeff):
print('ERROR: basis is not size reduced')
# print( coeff )
return False
for i in range(n - 1):
if not Lovasz(op.norm(gsBasis[i]), op.norm(gsBasis[i + 1]),
coeff[i + 1][i], delta):
print('ERROR: basis fails Lovasz condition at ' + str(i + 1))
return False
return True
def enumerate( basis, gsBasis, coeff, n, level, x, R, shortVec, w, combination ):
eps = BKZ_constants.eps
if level < 0:
curVec = op.scale(basis[0],x[n-1])
for i in range(1,n):
curVec = op.add(curVec,op.scale(basis[i],x[n-1-i]))
r = op.norm(curVec)
if r > eps and R > r + eps:
# utils.printShortSolutions(x,curVec)
shortVec = curVec
R = r
combination = list(reversed(x))
# print( "Found a shorter vector of norm ", r )
return ( R, shortVec, combination )
else:
alpha = 0
for i in range(level+1,n):
alpha += x[n-1-i] * coeff[i][level]
all0 = op.allZeros(x)
lowerBound, upperBound = getBounds( R, w, op.norm(gsBasis[level]), alpha, all0 )
i = upperBound
while i >= lowerBound:
y = list(x)
y.append(i)
res = enumerate( basis, gsBasis, coeff, n, level-1, y, R, shortVec, w + ( (i+alpha)**2 )*op.normSq(gsBasis[level]), combination )
if res[0] + eps < R:
R = res[0]
shortVec = res[1]
combination = res[2]
lowerBound, upperBound = getBounds( R, w, op.norm(gsBasis[level]), alpha, all0 )
i -= 1
return R, shortVec, combination
def printShortSolutions(x, v):
n = len(x)
msg = 'Norm: ' + str(round(op.norm(v), 2)) + ' '
for i in range(n - 1, -1, -1):
if x[i] != 0:
msg = msg + 'x' + str(n - i) + '=' + str(x[i]) + ' '
msg = msg + ' gives ' + str(v)
print(msg)
def volume(B, orthogonal=False):
if not orthogonal:
B = gramSchmidtOrthogonalization(B)[0]
vol = 1.0
n = len(B)
for i in range(n):
vol *= (op.norm(B[i]))**(1.0 / n)
return vol
def printVectorWithNorm(v):
print('Norm = ' + str(round(op.norm(v), 2)) + ' Vec: ' + str(v))