def encode(message):
if not all(x < p for x in message):
raise Exception(
"Message is improperly encoded as integers < p. It was:\n%r" %
message)
def row(i, b):
return [Fp(i**(j)) for j in range(k)] + [Fp(b)]
system = [row(i, message[i]) for i in range(k)]
interpolated = someSolution(system, freeVariableValue=1)
thePoly = Poly(interpolated)
print("The polynomial encoding the message is:")
print(thePoly)
return [thePoly(Fp(i)) for i in range(n)]
def solveSystem(encodedMessage, debug=False):
for e in range(maxE, 0, -1):
ENumVars = e + 1
QNumVars = e + k
def row(i, a, b):
return ([b * a**j for j in range(ENumVars)] +
[-1 * a**j for j in range(QNumVars)] + [0]
) # the "extended" part of the linear system
system = (
[row(i, a, b) for (i, (a, b)) in enumerate(encodedMessage)] +
[[0] * (ENumVars - 1) + [1] + [0] * (QNumVars) + [1]])
# ensure coefficient of x^e in E(x) is 1
if debug:
print("\ne is %r" % e)
print("\nsystem is:\n\n")
for row in system:
print("\t%r" % (row, ))
solution = someSolution(system, freeVariableValue=1)
E = Poly([solution[j] for j in range(e + 1)])
Q = Poly([solution[j] for j in range(e + 1, len(solution))])
if debug:
print("\nreduced system is:\n\n")
for row in system:
print("\t%r" % (row, ))
print("solution is %r" % (solution, ))
print("Q is %r" % (Q, ))
print("E is %r" % (E, ))
P, remainder = Q.__divmod__(E)
if remainder == 0:
return Q, E
raise Exception("found no divisors!")
def solveSystem(encodedMessage, debug=True):
for e in range(maxE, 0, -1):
ENumVars = e
QNumVars = e + k
def row(i, x, r):
return ([r * x**j for j in range(ENumVars)] +
[-1 * x**j
for j in range(QNumVars)] + [-r * x**ENumVars]
) # the "extended" part of the linear system
system = ([
row(i, a, b) for (i, (a, b)) in enumerate(encodedMessage)
])
# Add one more row in the end ensure coefficient of x^e in E(x) is 1
if debug:
print("\nSystem of equations is:\n\n")
for row in system:
print("\t%r" % (row, ))
solution = someSolution(system, freeVariableValue=1)
E = Poly([solution[j] for j in range(ENumVars)] + [Fp(1)])
Q = Poly([solution[j] for j in range(ENumVars, len(solution))])
if debug:
print("\nReduced system is:\n\n")
for row in system:
print("\t%r" % (row, ))
print("Solution is %r" % (solution, ))
print("Q is %r" % (Q, ))
print("E is %r" % (E, ))
P, remainder = Q.__divmod__(E)
if remainder == 0:
return Q, E
raise Exception("found no divisors!")
def solveSystem(encodedMessage, debug=False):
for e in range(maxE, 0, -1):
ENumVars = e+1
QNumVars = e+k
def row(i, a, b):
return ([b * a**j for j in range(ENumVars)] +
[-1 * a**j for j in range(QNumVars)] +
[0]) # the "extended" part of the linear system
system = ([row(i, a, b) for (i, (a,b)) in enumerate(encodedMessage)] +
[[0] * (ENumVars-1) + [1] + [0] * (QNumVars) + [1]])
# ensure coefficient of x^e in E(x) is 1
if debug:
print("\ne is %r" % e)
print("\nsystem is:\n\n")
for row in system:
print("\t%r" % (row,))
solution = someSolution(system, freeVariableValue=1)
E = Poly([solution[j] for j in range(e + 1)])
Q = Poly([solution[j] for j in range(e + 1, len(solution))])
if debug:
print("\nreduced system is:\n\n")
for row in system:
print("\t%r" % (row,))
print("solution is %r" % (solution,))
print("Q is %r" % (Q,))
print("E is %r" % (E,))
P, remainder = Q.__divmod__(E)
if remainder == 0:
return Q, E
raise Exception("found no divisors!")
from finitefield.finitefield import FiniteField
from linearsolver.linearsolver import someSolution
FF = FiniteField(13)
A = [
[FF(4), FF(2), FF(9), FF(1)],
[FF(4), FF(3), FF(6), FF(1)],
[FF(2), FF(11), FF(5), FF(1)],
]
B = someSolution(A)
from finitefield.finitefield import FiniteField from linearsolver.linearsolver import someSolution FF = FiniteField(13) A = [ [FF(4), FF(2), FF(9), FF(1)], [FF(4), FF(3), FF(6), FF(1)], [FF(2), FF(11), FF(5), FF(1)], ] B = someSolution(A)