def _small_irriducible(char, degree):
"""
Return an irreducible polynomial of self.degree with a small
number of non-zero coefficients.
"""
cardinality = char**degree
basefield = FinitePrimeField.getInstance(char)
top = uniutil.polynomial({degree: 1}, coeffring=basefield)
for seed in range(degree - 1):
for const in range(1, char):
coeffs = [const] + arith1.expand(seed, 2)
cand = uniutil.polynomial(enumerate(coeffs),
coeffring=basefield) + top
if cand.isirreducible():
_log.debug(cand.order.format(cand))
return cand
for subdeg in range(degree):
subseedbound = char**subdeg
for subseed in range(subseedbound + 1, char * subseedbound):
if not subseed % char:
continue
seed = subseed + cardinality
cand = uniutil.polynomial(enumerate(
arith1.expand(seed, cardinality)),
coeffring=basefield)
if cand.isirreducible():
return cand
def _random_irriducible(char, degree):
"""
Return randomly chosen irreducible polynomial of self.degree.
"""
cardinality = char ** degree
basefield = FinitePrimeField.getInstance(char)
seed = bigrandom.randrange(1, char) + cardinality
cand = uniutil.FiniteFieldPolynomial(enumerate(arith1.expand(seed, cardinality)), coeffring=basefield)
while cand.degree() < degree or not cand.isirreducible():
seed = bigrandom.randrange(1, cardinality) + cardinality
cand = uniutil.FiniteFieldPolynomial(enumerate(arith1.expand(seed, cardinality)), coeffring=basefield)
_log.debug(cand.order.format(cand))
return cand
def mul(Q, x, C, n):
"""
Return x*Q on the curve C mod n.
m*Q and (m+1)*Q are being tracked in the main loop.
"""
if x == 0:
return POINT_AT_INFINITY
if x == 1:
return Q
if x == 2:
return double(Q, C, n)
minor = Q
major = double(Q, C, n)
binary = _arith1.expand(x, 2)
lastbit, binary = binary[0], binary[1:]
while binary:
if binary.pop() == 1:
minor = add(major, minor, Q, n)
major = double(major, C, n)
else:
major = add(minor, major, Q, n)
minor = double(minor, C, n)
if lastbit:
return add(minor, major, Q, n)
return double(minor, C, n)
def mul(Q, x, C, n):
"""
Return x*Q on the curve C mod n.
m*Q and (m+1)*Q are being tracked in the main loop.
"""
if x == 0:
return POINT_AT_INFINITY
if x == 1:
return Q
if x == 2:
return double(Q, C, n)
minor = Q
major = double(Q, C, n)
binary = _arith1.expand(x, 2)
lastbit, binary = binary[0], binary[1:]
while binary:
if binary.pop() == 1:
minor = add(major, minor, Q, n)
major = double(major, C, n)
else:
major = add(minor, major, Q, n)
minor = double(minor, C, n)
if lastbit:
return add(minor, major, Q, n)
return double(minor, C, n)
def createElement(self, seed):
"""
Create an element of the field.
"""
if isinstance(seed, int):
expansion = arith1.expand(seed, self.char)
return FiniteExtendedFieldElement(
FinitePrimeFieldPolynomial(enumerate(expansion),
self.basefield), self)
elif isinstance(seed, FinitePrimeFieldPolynomial):
return FiniteExtendedFieldElement(seed, self)
elif isinstance(seed, FinitePrimeFieldElement
) and seed.m == self.getCharacteristic():
return FiniteExtendedFieldElement(
FinitePrimeFieldPolynomial([(0, seed)], self.basefield), self)
elif seed in self:
# seed is in self, return only embedding
return self.zero + seed
else:
try:
# lastly check sequence
return FiniteExtendedFieldElement(
FinitePrimeFieldPolynomial(enumerate(seed),
self.basefield), self)
except TypeError:
raise TypeError("seed %s is not an appropriate object." %
str(seed))
def createElement(self, seed):
"""
Create an element of the field.
"""
if isinstance(seed, int):
expansion = arith1.expand(seed, card(self.basefield))
return ExtendedFieldElement(
uniutil.FiniteFieldPolynomial(enumerate(expansion),
self.basefield), self)
elif seed in self.basefield:
return ExtendedFieldElement(
uniutil.FiniteFieldPolynomial([(0, seed)], self.basefield),
self)
elif seed in self:
# seed is in self, return only embedding
return self.zero + seed
elif (isinstance(seed, uniutil.FiniteFieldPolynomial)
and seed.getCoefficientRing() is self.basefield):
return ExtendedFieldElement(seed, self)
else:
try:
# lastly check sequence
return ExtendedFieldElement(
uniutil.FiniteFieldPolynomial(enumerate(seed),
self.basefield), self)
except TypeError:
raise TypeError("seed %s is not an appropriate object." %
str(seed))
def createElement(self, seed):
"""
Create an element of the field.
"""
if isinstance(seed, (int, long)):
expansion = arith1.expand(seed, card(self.basefield))
return ExtendedFieldElement(
uniutil.FiniteFieldPolynomial(enumerate(expansion), self.basefield),
self)
elif seed in self.basefield:
return ExtendedFieldElement(
uniutil.FiniteFieldPolynomial([(0, seed)], self.basefield),
self)
elif seed in self:
# seed is in self, return only embedding
return self.zero + seed
elif (isinstance(seed, uniutil.FiniteFieldPolynomial) and
seed.getCoefficientRing() is self.basefield):
return ExtendedFieldElement(seed, self)
else:
try:
# lastly check sequence
return ExtendedFieldElement(
uniutil.FiniteFieldPolynomial(enumerate(seed), self.basefield),
self)
except TypeError:
raise TypeError("seed %s is not an appropriate object." % str(seed))
def _random_irriducible(char, degree):
"""
Return randomly chosen irreducible polynomial of self.degree.
"""
cardinality = char**degree
basefield = FinitePrimeField.getInstance(char)
seed = bigrandom.randrange(1, char) + cardinality
cand = uniutil.FiniteFieldPolynomial(enumerate(
arith1.expand(seed, cardinality)),
coeffring=basefield)
while cand.degree() < degree or not cand.isirreducible():
seed = bigrandom.randrange(1, cardinality) + cardinality
cand = uniutil.FiniteFieldPolynomial(enumerate(
arith1.expand(seed, cardinality)),
coeffring=basefield)
_log.debug(cand.order.format(cand))
return cand
def __init__(self, characteristic, n_or_modulus):
"""
FiniteExtendedField(p, n_or_modulus) creates a finite field.
characteristic must be prime. n_or_modulus can be:
1) an integer greater than 1, or
2) a polynomial in a polynomial ring of F_p with degree
greater than 1.
"""
FiniteField.__init__(self, characteristic)
self.basefield = FinitePrimeField.getInstance(self.char)
if isinstance(n_or_modulus, int):
if n_or_modulus <= 1:
raise ValueError("degree of extension must be > 1.")
self.degree = n_or_modulus
# randomly chosen irreducible polynomial
top = FinitePrimeFieldPolynomial({self.degree: 1},
coeffring=self.basefield)
seed = bigrandom.randrange(1, self.char)
cand = FinitePrimeFieldPolynomial(enumerate(
arith1.expand(seed, self.char)),
coeffring=self.basefield) + top
while cand.degree() < self.degree or not cand.isirreducible():
seed = bigrandom.randrange(1, self.char**self.degree)
cand = FinitePrimeFieldPolynomial(
enumerate(arith1.expand(seed, self.char)),
coeffring=self.basefield) + top
self.modulus = cand
elif isinstance(n_or_modulus, FinitePrimeFieldPolynomial):
if isinstance(n_or_modulus.getCoefficientRing(), FinitePrimeField):
if n_or_modulus.degree() > 1 and n_or_modulus.isirreducible():
self.degree = n_or_modulus.degree()
self.modulus = n_or_modulus
else:
raise ValueError(
"modulus must be of degree greater than 1.")
else:
raise TypeError("modulus must be F_p polynomial.")
else:
raise TypeError("degree or modulus must be supplied.")
self.registerModuleAction(rational.theIntegerRing, self._int_mul)
self.registerModuleAction(FinitePrimeField.getInstance(self.char),
self._fp_mul)
def mul(self, k, P):
"""
this returns [k]*P
"""
if k >= 0:
l = arith1.expand(k, 2)
Q = [0]
for j in range(len(l) - 1, -1, -1):
Q = self.add(Q, Q)
if l[j] == 1:
Q = self.add(Q, P)
return Q
else:
return self.sub([0], self.mul(-k, P))
def mul(self, k, P):
"""
this returns [k]*P
"""
if k >= 0:
l = arith1.expand(k, 2)
Q = [0]
for j in range(len(l) - 1, -1, -1):
Q = self.add(Q, Q)
if l[j] == 1:
Q = self.add(Q, P)
return Q
else:
return self.sub([0], self.mul(-k, P))
def _small_irriducible(char, degree):
"""
Return an irreducible polynomial of self.degree with a small
number of non-zero coefficients.
"""
cardinality = char ** degree
basefield = FinitePrimeField.getInstance(char)
top = uniutil.polynomial({degree: 1}, coeffring=basefield)
for seed in range(degree - 1):
for const in range(1, char):
coeffs = [const] + arith1.expand(seed, 2)
cand = uniutil.polynomial(enumerate(coeffs), coeffring=basefield) + top
if cand.isirreducible():
_log.debug(cand.order.format(cand))
return cand
for subdeg in range(degree):
subseedbound = char ** subdeg
for subseed in range(subseedbound + 1, char * subseedbound):
if not subseed % char:
continue
seed = subseed + cardinality
cand = uniutil.polynomial(enumerate(arith1.expand(seed, cardinality)), coeffring=basefield)
if cand.isirreducible():
return cand
def _lucas_chain(n, f, g, x_0, x_1):
"""
Given an integer n, two functions f and g, and initial value (x_0, x_1),
compute (x_n, x_{n+1}), where the sequence {x_i} is defined as:
x_{2i} = f(x_i)
x_{2i+1} = g(x_i, x_{i+1})
"""
binary = arith1.expand(n, 2)
u = x_0
v = x_1
while binary:
if 1 == binary.pop():
u, v = g(u, v), f(v)
else:
u, v = f(u), g(u, v)
return u, v
def Lucas_chain(n, f, g, x_0, x_1):
"""
Given an integer n, two functions f and g, and initial value (x_0, x_1),
compute (x_n, x_{n+1}), where the sequence {x_i} is defined as:
x_{2i} = f(x_i)
x_{2i+1} = g(x_i, x_{i+1})
"""
binary = arith1.expand(n, 2)
u = x_0
v = x_1
while binary:
if 1 == binary.pop():
u, v = g(u, v), f(v)
else:
u, v = f(u), g(u, v)
return u, v
def __init__(self, p, initial, power):
"""
PadicInteger(p, initial, power)
The second argument initial can be either integer or iterable.
If initial is an integer, it will be expanded p-adically.
In iterable case, it will be treated as a p-adic expansion.
The third argument power is a power index of modulus, by which
the representation is finite.
"""
if isinstance(initial, int):
cutoff = p**power
# taking modulo cutoff is useful for nagative integers.
self.expansion = tuple(arith1.expand(initial % cutoff, p))
else:
# take only first 'power' elements of initial.
self.expansion = tuple([e for i, e in zip(range(power), initial)])
self.p = p
def _three_pow(self, element, index):
"""
powering by using left-right base 3 method.
"""
digits = arith1.expand(index, 3)
e = len(digits) - 1
f = e
# Precomputation
pre_table = [element, self.square(element)]
# Main Loop
while f >= 0:
tit = digits[f]
if f == e:
sol = pre_table[tit - 1]
else:
sol = self.mul(self.square(sol), sol)
if tit:
sol = self.mul(sol, pre_table[tit - 1])
f -= 1
return sol
def createElement(self, seed):
"""
Create an element of the field.
"""
# FIXME: Undefined variable 'FinitFieldExtensionElement'
if isinstance(seed, int):
expansion = arith1.expand(seed, card(self.basefield))
return FinitFieldExtensionElement(
polynomial.OneVariableDensePolynomial(expansion, "#1",
self.basefield), self)
elif isinstance(seed, polynomial.OneVariablePolynomial):
return FiniteFieldExtensionElement(seed("#1"), self)
else:
try:
# lastly check sequence
return FiniteFieldExtensionElement(
polynomial.OneVariableDensePolynomial(
list(seed), "#1", self.basefield), self)
except TypeError:
raise TypeError("seed %s is not an appropriate object." %
str(seed))
def testExpand(self):
self.assertEqual(
[0, 0, 0, 0, 0, 0, 1, 0, 0, 1, 0, 0, 0, 0, 1, 0, 1, 1, 1, 1],
arith1.expand(10**6, 2))
self.assertEqual([1, 1, 3, 3, 3, 3, 1, 1], arith1.expand(10**6, 7))