def ghf(cls, m, p, klen_list, prec=53):
"""
Estimate norm of shortest vector according to Gaussian Heuristic.
:param m: number of samples
:param p: ECDSA modulus
:param klen_list: list of lengths of key to recover
:param prec: precision to use
"""
# NOTE: The Gaussian Heuristic does not hold in small dimensions
w = 2 ** (max(klen_list) - 1)
RR = RealField(prec)
w = RR(w)
f_list = [Integer(w / (2 ** (klen - 1))) for klen in klen_list]
d = m + 1
log_vol = log(p) * (m - 1) + sum(map(log, f_list)) + log(w)
lgh = log_gamma(1 + d / 2.0) * (1.0 / d) - log(sqrt(pi)) + log_vol * (1.0 / d)
return RR(exp(lgh))
def _sage_(self):
import sage.all as sage
return sage.log_gamma(self.args[0]._sage_())
def test_sage_conversions():
try:
import sage.all as sage
except ImportError:
return
x, y = sage.SR.var('x y')
x1, y1 = symbols('x, y')
# Symbol
assert x1._sage_() == x
assert x1 == sympify(x)
# Integer
assert Integer(12)._sage_() == sage.Integer(12)
assert Integer(12) == sympify(sage.Integer(12))
# Rational
assert (Integer(1) / 2)._sage_() == sage.Integer(1) / 2
assert Integer(1) / 2 == sympify(sage.Integer(1) / 2)
# Operators
assert x1 + y == x1 + y1
assert x1 * y == x1 * y1
assert x1**y == x1**y1
assert x1 - y == x1 - y1
assert x1 / y == x1 / y1
assert x + y1 == x + y
assert x * y1 == x * y
# Doesn't work in Sage 6.1.1ubuntu2
# assert x ** y1 == x ** y
assert x - y1 == x - y
assert x / y1 == x / y
# Conversions
assert (x1 + y1)._sage_() == x + y
assert (x1 * y1)._sage_() == x * y
assert (x1**y1)._sage_() == x**y
assert (x1 - y1)._sage_() == x - y
assert (x1 / y1)._sage_() == x / y
assert x1 + y1 == sympify(x + y)
assert x1 * y1 == sympify(x * y)
assert x1**y1 == sympify(x**y)
assert x1 - y1 == sympify(x - y)
assert x1 / y1 == sympify(x / y)
# Functions
assert sin(x1) == sin(x)
assert sin(x1)._sage_() == sage.sin(x)
assert sin(x1) == sympify(sage.sin(x))
assert cos(x1) == cos(x)
assert cos(x1)._sage_() == sage.cos(x)
assert cos(x1) == sympify(sage.cos(x))
assert function_symbol('f', x1, y1)._sage_() == sage.function('f', x, y)
assert function_symbol('f', 2 * x1,
x1 + y1).diff(x1)._sage_() == sage.function(
'f', 2 * x, x + y).diff(x)
# For the following test, sage needs to be modified
# assert sage.sin(x) == sage.sin(x1)
# Constants
assert pi._sage_() == sage.pi
assert E._sage_() == sage.e
assert I._sage_() == sage.I
assert pi == sympify(sage.pi)
assert E == sympify(sage.e)
# SympyConverter does not support converting the following
# assert I == sympify(sage.I)
# Matrix
assert DenseMatrix(1, 2, [x1, y1])._sage_() == sage.matrix([[x, y]])
# SympyConverter does not support converting the following
# assert DenseMatrix(1, 2, [x1, y1]) == sympify(sage.matrix([[x, y]]))
# Sage Number
a = sage.Mod(2, 7)
b = PyNumber(a, sage_module)
a = a + 8
b = b + 8
assert isinstance(b, PyNumber)
assert b._sage_() == a
a = a + x
b = b + x
assert isinstance(b, Add)
assert b._sage_() == a
# Sage Function
e = x1 + wrap_sage_function(sage.log_gamma(x))
assert str(e) == "x + log_gamma(x)"
assert isinstance(e, Add)
assert e + wrap_sage_function(
sage.log_gamma(x)) == x1 + 2 * wrap_sage_function(sage.log_gamma(x))
f = e.subs({x1: 10})
assert f == 10 + log(362880)
f = e.subs({x1: 2})
assert f == 2
f = e.subs({x1: 100})
v = f.n(53, real=True)
assert abs(float(v) - 459.13420537) < 1e-7
f = e.diff(x1)
def test_sage_conversions():
x, y = sage.SR.var('x y')
x1, y1 = symbols('x, y')
# Symbol
assert x1._sage_() == x
assert x1 == sympify(x)
# Integer
assert Integer(12)._sage_() == sage.Integer(12)
assert Integer(12) == sympify(sage.Integer(12))
# Rational
assert (Integer(1) / 2)._sage_() == sage.Integer(1) / 2
assert Integer(1) / 2 == sympify(sage.Integer(1) / 2)
# Operators
assert x1 + y == x1 + y1
assert x1 * y == x1 * y1
assert x1**y == x1**y1
assert x1 - y == x1 - y1
assert x1 / y == x1 / y1
assert x + y1 == x + y
assert x * y1 == x * y
# Doesn't work in Sage 6.1.1ubuntu2
# assert x ** y1 == x ** y
assert x - y1 == x - y
assert x / y1 == x / y
# Conversions
assert (x1 + y1)._sage_() == x + y
assert (x1 * y1)._sage_() == x * y
assert (x1**y1)._sage_() == x**y
assert (x1 - y1)._sage_() == x - y
assert (x1 / y1)._sage_() == x / y
assert x1 + y1 == sympify(x + y)
assert x1 * y1 == sympify(x * y)
assert x1**y1 == sympify(x**y)
assert x1 - y1 == sympify(x - y)
assert x1 / y1 == sympify(x / y)
# Functions
assert sin(x1) == sin(x)
assert sin(x1)._sage_() == sage.sin(x)
assert sin(x1) == sympify(sage.sin(x))
assert cos(x1) == cos(x)
assert cos(x1)._sage_() == sage.cos(x)
assert cos(x1) == sympify(sage.cos(x))
assert function_symbol('f', x1, y1)._sage_() == sage.function('f')(x, y)
assert (function_symbol('f', 2 * x1,
x1 + y1).diff(x1)._sage_() == sage.function('f')(
2 * x, x + y).diff(x))
assert LambertW(x1) == LambertW(x)
assert LambertW(x1)._sage_() == sage.lambert_w(x)
assert KroneckerDelta(x1, y1) == KroneckerDelta(x, y)
assert KroneckerDelta(x1, y1)._sage_() == sage.kronecker_delta(x, y)
assert erf(x1) == erf(x)
assert erf(x1)._sage_() == sage.erf(x)
assert lowergamma(x1, y1) == lowergamma(x, y)
assert lowergamma(x1, y1)._sage_() == sage.gamma_inc_lower(x, y)
assert uppergamma(x1, y1) == uppergamma(x, y)
assert uppergamma(x1, y1)._sage_() == sage.gamma_inc(x, y)
assert loggamma(x1) == loggamma(x)
assert loggamma(x1)._sage_() == sage.log_gamma(x)
assert beta(x1, y1) == beta(x, y)
assert beta(x1, y1)._sage_() == sage.beta(x, y)
assert floor(x1) == floor(x)
assert floor(x1)._sage_() == sage.floor(x)
assert ceiling(x1) == ceiling(x)
assert ceiling(x1)._sage_() == sage.ceil(x)
assert conjugate(x1) == conjugate(x)
assert conjugate(x1)._sage_() == sage.conjugate(x)
# For the following test, sage needs to be modified
# assert sage.sin(x) == sage.sin(x1)
# Constants and Booleans
assert pi._sage_() == sage.pi
assert E._sage_() == sage.e
assert I._sage_() == sage.I
assert GoldenRatio._sage_() == sage.golden_ratio
assert Catalan._sage_() == sage.catalan
assert EulerGamma._sage_() == sage.euler_gamma
assert oo._sage_() == sage.oo
assert zoo._sage_() == sage.unsigned_infinity
assert nan._sage_() == sage.NaN
assert true._sage_() == True
assert false._sage_() == False
assert pi == sympify(sage.pi)
assert E == sympify(sage.e)
assert GoldenRatio == sympify(sage.golden_ratio)
assert Catalan == sympify(sage.catalan)
assert EulerGamma == sympify(sage.euler_gamma)
assert oo == sympify(sage.oo)
assert zoo == sympify(sage.unsigned_infinity)
assert nan == sympify(sage.NaN)
# SympyConverter does not support converting the following
# assert I == sympify(sage.I)
# Matrix
assert DenseMatrix(1, 2, [x1, y1])._sage_() == sage.matrix([[x, y]])
# SympyConverter does not support converting the following
# assert DenseMatrix(1, 2, [x1, y1]) == sympify(sage.matrix([[x, y]]))
# Sage Number
a = sage.Mod(2, 7)
b = PyNumber(a, sage_module)
a = a + 8
b = b + 8
assert isinstance(b, PyNumber)
assert b._sage_() == a
assert str(a) == str(b)
# Sage Function
e = x1 + wrap_sage_function(sage.log_gamma(x))
assert str(e) == "x + log_gamma(x)"
assert isinstance(e, Add)
assert (e + wrap_sage_function(sage.log_gamma(x)) == x1 +
2 * wrap_sage_function(sage.log_gamma(x)))
f = e.subs({x1: 10})
assert f == 10 + log(362880)
f = e.subs({x1: 2})
assert f == 2
f = e.subs({x1: 100})
v = f.n(53, real=True)
assert abs(float(v) - 459.13420537) < 1e-7
f = e.diff(x1)
def _sage_(self):
import sage.all as sage
return sage.log_gamma(self.args[0]._sage_())
def test_sage_conversions():
try:
import sage.all as sage
except ImportError:
return
x, y = sage.SR.var('x y')
x1, y1 = symbols('x, y')
# Symbol
assert x1._sage_() == x
assert x1 == sympify(x)
# Integer
assert Integer(12)._sage_() == sage.Integer(12)
assert Integer(12) == sympify(sage.Integer(12))
# Rational
assert (Integer(1) / 2)._sage_() == sage.Integer(1) / 2
assert Integer(1) / 2 == sympify(sage.Integer(1) / 2)
# Operators
assert x1 + y == x1 + y1
assert x1 * y == x1 * y1
assert x1 ** y == x1 ** y1
assert x1 - y == x1 - y1
assert x1 / y == x1 / y1
assert x + y1 == x + y
assert x * y1 == x * y
# Doesn't work in Sage 6.1.1ubuntu2
# assert x ** y1 == x ** y
assert x - y1 == x - y
assert x / y1 == x / y
# Conversions
assert (x1 + y1)._sage_() == x + y
assert (x1 * y1)._sage_() == x * y
assert (x1 ** y1)._sage_() == x ** y
assert (x1 - y1)._sage_() == x - y
assert (x1 / y1)._sage_() == x / y
assert x1 + y1 == sympify(x + y)
assert x1 * y1 == sympify(x * y)
assert x1 ** y1 == sympify(x ** y)
assert x1 - y1 == sympify(x - y)
assert x1 / y1 == sympify(x / y)
# Functions
assert sin(x1) == sin(x)
assert sin(x1)._sage_() == sage.sin(x)
assert sin(x1) == sympify(sage.sin(x))
assert cos(x1) == cos(x)
assert cos(x1)._sage_() == sage.cos(x)
assert cos(x1) == sympify(sage.cos(x))
assert function_symbol('f', x1, y1)._sage_() == sage.function('f', x, y)
assert function_symbol('f', 2 * x1, x1 + y1).diff(x1)._sage_() == sage.function('f', 2 * x, x + y).diff(x)
# For the following test, sage needs to be modified
# assert sage.sin(x) == sage.sin(x1)
# Constants
assert pi._sage_() == sage.pi
assert E._sage_() == sage.e
assert I._sage_() == sage.I
assert pi == sympify(sage.pi)
assert E == sympify(sage.e)
# SympyConverter does not support converting the following
# assert I == sympify(sage.I)
# Matrix
assert DenseMatrix(1, 2, [x1, y1])._sage_() == sage.matrix([[x, y]])
# SympyConverter does not support converting the following
# assert DenseMatrix(1, 2, [x1, y1]) == sympify(sage.matrix([[x, y]]))
# Sage Number
a = sage.Mod(2, 7)
b = PyNumber(a, sage_module)
a = a + 8
b = b + 8
assert isinstance(b, PyNumber)
assert b._sage_() == a
a = a + x
b = b + x
assert isinstance(b, Add)
assert b._sage_() == a
# Sage Function
e = x1 + wrap_sage_function(sage.log_gamma(x))
assert str(e) == "x + log_gamma(x)"
assert isinstance(e, Add)
assert e + wrap_sage_function(sage.log_gamma(x)) == x1 + 2*wrap_sage_function(sage.log_gamma(x))
f = e.subs({x1 : 10})
assert f == 10 + log(362880)
f = e.subs({x1 : 2})
assert f == 2
f = e.subs({x1 : 100});
v = f.n(53, real=True);
assert abs(float(v) - 459.13420537) < 1e-7
f = e.diff(x1)