def test_isqrt():
from math import sqrt as _sqrt
limit = 17984395633462800708566937239551
assert int(_sqrt(limit)) == integer_nthroot(limit, 2)[0]
assert int(_sqrt(limit + 1)) != integer_nthroot(limit + 1, 2)[0]
assert isqrt(limit + 1) == integer_nthroot(limit + 1, 2)[0]
assert isqrt(limit + 1 + S.Half) == integer_nthroot(limit + 1, 2)[0]
def test_isqrt():
from math import sqrt as _sqrt
limit = 17984395633462800708566937239551
assert int(_sqrt(limit)) == integer_nthroot(limit, 2)[0]
assert int(_sqrt(limit + 1)) != integer_nthroot(limit + 1, 2)[0]
assert isqrt(limit + 1) == integer_nthroot(limit + 1, 2)[0]
assert isqrt(limit + 1 + S.Half) == integer_nthroot(limit + 1, 2)[0]
def cholesky(matlist, K):
"""
Performs the cholesky decomposition of a Hermitian matrix and
returns L and it's conjugate transpose.
Examples
========
>>> from sympy.matrices.densesolve import cholesky
>>> from sympy import QQ
>>> cholesky([[QQ(25), QQ(15), QQ(-5)], [QQ(15), QQ(18), QQ(0)], [QQ(-5), QQ(0), QQ(11)]], QQ)
([[5, 0, 0], [3, 3, 0], [-1, 1, 3]], [[5, 3, -1], [0, 3, 1], [0, 0, 3]])
See Also
========
cholesky_solve
"""
new_matlist = copy.deepcopy(matlist)
nrow = len(new_matlist)
L = eye(nrow, K)
for i in range(nrow):
for j in range(i + 1):
a = K.zero
for k in range(j):
a += L[i][k] * L[j][k]
if i == j:
L[i][j] = isqrt(new_matlist[i][j] - a)
else:
L[i][j] = (new_matlist[i][j] - a) / L[j][j]
return L, conjugate_transpose(L, K)
def cholesky(matlist, K):
"""
Performs the cholesky decomposition of a Hermitian matrix and
returns L and it's conjugate transpose.
Examples
========
>>> from sympy.matrices.densesolve import cholesky
>>> from sympy import QQ
>>> cholesky([[QQ(25), QQ(15), QQ(-5)], [QQ(15), QQ(18), QQ(0)], [QQ(-5), QQ(0), QQ(11)]], QQ)
([[5, 0, 0], [3, 3, 0], [-1, 1, 3]], [[5, 3, -1], [0, 3, 1], [0, 0, 3]])
See Also
========
cholesky_solve
"""
new_matlist = copy.deepcopy(matlist)
nrow = len(new_matlist)
L = eye(nrow, K)
for i in range(nrow):
for j in range(i + 1):
a = K.zero
for k in range(j):
a += L[i][k]*L[j][k]
if i == j:
L[i][j] = isqrt(new_matlist[i][j] - a)
else:
L[i][j] = (new_matlist[i][j] - a)/L[j][j]
return L, conjugate_transpose(L, K)
def count_brute(n):
if n > 10 ** 8: return None
count = 0
for y in range(isqrt(n)+2):
y2 = y*y
for x in range(y+1):
if y2 + y*x + x*x == n:
count += 1
return count
def count_circle(n):
# Circle walk x^2 + y^2 + x*y
if n > (22 * 10 ** 7) ** 2:
return None
count = 0
# 0 <= x <= y
# Largest value is at (0, y) => y^2
# Smallest value is at (y, y) => 3*y^2
minY = isqrt(n // 3)
maxY = isqrt(n)
x = 0
for y in reversed(range(minY, maxY+1)):
test = x*x + x*y + y*y
while test < n and x < y:
x += 1
test = x*x + x*y + y*y
if test == n:
count += 1
# print("\t", test, " ", x, y)
return count
def _discrete_log_shanks_steps(n, a, b, order=None):
"""
Baby-step giant-step algorithm for computing the discrete logarithm of
``a`` to the base ``b`` modulo ``n``.
The algorithm is a time-memory trade-off of the method of exhaustive
search. It uses `O(sqrt(m))` memory, where `m` is the group order.
Examples
========
>>> from sympy.ntheory.residue_ntheory import _discrete_log_shanks_steps
>>> _discrete_log_shanks_steps(41, 15, 7)
3
See Also
========
discrete_log
References
==========
.. [1] "Handbook of applied cryptography", Menezes, A. J., Van, O. P. C., &
Vanstone, S. A. (1997).
"""
a %= n
b %= n
if order is None:
order = n_order(b, n)
m = isqrt(order) + 1
T = dict()
x = 1
for i in range(m):
T[x] = i
x = x * b % n
z = mod_inverse(b, n)
z = pow(z, m, n)
x = a
for i in range(m):
if x in T:
return i * m + T[x]
x = x * z % n
raise ValueError("Log does not exist")
def _discrete_log_shanks_steps(n, a, b, order=None):
"""
Baby-step giant-step algorithm for computing the discrete logarithm of
``a`` to the base ``b`` modulo ``n``.
The algorithm is a time-memory trade-off of the method of exhaustive
search. It uses `O(sqrt(m))` memory, where `m` is the group order.
References
==========
.. [1] "Handbook of applied cryptography", Menezes, A. J., Van, O. P. C., &
Vanstone, S. A. (1997).
Examples
========
>>> from sympy.ntheory.residue_ntheory import _discrete_log_shanks_steps
>>> _discrete_log_shanks_steps(41, 15, 7)
3
See also
========
discrete_log
"""
a %= n
b %= n
if order is None:
order = n_order(b, n)
m = isqrt(order) + 1
T = dict()
x = 1
for i in range(m):
T[x] = i
x = x * b % n
z = mod_inverse(b, n)
z = pow(z, m, n)
x = a
for i in range(m):
if x in T:
return i * m + T[x]
x = x * z % n
raise ValueError("Log does not exist")