def m_vectors(alms, l):
"""
Get multipole vectors in spherical coordinates for a multipole given a map.
Args:
alms (complex array): multipole moments from a map (Healpy indexing).
l (int): multipole.
Returns:
Float array [theta, phi] in radians.
"""
# Context for MPSolve:
ctx = mp.Context()
poly = mp.MonomialPoly(ctx, 2 * l)
# Get maximum multipole moment given an array.
lmax = hp.sphtfunc.Alm.getlmax(len(alms))
# Polynomial indexes:
m_array = np.arange(-l, l + 1, 1) # m values.
neg_m = m_array[0:l] # Negative m values.
pos_m = m_array[l:] # Positive m values.
# Binomial terms:
binom_term = [sqrt(bincoef(2 * l, l + i)) for i in range(-l, l + 1)]
# Indexes for all alms list:
index = hp.sphtfunc.Alm.getidx(lmax, l, abs(m_array))
alm_ell_real = np.real(alms[index])
alm_ell_imag = np.imag(alms[index])
# Negative imaginary part for m:
neg_rea_m_term = [(-1)**int(i) for i in neg_m]
# Positive imaginary part for m:
neg_imag_m_term = [-((-1)**int(i)) for i in neg_m]
# Join all real signs:
rea_sign = np.concatenate((neg_rea_m_term, np.ones(l + 1)))
# Join all imaginary signs:
imag_sign = np.concatenate((neg_imag_m_term, np.ones(l + 1)))
rea = rea_sign * alm_ell_real * binom_term # Real terms.
ima = imag_sign * alm_ell_imag * binom_term # Imaginary terms.
# Set coefficients:
[
poly.set_coefficient(i, str(rea[i]), str(ima[i]))
for i in range(2 * l + 1)
]
# Getting roots:
ctx.solve(poly, algorithm=mp.Algorithm.SECULAR_GA)
roots = np.array(ctx.get_roots())
phi = np.angle(roots)
r = np.absolute(roots)
theta = 2 * np.arctan(1 / r)
return np.vstack((theta, phi)).T
def calcPopDistribution(size, totalSize):
popDistribution = np.zeros(size)
for state in range(0, size):
binomialCoef = gmpy2.bincoef(int(size - 1), state)
popDistribution[state] = totalSize * (binomialCoef /
(math.pow(2, size)))
if not np.sum(popDistribution) == totalSize:
factor = totalSize / np.sum(popDistribution)
for i in range(0, len(popDistribution)):
popDistribution[i] = popDistribution[i] * factor
return popDistribution
def proba(N, K, W, L, P, EPS):
s = mpz(0)
for p in range(0, P + 2, 2):
P1 = p // 2
P2 = p - P1
N1 = (K + L) // 2
N2 = K + L - N1
for i1 in range(P1 + 1):
for i2 in range(P2 + 1):
s += bincoef(N1 - EPS, P1 - i1) * bincoef(
N2 - EPS, P2 - i2) * bincoef(2 * EPS, i1 + i2) * bincoef(
N - K - L, W - p)
if bincoef(N, W) < 2**(N - K):
s /= bincoef(N, W)
else:
s /= 2**(N - K)
return s
def main():
# Test local versions of libraries
utils.test_python_version()
utils.test_gmpy2_version()
# Parse command line arguments
parser = argparse.ArgumentParser(
description=
"From a table of draws, output a seed of appropriate length.")
parser.add_argument("input_draw_file", help="""Text file with draws.""")
parser.add_argument(
"output_seed_file",
help=
"""JSON file where we can store the seed computed from the draws.""")
parser.add_argument(
"entropy_to_gather",
help="""Minimum entropy to extract before drawing lone bits.""")
parser.add_argument("--nbr_lone_bits",
type=int,
help="""Number of lone bits to extract.""",
default=0)
args = parser.parse_args()
# Check arguments
output_seed_file = args.output_seed_file
if os.path.exists(output_seed_file):
utils.exit_error("The output file '%s' already exists. Exiting." %
(output_seed_file))
# Declare a few important variables
two_pow_entropy_to_gather = (1 << int(args.entropy_to_gather))
seed = 0
L = 1 # before lone bits are drawn, seed lies in [0,L - 1]
lone_bits_part = 0
nbr_lone_bits = args.nbr_lone_bits
# Scan the input file, construct the seed
with open(args.input_draw_file, "r") as f:
for line in f:
if not line or line.strip() == "" or line.startswith("#"):
continue
(draw_id, m, n, draw) = re.split("\s+", line.strip(), maxsplit=3)
if draw == "None":
continue
m = int(m)
n = int(n)
draw = [int(x) for x in draw.split(",")]
index = index_from_draw(draw, m)
if L < two_pow_entropy_to_gather:
print("Draw %s used to extract entropy" % (draw_id))
seed = gmpy2.bincoef(n, m) * seed + index
L *= gmpy2.bincoef(n, m)
else:
print("Draw %s used to extract a lone bit" % (draw_id))
b = index & 1
seed += L * (b << (args.nbr_lone_bits - nbr_lone_bits))
nbr_lone_bits -= 1
if L >= two_pow_entropy_to_gather and nbr_lone_bits == 0:
break
if nbr_lone_bits > 0 or L < two_pow_entropy_to_gather:
utils.exit_error(
"There wasn't enough draws to collect to request quantity of entropy and lone bits."
)
seed_upper_bound = L * 2**(args.nbr_lone_bits)
seed_entropy = math.floor(gmpy2.log2(seed_upper_bound))
print(
"The seed contains more than %d bits of entropy (including the %s lone bits)."
% (seed_entropy, args.nbr_lone_bits))
print("The seed is %d" % (seed))
print("Saving the seed to %s" % (output_seed_file))
with open(output_seed_file, "w") as f:
json.dump(
{
"seed": int(seed),
"seed_upper_bound": int(seed_upper_bound),
"approx_seed_entropy": int(seed_entropy),
"lone_bits": int(args.nbr_lone_bits)
},
f,
sort_keys=True)
def index_from_draw(draw, m):
index = 0
draw = sorted(draw)
for i in range(m):
index += gmpy2.bincoef(draw[i] - 1, i + 1)
return index
from gmpy2 import bincoef
from gmpy2 import next_prime
numbers = set()
for n in range(50):
for i in range(n + 1):
numbers.add(bincoef(n, i))
maxnum = max(numbers)
p = 2
while p * p < max(numbers):
numbers = [x for x in numbers if x % (p * p) != 0]
p = next_prime(p)
total = sum(numbers)
print(total)
def main():
# Test local versions of libraries
utils.test_python_version()
utils.test_gmpy2_version()
# Parse command line arguments
parser = argparse.ArgumentParser(description="From a table of draws, output a seed of appropriate length.")
parser.add_argument("input_draw_file", help="""Text file with draws.""")
parser.add_argument("output_seed_file", help="""JSON file where we can store the seed computed from the draws.""")
parser.add_argument("entropy_to_gather", help="""Minimum entropy to extract before drawing lone bits.""")
parser.add_argument("--nbr_lone_bits", type=int, help="""Number of lone bits to extract.""", default=0)
args = parser.parse_args()
# Check arguments
output_seed_file = args.output_seed_file
if os.path.exists(output_seed_file):
utils.exit_error("The output file '%s' already exists. Exiting."%(output_seed_file))
# Declare a few important variables
two_pow_entropy_to_gather = (1<<int(args.entropy_to_gather))
seed = 0
L = 1 # before lone bits are drawn, seed lies in [0,L - 1]
lone_bits_part = 0
nbr_lone_bits = args.nbr_lone_bits
# Scan the input file, construct the seed
with open(args.input_draw_file, "r") as f:
for line in f:
if not line or line.strip() == "" or line.startswith("#"):
continue
(draw_id, m, n, draw) = re.split("\s+", line.strip(), maxsplit=3)
if draw == "None":
continue
m = int(m)
n = int(n)
draw = [ int(x) for x in draw.split(",") ]
index = index_from_draw(draw,m)
if L < two_pow_entropy_to_gather:
print("Draw %s used to extract entropy"%(draw_id))
seed = gmpy2.bincoef(n,m)*seed + index
L *= gmpy2.bincoef(n,m)
else:
print("Draw %s used to extract a lone bit"%(draw_id))
b = index & 1
seed += L * (b << (args.nbr_lone_bits - nbr_lone_bits))
nbr_lone_bits -= 1
if L >= two_pow_entropy_to_gather and nbr_lone_bits == 0:
break
if nbr_lone_bits > 0 or L < two_pow_entropy_to_gather:
utils.exit_error("There wasn't enough draws to collect to request quantity of entropy and lone bits.")
seed_upper_bound = L * 2**(args.nbr_lone_bits)
seed_entropy = math.floor(gmpy2.log2(seed_upper_bound))
print("The seed contains more than %d bits of entropy (including the %s lone bits)."%(seed_entropy,args.nbr_lone_bits))
print("The seed is %d"%(seed))
print("Saving the seed to %s"%(output_seed_file))
with open(output_seed_file, "w") as f:
json.dump({"seed": int(seed),
"seed_upper_bound": int(seed_upper_bound),
"approx_seed_entropy": int(seed_entropy),
"lone_bits": int(args.nbr_lone_bits)},
f,
sort_keys=True)
def index_from_draw(draw,m):
index = 0
draw = sorted(draw)
for i in range(m):
index += gmpy2.bincoef(draw[i]-1, i+1)
return index
def __call__(self, i):
return gmpy2.mpz(gmpy2.round_away(gmpy2.bincoef(2 * i, i) / (i + 1)))
from gmpy2 import bincoef
count_greater = 0
for n in range(1, 101):
for r in range(n):
if bincoef(n, r) > 10**6:
count_greater += 1
print(count_greater)
# -*- coding: utf-8 -*-
#from decimal import Decimal, getcontext
#getcontext().prec = 1000000
#print sum(1/Decimal(16)**k *
#(Decimal(4)/(8*k+1) -
#Decimal(2)/(8*k+4) -
#Decimal(1)/(8*k+5) -
#Decimal(1)/(8*k+6)) for k in range(100))
import gmpy2
# See https://gmpy2.readthedocs.org/en/latest/mpfr.html
gmpy2.get_context().precision = 10000
pi = 0
for n in range(1000000):
# Formula from http://en.wikipedia.org/wiki/Calculating_pi#Arctangent
numer = pow(2, n + 1)
denom = gmpy2.bincoef(n + n, n) * (n + n + 1)
frac = gmpy2.mpq(numer, denom)
pi += frac
# Print every 1000 iterations
if n % 1000 == 0:
print(gmpy2.mpfr(pi))