def symbolic_sis(n, alpha, q, m=None, epsilon=None):
if epsilon is None:
epsilon = var("epsilon")
assume(epsilon>0)
assume(epsilon<ZZ(1)/2)
delta_0 = var("delta_0")
assume(delta_0>=1.0)
e = alpha*q/sqrt(2*pi)
if m is None:
m = sqrt(n*log(q)/log(delta_0))
v = e * delta_0**m * q**(n/m) # norm of the vector
# epsilon = exp(-pi*(|v|^2/q^2))
f = log(1/epsilon)/pi == (v/q)**2
# solve
f = 2* q**2 * m * f * pi
f = f.simplify_full()
f = f.solve(delta_0**(2*m))[0]
f = f.log().canonicalize_radical()
f = f.solve(log(delta_0))[0]
f = f.simplify_log()
return f
def number_of_coefficients_needed(Q, kappa_fe, lambda_fe, max_t):
# TODO: This doesn't work. Trouble when computing t0
# We completely mimic what lcalc does when it decides whether
# to print a warning.
DIGITS = 14 # These are names of lcalc parameters, and we are
DIGITS2 = 2 # mimicking them.
logger.debug("Start NOC")
theta = sum(kappa_fe)
c = DIGITS2 * log(10.0)
a = len(kappa_fe)
c1 = 0.0
for j in range(a):
logger.debug("In loop NOC")
t0 = kappa_fe[j] * max_t + complex(lambda_fe[j]).imag()
logger.debug("In loop 2 NOC")
if abs(t0) < 2 * c / (math.pi * a):
logger.debug("In loop 3_1 NOC")
c1 += kappa_fe[j] * pi / 2.0
else:
c1 += kappa_fe[j] * abs(c / (t0 * a))
logger.debug("In loop 3_2 NOC")
return int(round(Q * exp(log(2.3 * DIGITS * theta / c1) * theta) + 10))
def symbolic_modulus_switching(n, alpha, q, h, m=None, epsilon=None):
if epsilon is None:
epsilon = var("epsilon")
assume(epsilon>0)
assume(epsilon<ZZ(1)/2)
delta_0 = var("delta_0")
assume(delta_0>=1.0)
if m is None:
m = sqrt(n*log(q)/log(delta_0))
e = alpha*q/sqrt(2*pi)
c = e * sqrt(m-n)/sqrt(h)
v = delta_0**m * (q/c)**(n/m) # norm of the vector
v_ = v**2/m # variance of each component
v_r = (m-n) * e**2 *v_ # noise contribution
v_l = h * v_ * c**2 # nose contribution of rounding noise
# epsilon = exp(-pi*(|v|^2/q^2))
f = log(1/epsilon)/pi == (v_l + v_r)/q**2
# solve
f = 2* q**2 * m * f * pi
f = f.simplify_full()
f = f.solve(delta_0**(2*m))[0]
f = f.log().canonicalize_radical()
f = f.solve(log(delta_0))[0]
f = f.simplify_log()
return f
def get_type_2_bound(K):
"""The bound in the proof of Theorem 6.4 of Larson/Vaintrob, souped up with
Theorem 5.1 of Bach and Sorenson."""
# The Bach and Sorenson parameters
A = 4
B = 2.5
C = 5
n_K = K.degree()
delta_K = K.discriminant().abs()
D = 2 * A * n_K
E = 4 * A * log(delta_K) + 2 * A * n_K * log(12) + 4 * B * n_K + C + 1
f = x - (D * log(x) + E)**4
try:
bound = find_root(f, 10, GENERIC_UPPER_BOUND)
return ceil(bound)
except RuntimeError:
warning_msg = ("Type 2 bound for quadratic field with "
"discriminant {} failed. Returning generic upper bound"
).format(delta_K)
logger.warning(warning_msg)
return GENERIC_UPPER_BOUND
def symbolic_sis(n, alpha, q, m=None, epsilon=None):
if epsilon is None:
epsilon = var("epsilon")
assume(epsilon > 0)
assume(epsilon < ZZ(1) / 2)
delta_0 = var("delta_0")
assume(delta_0 >= 1.0)
e = alpha * q / sqrt(2 * pi)
if m is None:
m = sqrt(n * log(q) / log(delta_0))
v = e * delta_0**m * q**(n / m) # norm of the vector
# epsilon = exp(-pi*(|v|^2/q^2))
f = log(1 / epsilon) / pi == (v / q)**2
# solve
f = 2 * q**2 * m * f * pi
f = f.simplify_full()
f = f.solve(delta_0**(2 * m))[0]
f = f.log().canonicalize_radical()
f = f.solve(log(delta_0))[0]
f = f.simplify_log()
return f
def test_manifoldhp(M):
qd_equiv, snap_high = 209, 2048
M_hp = M.high_precision()
M_snap_low = M.copy()
M_snap_high = M.copy()
shapes_qd = vector(M_hp.tetrahedra_shapes('rect'))
log_shapes_qd = vector(M_hp.tetrahedra_shapes('log'))
shapes_snap_low = vector(polished_tetrahedra_shapes(M_snap_low, bits_prec=qd_equiv))
shapes_snap_low = shapes_snap_low.change_ring(CC)
log_shapes_snap_low = vector([log(s) for s in
polished_tetrahedra_shapes(M_snap_low, bits_prec=qd_equiv)])
shapes_snap_high = vector(polished_tetrahedra_shapes(M_snap_high, bits_prec=snap_high))
log_shapes_snap_high = vector([log(s) for s in
polished_tetrahedra_shapes(M_snap_high, bits_prec=snap_high)])
print(" ManifoldHP shape errors:" , log_infinity_norm(shapes_qd - shapes_snap_high))
print(" ManifoldHP log shape errors:" , log_infinity_norm(log_shapes_qd - log_shapes_snap_high))
print(" Snap @ 212 bits shape errors:", log_infinity_norm(shapes_snap_low - shapes_snap_high))
print(" Snap @ 212 bits log_shape errors:", log_infinity_norm(log_shapes_snap_low - log_shapes_snap_high))
fgargs = [False, False, False]
G_qd = to_matrix_gens(M_hp.fundamental_group(*fgargs))
G_snap_low = to_matrix_gens(polished_holonomy(M_snap_low, bits_prec=qd_equiv,
fundamental_group_args=fgargs))
G_snap_high = to_matrix_gens(polished_holonomy(M_snap_high, bits_prec=snap_high,
fundamental_group_args=fgargs))
print(" ManifoldHP matrix errors:", compare_matrices(G_qd, G_snap_high))
print(" Snap @ 212 bits matrix errors:", compare_matrices(G_snap_low, G_snap_high))
def symbolic_modulus_switching(n, alpha, q, h, m=None, epsilon=None):
if epsilon is None:
epsilon = var("epsilon")
assume(epsilon > 0)
assume(epsilon < ZZ(1) / 2)
delta_0 = var("delta_0")
assume(delta_0 >= 1.0)
if m is None:
m = sqrt(n * log(q) / log(delta_0))
e = alpha * q / sqrt(2 * pi)
c = e * sqrt(m - n) / sqrt(h)
v = delta_0**m * (q / c)**(n / m) # norm of the vector
v_ = v**2 / m # variance of each component
v_r = (m - n) * e**2 * v_ # noise contribution
v_l = h * v_ * c**2 # nose contribution of rounding noise
# epsilon = exp(-pi*(|v|^2/q^2))
f = log(1 / epsilon) / pi == (v_l + v_r) / q**2
# solve
f = 2 * q**2 * m * f * pi
f = f.simplify_full()
f = f.solve(delta_0**(2 * m))[0]
f = f.log().canonicalize_radical()
f = f.solve(log(delta_0))[0]
f = f.simplify_log()
return f
def instances(GD, N, MD, p):
"""
The formula from "Optimizing the oracle under a depth limit". Assuming single-target, t = 1.
:params GD: Grover's oracle depth
:params N: keyspace size
:params MD: MAXDEPTH
:params p: target success probability
Assuming p = 1
In depth MD can fit j = floor(MD/GD) iterations.
These give probability 1 for a search space of size M.
p(j) = sin((2j+1)theta)**2
1 = sin((2j+1)theta)**2
1 = sin((2j+1)theta)
(2j+1)theta = pi/2
theta = pi/(2(2j+1)) = sqrt(t/M) = 1/sqrt(M).
sqrt(M) = 2(2j+1)/pi
M = (2(2j+1)/pi)**2
Hence need S = ceil(N/M) machines.
S = ceil(N/(2(2*floor(MD/GD)+1)/pi)**2)
Else
Could either lower each individual computer's success prob, since the target is inside only one computer's state.
Then given a requested p, we have
p = sin((2j+1)theta)**2
arcsine(sqrt(p)) = (2j+1)theta = (2j+1)/sqrt(M)
M = (2j+1)**2/arcsine(sqrt(p))**2
S = ceil(N*(arcsine(sqrt(p))/(2j+1))**2)
Or could just run full depth but have less machines.
For a target p, one would choose to have ceil(p*S) machines, where S is chosen as in the p = 1 case.
Then look at which of both strategies gives lower cost.
"""
# compute the p=1 case first
S1 = ceil(N / (2 * (2 * floor(MD / GD) + 1) / pi)**2)
# An alternative reasoning giving the same result for p == 1 (up to a very small difference):
# Inner parallelisation gives sqrt(S) speedup without loosing success prob.
# Set ceil(sqrt(N) * pi/4) * GD/sqrt(S) = MAXDEPTH
# S1 = float(ceil(((pi*sqrt(N)/4) * GD / MD)**2))
if p == 1:
return S1
else:
Sp = ceil(N * (arcsin(sqrt(R(p))) / (2 * floor(MD / GD) + 1))**2)
if ceil(p * S1) == Sp:
print(
"NOTE: for (GD, log2(N), log2(MD), p) == (%d, %.2f, %.2f, %.2f) naive reduction of parallel machines is not worse!"
% (GD, log(N, 2).n(), log(MD, 2).n(), p))
elif ceil(p * S1) < Sp:
print(
"NOTE: for (GD, log2(N), log2(MD), p) == (%d, %.2f, %.2f, %.2f) naive reduction of parallel machines is better!"
% (GD, log(N, 2).n(), log(MD, 2).n(), p))
res = min(Sp, ceil(p * S1))
return res
def realball_to_mid_rad_str(elt, extra_digits=9, max_rad=2**-103):
# conversion back and forth from binary to decimal may incur various losses
# 9 extra digits seems to make the trick
if elt.mid() != 0:
digits = ceil(log(elt.mid().abs()/(2*elt.rad()))/log(10)) + extra_digits
mid = elt.mid().str(digits=digits)
else:
mid = '0'
rad = elt.rad().str()
assert elt in elt.parent()(mid, float(rad))
return mid, rad
def compare_formulas_2(D, k):
d1 = old_div(RR(abs(D)), RR(6))
if D < 0:
D = -D
s1 = RR(
sqrt(abs(D)) * sum([
log(d) for d in divisors(D) if is_fundamental_discriminant(-d)
and kronecker(-d, old_div(D, d)) == 1
]))
d2 = RR((old_div(2, (sqrt(3) * pi))) * s1)
return d1 - d2, d2, RR(2 * sqrt(D) * log(D) / pi)
def estimate(nlen=256, m=85, klen=254, skip=None):
"""
Estimate the cost of solving HNP for an ECDSA with biased nonces instance.
:param nlen:
:param m:
:param klen:
:param compute:
:returns:
:rtype:
EXAMPLES::
sage: estimate(256, m=85, klen=254)
sage: estimate(160, m=85, klen=158)
"""
from usvp import solvers
if skip is None:
skip = []
ecdsa = ECDSA(nbits=nlen)
klen_list = make_klen_list(klen, m)
gh = ECDSASolver.ghf(m, ecdsa.n, klen_list, prec=nlen // 2)
vol = ECDSASolver.volf(m, ecdsa.n, klen_list, prec=nlen // 2)
target_norm = ECDSASolver.evf(m, max(klen_list), prec=nlen // 2)
print(
("% {t:s} {h:s}, nlen: {nlen:3d}, m: {m:2d}, klen: {klen:.3f}").format(
t=str(datetime.datetime.now()), h=socket.gethostname(), nlen=nlen, m=m, klen=float(mean(klen_list))
)
)
print(" E[|b[0]|]: 2^{v:.2f}".format(v=float(RR(log(gh, 2)))))
print(" E[|v|]: 2^{v:.2f}".format(v=float(RR(log(target_norm, 2)))))
print(" E[v]/E[b[0]]: %.4f" % float(target_norm / gh))
print("")
for solver in solvers:
if solver in skip:
continue
cost, params = solvers[solver].estimate((2 * log(vol), m + 1), target_norm ** 2)
if cost is None:
print(" {solver:20s} not applicable".format(solver=solver))
continue
else:
print(
" {solver:20s} cost: 2^{c:.1f} cycles ≈ {t:12.4f}h, aux data: {params}".format(
solver=solver, c=float(log(cost, 2)), t=cost / (2.0 * 10.0 ** 9 * 3600.0), params=params
)
)
def run_instance(L, block_size, tours, evec):
from fpylll import BKZ, LLL, GSO, IntegerMatrix
from fpylll.algorithms.bkz2 import BKZReduction as BKZ2
from sage.all import e
A = IntegerMatrix.from_matrix(L)
block_size = ZZ(block_size)
par = BKZ.Param(block_size=block_size,
strategies=BKZ.DEFAULT_STRATEGY,
flags=BKZ.VERBOSE)
block_size = ZZ(block_size)
delta_0 = (block_size / (2 * pi * e) *
(pi * block_size)**(1 / block_size))**(1 / (2 * block_size - 1))
n = ZZ(L.nrows())
alpha = delta_0**(-2 * n / (n - 1))
if len(evec) == n - 1:
evec = vector(list(evec) + [1])
LLL.reduction(A)
M = GSO.Mat(A)
M.update_gso()
vol = sqrt(prod([RR(M.get_r(i, i)) for i in range(n)]))
norms = [
map(lambda x: RR(log(x, 2)),
[(alpha**i * delta_0**n * vol**(1 / n))**2 for i in range(n)])
]
def proj(v, i):
return v - vector(RR, M.to_canonical(list(M.from_canonical(v, 0, i))))
# norms += [map(lambda x: RR(log(x,2)),
# [(stddev*sqrt(n-i))**2 for i in range(n)])]
norms += [
map(lambda x: RR(log(x, 2)),
[proj(evec, i).norm()**2 for i in range(1, n - 1)])
]
norms += [[log(RR(M.get_r(i, i)), 2) for i in range(n)]]
bkz = BKZ2(M)
for i in range(tours):
bkz.tour(par)
norms += [[log(M.get_r(i, i), 2) for i in range(n)]]
return A.to_matrix(matrix(ZZ, n, n)), norms
def helib_find_q(n,
target_cost,
helib_offset=log(2.33 * 10**9, 2),
q=None,
step_size=2,
sigma=8.0):
if q is None:
q = n**2
while True:
cost = sis(n, sigma / q, q, optimisation_target="lp")
if log(cost["lp"], 2.) - helib_offset < target_cost:
return q / step_size
else:
q *= step_size
def compute_kernel(args):
if args.seed is not None:
set_random_seed(args.seed)
FPLLL.set_random_seed(args.seed)
ecdsa = ECDSA(nbits=args.nlen)
lines, k_list, _ = ecdsa.sample(m=args.m, klen_list=args.klen_list, seed=args.seed, errors=args.e)
w_list = [2 ** (klen - 1) for klen in args.klen_list]
f_list = [Integer(max(w_list) / wi) for wi in w_list]
targetvector = vector([(k - w) * f for k, w, f in zip(k_list, w_list, f_list)] + [max(w_list)])
try:
solver = ECDSASolver(ecdsa, lines, m=args.m, d=args.d, threads=args.threads)
except KeyError:
raise ValueError("Algorithm {alg} unknown".format(alg=args.alg))
expected_length = solver.evf(args.m, max(args.klen_list), prec=args.nlen // 2)
gh = solver.ghf(args.m, ecdsa.n, args.klen_list, prec=args.nlen // 2)
params = args.params if args.params else {}
key, res = solver(solver=args.algorithm, flavor=args.flavor, **params)
RR = RealField(args.nlen // 2)
logging.info(
(
"try: {i:3d}, tag: 0x{tag:016x}, success: {success:1d}, "
"|v|: 2^{v:.2f}, |b[0]|: 2^{b0:.2f}, "
"|v|/|b[0]|: {b0r:.3f}, "
"E|v|/|b[0]|: {eb0r:.3f}, "
"|v|/E|b[0]|: {b0er:.3f}, "
"cpu: {cpu:10.1f}s, "
"wall: {wall:10.1f}s, "
"work: {total:d}"
).format(
i=args.i,
tag=args.tag,
success=int(res.success),
v=float(log(RR(targetvector.norm()), 2)),
b0=float(log(RR(res.b0), 2)),
b0r=float(RR(targetvector.norm()) / RR(res.b0)),
eb0r=float(RR(expected_length) / RR(res.b0)),
b0er=float(RR(targetvector.norm()) / gh),
cpu=float(res.cputime),
wall=float(res.walltime),
total=res.ntests,
)
)
return key, res, float(targetvector.norm())
def volf(cls, m, p, klen_list, prec=53):
"""
Lattice volume.
:param m: number of samples
:param p: ECDSA modulus
:param klen_list: list of lengths of key to recover
:param prec: precision to use
"""
w = 2 ** (max(klen_list) - 1)
RR = RealField(prec)
f_list = [Integer(w / (2 ** (klen - 1))) for klen in klen_list]
return RR(exp(log(p) * (m - 1) + sum(map(log, f_list)) + log(w)))
def fit_to_power_of_log(v):
"""
INPUT:
- v -- a list of (x,y) values, with x increasing.
OUTPUT:
- number a such that data is "approximated" by b*log(x)^a.
"""
# ALGORITHM: transform data to (log(log(x)), log(y)) and find the slope
# of the best line that fits this transformed data.
# This is the right thing to do, since if y = log(x)^a, then log(y) = a*log(log(x)).
from math import log, exp
w = [(log(log(x)), log(y)) for x, y in v]
a, b = least_squares_fit(w)
return float(a), exp(float(b))
def delta_rank(f):
"""Returns the Gamma-rank of the function with LUT f.
The Gamma-rank is the rank of the 2^{2n} \times 2^{2n} binary
matrix M defined by
M[x][y] = 1 if and only if x + y \in \Delta,
where \Delta is defined as
\Delta = \{ (a, b), DDT_f[a][b] == 2 \} ~.
"""
n = int(log(len(f), 2))
dim = 2**(2 * n)
d = ddt(f)
gamma = [(a << n) | b
for a, b in itertools.product(xrange(1, 2**n), xrange(0, 2**n))
if d[a][b] == 2]
mat_content = []
for x in xrange(0, dim):
row = [0 for j in xrange(0, dim)]
for y in gamma:
row[oplus(x, y)] = 1
mat_content.append(row)
mat_gf2 = Matrix(GF(2), dim, dim, mat_content)
return mat_gf2.rank()
def central_char_function(self):
dim = self.dimension()
dfactor = (-1)**dim
# doubling insures integers below
# we could test for when we need it, but then we carry the "if"
# throughout
charf = 2*self.character_field()
localfactors = self.local_factors_table()
bad = [0 if dim+1>len(z) else 1 for z in localfactors]
localfactors = [self.from_conjugacy_class_index_to_polynomial(j+1) for j in range(len(localfactors))]
localfactors = [z.leading_coefficient()*dfactor for z in localfactors]
# Now take logs to figure out what power these are
mypi = RealField(100)(pi)
localfactors = [charf*log(z)/(2*I*mypi) for z in localfactors]
localfactorsa = [z.real().round() % charf for z in localfactors]
# Test to see if we are ok?
localfactorsa = [localfactorsa[j] if bad[j]>0 else -1 for j in range(len(localfactorsa))]
def myfunc(inp, n):
fn = list(factor(inp))
pvals = [[localfactorsa[self.any_prime_to_cc_index(z[0])-1], z[1]] for z in fn]
# -1 is the marker that the prime divides the conductor
for j in range(len(pvals)):
if pvals[j][0] < 0:
return -1
pvals = sum([z[0]*z[1] for z in pvals])
return (pvals % n)
return myfunc
def ea_classes_in_the_ccz_class_of(f, include_start=False):
"""Returns an iterable that, when iterated over, will yield at least
one function from each of the EA-classes constituting the
CCZ-class of `f`.
Note that several functions in the same EA-class may be
returned. Solving this problem is in fact an open *research*
problem.
If `include_start` is set to False then the ea class of f is
hopefully not returned. More precisely, the spaces with thickness
0 are not considered.
"""
N = int(log(len(f), 2))
mask = sum(int(1 << i) for i in xrange(0, N))
graph_f = [(x << N) | f[x] for x in xrange(0, 2**N)]
z = lat_zeroes(f)
for b in vector_spaces_bases_iterator(z, N, 2 * N):
if include_start or thickness(b, 2 * N) > 0:
L_map = FastLinearMapping(
get_generating_matrix(b, 2 * N).transpose())
graph_g = [L_map(word) for word in graph_f]
g = [-1 for x in xrange(0, 2**N)]
for word in graph_g:
x, y = word >> N, word & mask
g[x] = y
if -1 in g:
raise Exception("CCZ map is ill defined!")
else:
yield g
def enumerate_ea_classes(f):
"""Returns a list containing at least one function from each of the
EA-classes constituting the CCZ-class of `f`.
Note that several functions in the same EA-class may be
returned. Solving this problem is in fact an open *research*
problem.
"""
N = int(log(len(f), 2))
mask = sum(int(1 << i) for i in xrange(0, N))
graph_f = [(x << N) | f[x] for x in xrange(0, 2**N)]
bases = get_lat_zeroes_spaces(f)
result = []
for b in bases:
L_map = FastLinearMapping(get_generating_matrix(b, 2 * N).transpose())
graph_g = [L_map(word) for word in graph_f]
g = [-1 for x in xrange(0, 2**N)]
for word in graph_g:
x, y = word >> N, word & mask
g[x] = y
if -1 in g:
raise Exception("permutation ill defined!")
else:
result.append(g)
return result
def dirichlet_series_coeffs(self, prec, eps=1e-10):
"""
Return the coefficients of the Dirichlet series representation
of self, up to the given precision.
INPUT:
- prec -- positive integer
- eps -- None or a positive real; any coefficient with absolute
value less than eps is set to 0.
"""
# Use multiplicativity to compute the Dirichlet series
# coefficients, then make a DirichletSeries object.
zero = RDF(0)
coeffs = [RDF(0),RDF(1)] + [None]*(prec-2)
from sage.all import log, floor # TODO: slow
# prime-power indexed coefficients
for p in prime_range(2, prec):
B = floor(log(prec, p)) + 1
series = self._local_series(p, B)
p_pow = p
for i in range(1, B):
coeffs[p_pow] = series[i] if (eps is None or abs(series[i])>eps) else zero
p_pow *= p
# non-prime-powers
from sage.all import factor
for n in range(2, prec):
if coeffs[n] is None:
a = prod(coeffs[p**e] for p, e in factor(n))
coeffs[n] = a if (eps is None or abs(a) > eps) else zero
return coeffs
def central_char_function(self):
dim = self.dimension()
dfactor = (-1) ** dim
# doubling insures integers below
# we could test for when we need it, but then we carry the "if"
# throughout
charf = 2 * self.character_field()
localfactors = self.local_factors_table()
bad = [0 if dim + 1 > len(z) else 1 for z in localfactors]
localfactors = [self.from_conjugacy_class_index_to_polynomial(j + 1) for j in range(len(localfactors))]
localfactors = [z.leading_coefficient() * dfactor for z in localfactors]
# Now take logs to figure out what power these are
mypi = RealField(100)(pi)
localfactors = [charf * log(z) / (2 * I * mypi) for z in localfactors]
localfactorsa = [z.real().round() % charf for z in localfactors]
# Test to see if we are ok?
localfactorsa = [localfactorsa[j] if bad[j] > 0 else -1 for j in range(len(localfactorsa))]
def myfunc(inp, n):
fn = list(factor(inp))
pvals = [[localfactorsa[self.any_prime_to_cc_index(z[0]) - 1], z[1]] for z in fn]
# -1 is the marker that the prime divides the conductor
for j in range(len(pvals)):
if pvals[j][0] < 0:
return -1
pvals = sum([z[0] * z[1] for z in pvals])
return pvals % n
return myfunc
def dirichlet_series_coeffs(self, prec, eps=1e-10):
"""
Return the coefficients of the Dirichlet series representation
of self, up to the given precision.
INPUT:
- prec -- positive integer
- eps -- None or a positive real; any coefficient with absolute
value less than eps is set to 0.
"""
# Use multiplicativity to compute the Dirichlet series
# coefficients, then make a DirichletSeries object.
zero = RDF(0)
coeffs = [RDF(0), RDF(1)] + [None] * (prec - 2)
from sage.all import log, floor # TODO: slow
# prime-power indexed coefficients
for p in prime_range(2, prec):
B = floor(log(prec, p)) + 1
series = self._local_series(p, B)
p_pow = p
for i in range(1, B):
coeffs[p_pow] = series[i] if (
eps is None or abs(series[i]) > eps) else zero
p_pow *= p
# non-prime-powers
from sage.all import factor
for n in range(2, prec):
if coeffs[n] is None:
a = prod(coeffs[p**e] for p, e in factor(n))
coeffs[n] = a if (eps is None or abs(a) > eps) else zero
return coeffs
def tu_decomposition(s, v, verbose=False):
"""Using the knowledge that v is a subspace of Z_s of dimension n, a
TU-decomposition (as defined in [Perrin17]) of s is performed and T
and U are returned.
"""
N = int(log(len(s), 2))
# building B
basis = extract_basis(v[0], N)
t = len(basis)
basis = complete_basis(basis, N)
B = Matrix(GF(2), N, N, [tobin(x, N) for x in reversed(basis)])
if verbose:
print "B= (rank={})\n{}".format(B.rank(), B.str())
# building A
basis = complete_basis(extract_basis(v[1], N), N)
A = Matrix(GF(2), N, N, [tobin(x, N) for x in reversed(basis)])
if verbose:
print "A= (rank={})\n{}".format(A.rank(), A.str())
# building linear equivalent s_prime
s_prime = [
apply_bin_mat(s[apply_bin_mat(x, A.inverse())], B)
for x in xrange(0, 2**N)
]
# TU decomposition of s'
T, U = get_tu_open(s_prime, t)
if verbose:
print "T=["
for i in xrange(0, 2**(N - t)):
print " {} {}".format(T[i], is_permutation(T[i]))
print "]\nU=["
for i in xrange(0, 2**t):
print " {} {}".format(U[i], is_permutation(U[i]))
print "]"
return T, U
def make_seal_table_2(our_q=False, optimisation_target="sieve"):
if our_q:
params = [(1024, 47.9), (2048, 89.0), (4096, 170.8), (8192, 331.8),
(16384, 652.2)]
else:
params = [(1024, 47.9), (2048, 94.0), (4096, 190.0), (8192, 383.0),
(16384, 767.0)]
data = OrderedDict()
for param in params:
n, q = param
q = ceil(2**q)
alpha = 8.0 / q
res = sis(n, alpha, q, optimisation_target=optimisation_target)
data[(n, alpha, q, "dual")] = res
print "%6.1f &" % log(res[optimisation_target], 2.0),
print
for param in params:
n, q = param
q = ceil(2**q)
alpha = 8.0 / q
res_lll = sis_small_secret_mod_switch(
n,
alpha,
q,
secret_bounds=(-1, 1),
h=ceil(2 * n / 3),
optimisation_target=optimisation_target,
use_lll=True)
res = sis_small_secret_mod_switch(
n,
alpha,
q,
secret_bounds=(-1, 1),
h=ceil(2 * n / 3),
optimisation_target=optimisation_target,
use_lll=False)
if res_lll[optimisation_target] < res[optimisation_target]:
res = res_lll
data[(n, alpha, q, "sparse")] = res
print "%6.1f &" % log(res[optimisation_target], 2.0),
print
return data
def get_lat_zeroes_spaces(s, n_threads=DEFAULT_N_THREADS):
"""Returns a list containing the basis of each vector space of
dimension n contained in the LAT zeroes of `s`.
"""
return get_lat_zeroes_spaces_fast(s,
int(log(len(s), 2)),
int(n_threads))
def compare_formulas_2a(D, k):
d1 = dimension_new_cusp_forms(kronecker_character(D), k)
if D < 0:
D = -D
d2 = RR(1 / pi * sqrt(D) * sum([
log(d) * sigma(old_div(D, d), 0) for d in divisors(D) if Zmod(d)
(old_div(D, d)).is_square() and is_fundamental_discriminant(-d)
]))
return d1 - d2
def printS(P, w, d, N, t, Smax, M):
step = (Smax-1)//M
for i in range(M+1):
Si = 1 + step*i
p = p_succ_outer_inv(P, Si)
it = iterations(p, N, t)
c = cost_out_p(p, Si, w, d, N, t)
print ("S = %s, p = %.2f, iter = %.2f, cost = %.2f" % (Si, p, it, log(c, 2)))
def get_ccz_equivalent_function_cartesian(s, v, verbose=False):
"""Takes as input a vector space v of the set of zeroes in
the LAT of s, v being the cartesian product of two spaces, and
returns a function CCZ equivalent to s obtained using this CCZ
space.
"""
N = int(log(len(s), 2))
T_inv, U = tu_decomposition(s, v, verbose=verbose)
t = int(log(len(T_inv[0]), 2))
T = [inverse(row) for row in T_inv]
result = [0 for x in xrange(0, 2**N)]
for l in xrange(0, 2**t):
for r in xrange(0, 2**(N - t)):
x = (l << (N - t)) | r
y_r = T[r][l]
y_l = U[l][r]
result[x] = (y_l << t) | y_r
return result
def linear_function_lut_to_matrix(l):
"""Turns the look up table of a linear function into the
corresponding binary matrix."""
n = int(log(len(l), 2))
result = []
for i in xrange(0, n):
line = [(int(l[1 << (n - 1 - j)]) >> (n - 1 - i)) & 1
for j in xrange(0, n)]
result.append(line)
return Matrix(GF(2), n, n, result)
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 _number_of_bits(n):
"""
Description:
Returns the number of bits in the binary representation of n
Input:
n - integer
Output:
num_bits - number of bits
"""
if n == 0:
return 1
else:
return floor(log(n).n() / log(2).n()) + 1
def _number_of_bits(n):
"""
Description:
Returns the number of bits in the binary representation of n
Input:
n - integer
Output:
num_bits - number of bits
"""
if n == 0:
return 1
else:
return floor(log(n).n()/log(2).n()) + 1
def prop_int_pretty(n):
"""
This function should be called whenever displaying an integer in the
properties table so that we can keep the formatting consistent
"""
if abs(n) >= 10**12:
e = floor(log(abs(n),10))
return r'$%.3f\times 10^{%d}$' % (n/10**e, e)
else:
return '$%s$' % n
def _inc_weight(Q):
'''
Let D be the differential operator ass. to Q.
Let f_1, .., f_t be vector valued modular forms of determinant
weights k_1, ..., k_t.
If the determinant weight of D(f_1, ..., f_t) is equal to
k_1 + ... + k_t + k,
this function returns k.
'''
S = Q.parent()
R = S.base_ring()
u1, _ = S.gens()
rs = R.gens()
rdct = {}
for r11, r12, _ in group(rs, 3):
rdct[r11] = 4 * r11
rdct[r12] = 2 * r12
t = [t for t, v in Q.dict().iteritems() if v != 0][0]
a = Q.map_coefficients(lambda f: f.subs(rdct))[t] / Q.subs({u1: 2 * u1})[t]
return int(log(a) / log(2))
def DLMV(K):
"""Compute the DLMV bound"""
# First compute David's C_0
Delta_K = K.discriminant().abs()
h_K = K.class_number()
R_K = K.regulator()
r_K = K.unit_group().rank()
delta_K = log(2) / (r_K + 1)
C_1_K = r_K**(r_K + 1) * delta_K**(-(r_K - 1)) / 2
C_2_K = exp(24 * C_1_K * R_K)
CHEB_DEN_BOUND = (4 * log(Delta_K**h_K) + 5 * h_K + 5)**2
C_0 = ((CHEB_DEN_BOUND**(12 * h_K)) * C_2_K + CHEB_DEN_BOUND**(6 * h_K))**4
# Now the Type 1 and 2 bounds
type_1_bound = (1 + 3**(12 * h_K))**2
type_2_bound = get_type_2_bound(K)
return max(C_0, type_1_bound, type_2_bound)
def run(num_bits,k):
"""
Description:
Runs the Dupont-Enge-Morain method multiple times until a valid curve is found
Input:
num_bits - number of bits
k - an embedding degree
Output:
(q,t,r,k,D) - an elliptic curve;
if no curve is found, the algorithm returns (0,0,0,k,0)
"""
j,r,q,t = 0,0,0,0
num_generates = 512
h = num_bits/(euler_phi(k))
tried = [(0,0)] # keep track of random values tried for efficiency
for i in range(0,num_generates):
D = 0
y = 0
while (D,y) in tried: # find a pair that we have not tried
D = -randint(1, 1024) # pick a small D so that the CM method is fast
D = fundamental_discriminant(D)
m = 0.5*(h - log(-D).n()/(2*log(2)).n())
if m < 1:
m = 1
y = randint(floor(2**(m-1)), floor(2**m))
tried.append((D,y))
q,t,r,k,D = method(num_bits,k,D,y) # run DEM
if q != 0 and t != 0 and r != 0 and k != 0 and D != 0: # found an answer, so output it
assert is_valid_curve(q,t,r,k,D), 'Invalid output'
return q,t,r,k,D
return 0,0,0,k,0 # found nothing
def list_zeros(N=None,
t=None,
limit=None,
fmt=None,
download=None):
if N is None:
N = request.args.get("N", None, int)
if t is None:
t = request.args.get("t", 0, float)
if limit is None:
limit = request.args.get("limit", 100, int)
if fmt is None:
fmt = request.args.get("format", "plain")
if download is None:
download = request.args.get("download", "no")
if limit < 0:
limit = 100
if N is not None: # None is < 0!! WHAT THE WHAT!
if N < 0:
N = 0
if t < 0:
t = 0
if limit > 100000:
# limit = 100000
#
bread = [("L-functions", url_for("l_functions.l_function_top_page")),("Zeros of $\zeta(s)$", url_for(".zetazeros"))]
return render_template('single.html', title="Too many zeros", bread=bread, kid = "dq.zeros.zeta.toomany")
if N is not None:
zeros = zeros_starting_at_N(N, limit)
else:
zeros = zeros_starting_at_t(t, limit)
if fmt == 'plain':
response = flask.Response(("%d %s\n" % (n, nstr(z,31+floor(log(z,10))+1,strip_zeros=False,min_fixed=-inf,max_fixed=+inf)) for (n, z) in zeros))
response.headers['content-type'] = 'text/plain'
if download == "yes":
response.headers['content-disposition'] = 'attachment; filename=zetazeros'
else:
response = str(list(zeros))
return response
def make_curve(num_bits, num_curves=1):
"""
Description:
Finds num_curves Barreto-Naehrig curves with a prime order that is at least 2^num_bits.
Input:
num_bits - number of bits for the prime order of the curve
num_curves - number of curves to find
Output:
curves - list of the first num_curves BN curves each of prime order at least 2^num_bits;
each curve is represented as a tuple (q,t,r,k,D)
"""
def P(y):
x = Integer(y)
return 36*pow(x,4) + 36*pow(x,3) + 24*pow(x,2) + 6*x + 1
x = Integer(floor(pow(2, (num_bits)/4.0)/(sqrt(6))))
q = 0
r = 0
t = 0
curve_num = 0
curves = []
while curve_num < num_curves or (log(q).n()/log(2).n() < 2*num_bits and not (utils.is_suitable_q(q) and utils.is_suitable_r(r) and utils.is_suitable_curve(q,t,r,12,-3,num_bits))):
t = Integer(6*pow(x,2) + 1)
q = P(-x)
r = q + 1 - t
b = utils.is_suitable_q(q) and utils.is_suitable_r(r) and utils.is_suitable_curve(q,t,r,12,-3,num_bits)
if b:
try:
assert floor(log(r)/log(2)) + 1 >= num_bits, 'Subgroup not large enough'
curves.append((q,t,r,12,-3))
curve_num += 1
except AssertionError as e:
pass
if curve_num < num_curves or not b:
q = P(x)
r = q+1-t
if (utils.is_suitable_q(q) and utils.is_suitable_r(r) and utils.is_suitable_curve(q,t,r,12,-3,num_bits)):
try:
assert floor(log(r)/log(2)) + 1 >= num_bits, 'Subgroup not large enough'
curves.append((q,t,r,12,-3))
curve_num += 1
except AssertionError as e:
pass
x += 1
return curves
def _sage_(self):
import sage.all as sage
return sage.log(self.args[0]._sage_())
def affine_global_height(affine_point, abs_val=lambda x: x.abs()):
"""The affine logarithmic height function (works only over the rationals?)."""
# use .numerical_approx() if we want a float
return log(affine_height(affine_point, abs_val))
def dimension(self,k,ignore=False, debug = 0):
if k < 2 and not ignore:
raise NotImplementedError("k has to >= 2")
s = self._signature
if not (2*k in ZZ):
raise ValueError("k has to be integral or half-integral")
if (2*k+s)%2 != 0:
return 0
m = self._m
n2 = self._n2
if self._v2.has_key(0):
v2 = self._v2[0]
else:
v2 = 1
if self._g != None:
if not self._aniso_formula:
vals = self._g.values()
#else:
#print "using aniso_formula"
M = self._g
else:
vals = self._M.values()
M = self._M
if (2*k+s)%4 == 0:
d = Integer(1)/Integer(2)*(m+n2) # |dimension of the Weil representation on even functions|
self._d = d
self._alpha4 = 1/Integer(2)*(vals[0]+v2) # the codimension of SkL in MkL
else:
d = Integer(1)/Integer(2)*(m-n2) # |dimension of the Weil representation on odd functions|
self._d = d
self._alpha4 = 1/Integer(2)*(vals[0]-v2) # the codimension of SkL in MkL
prec = ceil(max(log(M.order(),2),52)+1)+17
#print prec
RR = RealField(prec)
CC = ComplexField(prec)
if debug > 0: print "d, m = {0}, {1}".format(d,m)
eps = exp( 2 * CC.pi() * CC(0,1) * (s + 2*k) / Integer(4) )
eps = round(real(eps))
if self._alpha3 is None or self._last_eps != eps:
self._last_eps = eps
if self._aniso_formula:
self._alpha4 = 1
self._alpha3 = -sum([BB(a)*mm for a,mm in self._v2.iteritems() if a != 0])
#print self._alpha3
self._alpha3 += Integer(d) - Integer(1) - self._g.beta_formula()
#print self._alpha3, self._g.a5prime_formula()
self._alpha3 = self._alpha3/RR(2)
else:
self._alpha3 = eps*sum([(1-a)*mm for a,mm in self._v2.iteritems() if a != 0])
if debug>0: print "alpha3t = ", self._alpha3
self._alpha3 += sum([(1-a)*mm for a,mm in vals.iteritems() if a != 0])
#print self._alpha3
self._alpha3 = self._alpha3 / Integer(2)
alpha3 = self._alpha3
alpha4 = self._alpha4
if debug > 0: print alpha3, alpha4
g1=M.char_invariant(1)
g1=CC(g1[0]*g1[1])
#print g1
g2=M.char_invariant(2)
g2=CC(g2[0]*g2[1])
if debug > 0: print g2, g2.parent()
g3=M.char_invariant(-3)
g3=CC(g3[0]*g3[1])
if debug > 0: print "eps = {0}".format(eps)
if debug > 0: print "d/4 = {0}, m/4 = {1}, e^(2pi i (2k+s)/8) = {2}".format(RR(d) / RR(4), sqrt(RR(m)) / RR(4), CC(exp(2 * CC.pi() * CC(0,1) * (2 * k + s) / Integer(8))))
if eps == 1:
g2_2 = real(g2)
else:
g2_2 = imag(g2)*CC(0,1)
alpha1 = RR(d) / RR(4) - sqrt(RR(m)) / RR(4) * CC(exp(2 * CC.pi() * CC(0,1) * (2 * k + s) / Integer(8)) * g2_2)
if debug > 0: print alpha1
alpha2 = RR(d) / RR(3) + sqrt(RR(m)) / (3 * sqrt(RR(3))) * real(exp(CC(2 * CC.pi() * CC(0,1) * (4 * k + 3 * s - 10) / 24)) * (g1 + eps*g3))
if debug > 0: print "alpha1 = {0}, alpha2 = {1}, alpha3 = {2}, g1 = {3}, g2 = {4}, g3 = {5}, d = {6}, k = {7}, s = {8}".format(alpha1, alpha2, alpha3, g1, g2, g3, d, k, s)
dim = real(d + (d * k / Integer(12)) - alpha1 - alpha2 - alpha3)
if debug > 0:
print "dimension:", dim
if abs(dim-round(dim)) > 1e-6:
raise RuntimeError("Error ({0}) too large in dimension formula for {1} and k={2}".format(abs(dim-round(dim)), self._M if self._M is not None else self._g, k))
dimr = dim
dim = Integer(round(dim))
if k >=2 and dim < 0:
raise RuntimeError("Negative dimension (= {0}, {1})!".format(dim, dimr))
return dim
def dimension(self,k,ignore=False, debug = 0):
if k < 2 and not ignore:
raise NotImplementedError("k has to >= 2")
s = self._signature
if not (2*k in ZZ):
raise ValueError("k has to be integral or half-integral")
if (2*k+s)%4 != 0 and not ignore:
raise NotImplementedError("2k has to be congruent to -signature mod 4")
if self._v2.has_key(0):
v2 = self._v2[0]
else:
v2 = 1
if self._g != None:
if not self._aniso_formula:
vals = self._g.values()
#else:
#print "using aniso_formula"
M = self._g
else:
vals = self._M.values()
M = self._M
prec = ceil(max(log(M.order(),2),52)+1)+17
#print prec
RR = RealField(prec)
CC = ComplexField(prec)
d = self._d
m = self._m
if debug > 0: print d,m
if self._alpha3 == None:
if self._aniso_formula:
self._alpha4 = 1
self._alpha3 = -sum([BB(a)*mm for a,mm in self._v2.iteritems() if a != 0])
#print self._alpha3
self._alpha3 += Integer(d) - Integer(1) - self._g.beta_formula()
#print self._alpha3, self._g.a5prime_formula()
self._alpha3 = self._alpha3/RR(2)
else:
self._alpha3 = sum([(1-a)*mm for a,mm in self._v2.iteritems() if a != 0])
#print self._alpha3
self._alpha3 += sum([(1-a)*mm for a,mm in vals.iteritems() if a != 0])
#print self._alpha3
self._alpha3 = self._alpha3 / Integer(2)
self._alpha4 = 1/Integer(2)*(vals[0]+v2) # the codimension of SkL in MkL
alpha3 = self._alpha3
alpha4 = self._alpha4
if debug > 0: print alpha3, alpha4
g1=M.char_invariant(1)
g1=CC(g1[0]*g1[1])
#print g1
g2=M.char_invariant(2)
g2=RR(real(g2[0]*g2[1]))
if debug > 0: print g2, g2.parent()
g3=M.char_invariant(-3)
g3=CC(g3[0]*g3[1])
if debug > 0: print RR(d) / RR(4), sqrt(RR(m)) / RR(4), CC(exp(2 * CC.pi() * CC(0,1) * (2 * k + s) / Integer(8)))
alpha1 = RR(d) / RR(4) - (sqrt(RR(m)) / RR(4) * CC(exp(2 * CC.pi() * CC(0,1) * (2 * k + s) / Integer(8))) * g2)
if debug > 0: print alpha1
alpha2 = RR(d) / RR(3) + sqrt(RR(m)) / (3 * sqrt(RR(3))) * real(exp(CC(2 * CC.pi() * CC(0,1) * (4 * k + 3 * s - 10) / 24)) * (g1+g3))
if debug > 0: print alpha1, alpha2, g1, g2, g3, d, k, s
dim = real(d + (d * k / Integer(12)) - alpha1 - alpha2 - alpha3)
if debug > 0:
print "dimension:", dim
if abs(dim-round(dim)) > 1e-6:
raise RuntimeError("Error ({0}) too large in dimension formula for {1} and k={2}".format(abs(dim-round(dim)), self._M if self._M is not None else self._g, k))
dimr = dim
dim = Integer(round(dim))
if k >=2 and dim < 0:
raise RuntimeError("Negative dimension (= {0}, {1})!".format(dim, dimr))
return dim
def silke(A, c, beta, h, m=None, scale=1, float_type="double"):
"""
:param A: LWE matrix
:param c: LWE vector
:param beta: BKW block size
:param m: number of samples to consider
:param scale: scale rhs of lattice by this factor
"""
from fpylll import BKZ, IntegerMatrix, LLL, GSO
from fpylll.algorithms.bkz2 import BKZReduction as BKZ2
if m is None:
m = A.nrows()
L = dual_instance1(A, scale=scale)
L = IntegerMatrix.from_matrix(L)
L = LLL.reduction(L, flags=LLL.VERBOSE)
M = GSO.Mat(L, float_type=float_type)
bkz = BKZ2(M)
t = 0.0
param = BKZ.Param(block_size=beta,
strategies=BKZ.DEFAULT_STRATEGY,
auto_abort=True,
max_loops=16,
flags=BKZ.VERBOSE|BKZ.AUTO_ABORT|BKZ.MAX_LOOPS)
bkz(param)
t += bkz.stats.total_time
H = copy(L)
import pickle
pickle.dump(L, open("L-%d-%d.sobj"%(L.nrows, beta), "wb"))
E = []
Y = set()
V = set()
y_i = vector(ZZ, tuple(L[0]))
Y.add(tuple(y_i))
E.append(apply_short1(y_i, A, c, scale=scale)[1])
v = L[0].norm()
v_ = v/sqrt(L.ncols)
v_r = 3.2*sqrt(L.ncols - A.ncols())*v_/scale
v_l = sqrt(h)*v_
fmt = u"{\"t\": %5.1fs, \"log(sigma)\": %5.1f, \"log(|y|)\": %5.1f, \"log(E[sigma]):\" %5.1f}"
print
print fmt%(t,
log(abs(E[-1]), 2),
log(L[0].norm(), 2),
log(sqrt(v_r**2 + v_l**2), 2))
print
for i in range(m):
t = cputime()
M = GSO.Mat(L, float_type=float_type)
bkz = BKZ2(M)
t = cputime()
bkz.randomize_block(0, L.nrows, stats=None, density=3)
LLL.reduction(L)
y_i = vector(ZZ, tuple(L[0]))
l_n = L[0].norm()
if L[0].norm() > H[0].norm():
L = copy(H)
t = cputime(t)
Y.add(tuple(y_i))
V.add(y_i.norm())
E.append(apply_short1(y_i, A, c, scale=scale)[1])
if len(V) >= 2:
fmt = u"{\"i\": %4d, \"t\": %5.1fs, \"log(|e_i|)\": %5.1f, \"log(|y_i|)\": %5.1f,"
fmt += u"\"log(sigma)\": (%5.1f,%5.1f), \"log(|y|)\": (%5.1f,%5.1f), |Y|: %5d}"
print fmt%(i+2, t, log(abs(E[-1]), 2), log(l_n, 2), log_mean(E), log_var(E), log_mean(V), log_var(V), len(Y))
return E
def log_var(X):
return log(variance(X).sqrt(), 2)
def log_mean(X):
return log(mean([abs(x) for x in X]), 2)