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@mimoo
Last active May 2, 2016 10:34
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plaid
# plaid ctf triplets
data = [
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]
import time
debug = True
# display stats on helpful vectors
def helpful_vectors(BB, modulus):
nothelpful = 0
for ii in range(BB.dimensions()[0]):
if BB[ii,ii] >= modulus:
nothelpful += 1
print nothelpful, "/", BB.dimensions()[0], " vectors are not helpful"
# display matrix picture with 0 and X
def matrix_overview(BB, bound):
for ii in range(BB.dimensions()[0]):
a = ('%02d ' % ii)
for jj in range(BB.dimensions()[1]):
a += '0' if BB[ii,jj] == 0 else 'X'
if BB.dimensions()[0] < 60:
a += ' '
if BB[ii, ii] >= bound:
a += '~'
print a
def boneh_durfee(pol, modulus, mm, tt, XX, YY):
"""
Boneh and Durfee revisited by Herrmann and May
finds a solution if:
* d < N^delta
* |x| < e^delta
* |y| < e^0.5
whenever delta < 1 - sqrt(2)/2 ~ 0.292
"""
# substitution (Herrman and May)
PR.<u, x, y> = PolynomialRing(ZZ)
Q = PR.quotient(x*y + 1 - u) # u = x*y + 1
polZ = Q(pol).lift()
UU = XX*YY + 1
# x-shifts
gg = []
for kk in range(mm + 1):
for ii in range(mm - kk + 1):
xshift = x^ii * modulus^(mm - kk) * polZ(u, x, y)^kk
gg.append(xshift)
gg.sort()
# x-shifts monomials
monomials = []
for polynomial in gg:
for monomial in polynomial.monomials():
if monomial not in monomials:
monomials.append(monomial)
monomials.sort()
# y-shifts (selected by Herrman and May)
for jj in range(1, tt + 1):
for kk in range(floor(mm/tt) * jj, mm + 1):
yshift = y^jj * polZ(u, x, y)^kk * modulus^(mm - kk)
yshift = Q(yshift).lift()
gg.append(yshift) # substitution
# y-shifts monomials
for jj in range(1, tt + 1):
for kk in range(floor(mm/tt) * jj, mm + 1):
monomials.append(u^kk * y^jj)
# construct lattice B
nn = len(monomials)
BB = Matrix(ZZ, nn)
for ii in range(nn):
BB[ii, 0] = gg[ii](0, 0, 0)
for jj in range(1, ii + 1):
if monomials[jj] in gg[ii].monomials():
BB[ii, jj] = gg[ii].monomial_coefficient(monomials[jj]) * monomials[jj](UU,XX,YY)
# check if vectors are helpful
if debug:
helpful_vectors(BB, modulus^mm)
# check if determinant is correctly bounded
if debug:
det = BB.det()
bound = modulus^(mm*nn)
if det >= bound:
print "We do not have det < bound. Solutions might not be found."
print "Try with highers m and t."
diff = (log(det) - log(bound)) / log(2)
print "size det(L) - size e^(m*n) = ", floor(diff)
return -1, -1
else:
print "det(L) < e^(m*n)"
# debug: display matrix
if debug:
matrix_overview(BB, modulus^mm)
# LLL
BB = BB.LLL()
# vector 1 & 2 -> polynomials 1 & 2
PR.<w,z> = PolynomialRing(ZZ)
pol1 = pol2 = 0
for jj in range(nn):
pol1 += monomials[jj](w*z+1,w,z) * BB[0, jj] / monomials[jj](UU,XX,YY)
pol2 += monomials[jj](w*z+1,w,z) * BB[1, jj] / monomials[jj](UU,XX,YY)
# resultant
PR.<q> = PolynomialRing(ZZ)
rr = pol1.resultant(pol2)
if rr.is_zero() or rr.monomials() == [1]:
print "failure"
return pol1, pol2
rr = rr(q, q)
# solutions
soly = rr.roots()
if len(soly) == 0:
print "your prediction (delta) is too small"
return 0, 0
soly = soly[0][0]
print "found for y_0:", soly
ss = pol1(q, soly)
solx = ss.roots()[0][0]
print "found for x_0:", solx
#
return solx, soly
############################################
# Plaid CTF
##########################################
start_time = time.time()
for index, triplet in enumerate(data):
print "\n\nTesting triplet #", index
N = triplet[0]
e = triplet[1]
# Problem put in equation (default)
P.<x,y> = PolynomialRing(ZZ)
A = int((N+1)/2)
pol = 1 + x * (A + y)
# Default values
# you should tweak delta and m. X should be OK as well
#
delta = 0.26 # < 0.292 (Boneh & Durfee's bound)
X = 2*floor(N^delta) # this _might_ be too much
Y = floor(N^(1/2)) # correct if p, q are ~ same size
# Checking bounds
print "=== checking values ==="
print "* size of e:", int(log(e)/log(2))
print "* size of N:", int(log(N)/log(2))
print "* delta (hypothesis):", delta
# boneh_durfee
print "=== running algorithm ==="
# we need to increase m
solx = -1
m = 2
while solx == -1:
m += 1
t = int((1-2*delta) * m) # optimization from Herrmann and May
print "* m: ", m, "and t:", t
solx, soly = boneh_durfee(pol, e, m, t, X, Y)
print solx
# Checking solutions (for the demo)
if solx != 0:
d = int(pol(solx, soly) / e)
print "found the private exponent d!"
print d
m = power_mod(triplet[2], d, N)
hex_string = "%x" % m
import binascii
print "the plaintext:", binascii.unhexlify(hex_string)
break
# Stats
print("Found the solution in %s seconds" % (time.time() - start_time))
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