plaid

plaidctf.sage

# plaid ctf triplets
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    [0xc0b7267fb9305ece55bb82e4bc2bd36226919aefff62338bdfa39a6c30f92c62a896af478d7d4f0293bfc4bcfb3a21d28568e33bf9daa57b04b1918777618d2d4dfee7a554f92ec79df0a7896a49fd14936607365034937290c14c5724ed4d4884cfc5b6a22e582596c96ff62cdd379b309ef7a283eb6decdb85e98713393a4f, 0xc219cc959bc1849c3c14f3300e44f9bf499d30b43698d36f2e3d7b676476651fded82391e4a50106fc68027c4c39dfe9fa3c51669b0d00f38ffbf49ccba38ec57e954a4dac7dd0ff0b508cef085fa6f0aeb1e3d90f3b7a268aca1583d0fec46565613d3eb256c266f56f07f8ab6c2aa302eed2f398b61f4f3f0aeb6806582045, 0xbcb5580a86e402f01e2c80809d4bc4e67f7229d9046a08184ca8a15135e69898b9db17fdbd3a95761bc70785352e64a644d10435849c6f7f57e25b4d01f6a6da0808eb28ee43158630ea2f7e25cd8215dda6dbcd5395249b07109949f71dcb138851500621d0590958202411630894fabf35ce1e89807240873b0de01c6a30b7]
    ]

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))