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@hellman hellman/0_solve.sage
Last active Apr 19, 2020

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PlaidCTF 2020 - MPKC (Crypto 350)
from sage.all import *
import sys
q,n,a,s = (3,59,10,25)
m = n+1-a+s
FF = GF(q)
R = PolynomialRing(FF, ["x{}".format(i) for i in range(n)])
xs = R.gens()
pk, ct = open("output").readlines()
ct = eval(ct)
try:
pk = load("pub")
print("load ok")
except IOError:
pk = eval(pk.replace("^", "**"), {}, {"x%d" % i: x for i, x in enumerate(xs)})
save(pk, "pub")
from itertools import *
# 1. find linear combinations of outputs which have no quadratic terms
order = [prod(c) for c in combinations_with_replacement(xs, 2)]
mono2i = {mono: i for i, mono in enumerate(order)}
m = matrix(FF, len(pk), len(order))
for y, poly in enumerate(pk):
for coef, mono in list(poly):
if mono.degree() == 2:
m[y,mono2i[mono]] = coef
# matrix to obtain linear functions
mdec = m.left_kernel().matrix()
# 2. convert affine polys to a matrix
mono2i = {x: i for i, x in enumerate(xs)}
polys = mdec * vector(R, pk)
m = matrix(FF, len(polys), n)
t = vector(FF, len(polys))
for y, poly in enumerate(polys):
for coef, mono in list(poly):
if mono.degree() == 1:
m[y,mono2i[mono]] = coef
elif mono.degree() == 0:
t[y] = coef
# generate kernel to exhaust solutions
zs = list(m.right_kernel())
print("kernel size", len(zs))
presol = []
for I, block0 in enumerate(ct[::-1]):
print("Block", I)
block = mdec * vector(FF, block0)
sol0 = m.solve_right(block - t)
for z in zs:
sol = sol0 + z
full_sol = list(presol) + list(sol[::-1])
s = "".join(map(str, full_sol))
s = int(s, 3) * 3**(n*(5-I))
# some hand calculations of sizes
if s >= 256**70//2: continue
test = s.to_bytes(70, "big")[:11*(I+1)].strip(b"\x00")
if sum((32 <= c < 127) for c in test) >= len(test):
full = s.to_bytes(70, "big").strip(b"\x00")
print("maybe", full)
for p, c in zip(pk[:10], block0):
if p(*sol) != c:
break
else:
presol += list(sol[::-1])
print("verified!")
break
# Jiahui Chen et al. cryptosystem, 80-bit security
# WARNING: very slow implementation.
import sys
q,n,a,s = (3,59,10,25)
m = n+1-a+s
FF = GF(q)
R = PolynomialRing(FF, ["x{}".format(i) for i in range(n)])
xs = R.gens()
def keygen():
while True:
C = random_matrix(FF, n+1, n)
if matrix(FF, [2*C[i]-2*C[i+1] for i in range(n)]).is_invertible():
break
FC = []
for i in range(n+1):
p = 0
for j in range(n):
p += (xs[j] - C[i][j])^2
FC.append(p)
while True:
S_lin = random_matrix(FF, n, n)
if S_lin.is_invertible():
break
S_trans = (FF^n).random_element()
S = (S_lin, S_trans)
while True:
T_lin = random_matrix(FF, m, m)
if T_lin.is_invertible():
break
T_trans = (FF^m).random_element()
T = (T_lin, T_trans)
G = []
for i in range(s):
G.append(R.random_element(degree=2, terms=Infinity))
F = FC[:n+1-a] + G
P = vector(xs)
P = S[0]*P
P += S[1]
v = []
for i in range(len(F)):
v.append(F[i](*P))
print("keygen {}/{}".format(i+1,len(F)), file=sys.stderr)
P = vector(v)
P = T[0]*P
P += T[1]
print("done keygen", file=sys.stderr)
return (P, (C, G, S, T))
def make_blocks(ss):
x = 0
for i in ss:
x = x*256+ord(i)
v = []
while x > 0:
v.append(FF(x%q))
x = x//q
v += [FF(0) for i in range(n - (len(v) % n))]
blocks = []
for i in range(0, len(v), n):
blocks.append(vector(v[i:i+n]))
return blocks
def combine_blocks(blocks):
x = 0
for i in blocks[::-1]:
for j in i[::-1]:
x = x*q+Integer(j)
ss = ""
while x > 0:
ss = chr(x % 256) + ss
x = x//256
return ss
def encrypt_block(plain, pk):
return pk(*plain)
def encrypt(plain, pk):
blocks = make_blocks(plain)
enc = []
for i in range(len(blocks)):
print("encrypt {}/{}".format(i+1,len(blocks)), file=sys.stderr)
enc.append(encrypt_block(blocks[i], pk))
return enc
def decrypt_block(cipher, sk):
C, G, S, T = sk
C2I = matrix(FF, [2*C[i]-2*C[i+1] for i in range(n)]).inverse()
cv = []
for i in range(n):
cc = 0
for j in range(n):
cc += C[i+1][j]^2 - C[i][j]^2
cv.append(cc)
cv = vector(cv)
g1 = T[0].inverse()*(cipher - T[1])
for g2 in FF^a:
print("decrypt: trying:", g2, file=sys.stderr)
g = vector(list(g1)[:n+1-a]+list(g2))
g_diff = vector([g[i+1]-g[i] for i in range(len(g)-1)])
d = C2I * (g_diff-cv)
for i,j in zip(G,g1[n+1-a:]):
if i(*d) != j:
break
else:
return S[0].inverse() * (d - S[1])
def decrypt(cipher, sk):
dec = []
for i in range(len(cipher)):
print("decrypt {}/{}".format(i+1,len(cipher)), file=sys.stderr)
dec.append(decrypt_block(cipher[i], sk))
return combine_blocks(dec)
pk, sk = keygen()
print(pk)
flag = open("flag.txt").read().strip()
print(encrypt(flag, pk))
(x0^2 - x1^2 - x2^2 - x0*x3 + x1*x3 + x2*x3 - x0*x4 + x1*x4 + x3*x4 - x4^2 - x0*x5 - x2*x5 - x4*x5 + x5^2 - x0*x6 + x1*x6 - x2*x6 - x3*x6 + x5*x6 - x6^2 + x0*x7 + x1*x7 - x2*x7 + x3*x7 + x4*x7 - x5*x7 - x6*x7 + x0*x8 + x1*x8 + x2*x8 + x3*x8 - x5*x8 - x6*x8 - x7*x8 + x1*x9 - x2*x9 - x3*x9 - x5*x9 - x6*x9 - x7*x9 + x9^2 - x2*x10 + x3*x10 + x4*x10 - x5*x10 + x6*x10 - x7*x10 + x10^2 - x1*x11 + x2*x11 - x3*x11 - x5*x11 - x10*x11 - x1*x12 + x2*x12 - x3*x12 - x4*x12 + x5*x12 - x6*x12 - x7*x12 - x8*x12 + x9*x12 - x11*x12 - x1*x13 - x2*x13 - x3*x13 - x4*x13 + x7*x13 - x8*x13 + x11*x13 + x13^2 - x0*x14 - x1*x14 + x2*x14 - x5*x14 + x6*x14 + x11*x14 - x13*x14 + x0*x15 + x1*x15 + x3*x15 - x5*x15 - x10*x15 - x11*x15 - x13*x15 + x14*x15 - x0*x16 - x1*x16 + x3*x16 + x4*x16 + x5*x16 + x6*x16 - x9*x16 - x11*x16 + x12*x16 - x13*x16 + x14*x16 + x16^2 + x0*x17 - x1*x17 - x4*x17 - x6*x17 - x8*x17 + x9*x17 - x10*x17 - x11*x17 - x13*x17 + x14*x17 + x15*x17 + x16*x17 - x0*x18 - x1*x18 - x3*x18 - x5*x18 + x7*x18 - x8*x18 + x12*x18 - x13*x18 - x14*x18 + x15*x18 + x0*x19 - x2*x19 + x6*x19 - x12*x19 - x14*x19 + x15*x19 - x16*x19 + x17*x19 + x18*x19 + x19^2 - x0*x20 - x1*x20 + x2*x20 + x3*x20 + x4*x20 + x6*x20 - x7*x20 - x8*x20 + x10*x20 + x11*x20 - x12*x20 - x14*x20 - x16*x20 - x18*x20 + x20^2 - x1*x21 + x2*x21 - x4*x21 - x5*x21 - x6*x21 + x7*x21 + x8*x21 - x9*x21 + x11*x21 - x13*x21 - x14*x21 + x16*x21 - x18*x21 - x20*x21 - x21^2 - x0*x22 - x1*x22 + x2*x22 - x3*x22 + x4*x22 + x5*x22 + x6*x22 - x7*x22 + x9*x22 + x10*x22 - x12*x22 - x13*x22 + x15*x22 + x17*x22 - x18*x22 - x19*x22 - x20*x22 + x21*x22 - x0*x23 - x1*x23 + x2*x23 - x3*x23 + x5*x23 + x7*x23 + x10*x23 - x11*x23 + x13*x23 + x14*x23 + x17*x23 - x19*x23 + x20*x23 + x21*x23 + x22*x23 + x23^2 + x1*x24 + x2*x24 - x4*x24 + x5*x24 - x7*x24 - x8*x24 + x9*x24 + x10*x24 - x11*x24 + x13*x24 + x15*x24 + x19*x24 + x21*x24 + x22*x24 + x23*x24 + x0*x25 + x1*x25 + x2*x25 - x3*x25 + x5*x25 - x6*x25 + x7*x25 + x8*x25 - x10*x25 + x11*x25 + x12*x25 + x13*x25 - x16*x25 - x17*x25 + x18*x25 - x19*x25 - x20*x25 - x21*x25 - x22*x25 + x24*x25 - x25^2 - x1*x26 + x3*x26 - x5*x26 - x7*x26 + x9*x26 - x17*x26 + x18*x26 + x22*x26 + x23*x26 - x24*x26 - x25*x26 - x3*x27 - x4*x27 + x7*x27 - x8*x27 - x9*x27 - x12*x27 + x14*x27 - x18*x27 + x19*x27 - x21*x27 + x22*x27 - x26*x27 + x1*x28 - x2*x28 + x3*x28 + x4*x28 - x5*x28 - x6*x28 + x7*x28 - x9*x28 + x10*x28 - x12*x28 + x13*x28 - x15*x28 - x16*x28 - x17*x28 + x20*x28 - x21*x28 + x23*x28 + x24*x28 - x26*x28 - x27*x28 - x28^2 - x0*x29 - x1*x29 + x4*x29 - x6*x29 + x7*x29 - x8*x29 - x9*x29 - x12*x29 - x15*x29 + x16*x29 + x17*x29 - x18*x29 - x19*x29 - x20*x29 + x23*x29 + x25*x29 - x26*x29 - x27*x29 - x28*x29 + x1*x30 + x4*x30 + x5*x30 + x7*x30 + x8*x30 - x9*x30 + x12*x30 - x16*x30 - x18*x30 - x19*x30 - x20*x30 + x23*x30 - x24*x30 + x25*x30 + x26*x30 + x30^2 - x0*x31 - x1*x31 - x2*x31 + x4*x31 + x5*x31 + x8*x31 + x12*x31 - x14*x31 + x15*x31 - x16*x31 - x17*x31 - x18*x31 + x20*x31 - x23*x31 + x27*x31 - x28*x31 + x30*x31 + x31^2 + x0*x32 - x1*x32 + x2*x32 + x3*x32 - x5*x32 - x8*x32 - x9*x32 + x11*x32 - x13*x32 - x14*x32 + x15*x32 - x16*x32 - x17*x32 + x18*x32 - x19*x32 + x20*x32 - x21*x32 - x23*x32 + x24*x32 + x26*x32 + x28*x32 - x29*x32 + x31*x32 - x0*x33 + x1*x33 - x3*x33 + x4*x33 - x5*x33 + x6*x33 - x8*x33 - x9*x33 - x11*x33 + x13*x33 - x14*x33 - x15*x33 - x16*x33 - x17*x33 - x18*x33 - x21*x33 - x22*x33 - x23*x33 + x25*x33 + x26*x33 - x27*x33 + x28*x33 - x29*x33 + x30*x33 + x32*x33 + x0*x34 + x2*x34 + x3*x34 + x4*x34 - x5*x34 + x6*x34 + x7*x34 + x8*x34 + x9*x34 - x13*x34 - x14*x34 - x15*x34 + x17*x34 + x19*x34 - x20*x34 + x22*x34 - x29*x34 - x30*x34 + x31*x34 + x32*x34 - x33*x34 - x34^2 + x1*x35 + x2*x35 + x4*x35 + x5*x35 - x6*x35 + x11*x35 + x13*x35 - x14*x35 + x17*x35 - x18*x35 - x24*x35 - x25*x35 + x27*x35 + x29*x35 + x32*x35 + x34*x35 - x35^2 - x0*x36 + x2*x36 + x5*x36 - x7*x36 - x8*x36 - x11*x36 + x12*x36 - x14*x36 + x15*x36 - x16*x36 - x17*x36 - x19*x36 - x20*x36 - x21*x36 + x22*x36 - x24*x36 - x25*x36 + x26*x36 + x28*x36 - x30*x36 - x35*x36 + x36^2 - x0*x37 - x1*x37 + x2*x37 - x4*x37 + x6*x37 + x7*x37 + x8*x37 - x10*x37 - x11*x37 + x12*x37 + x13*x37 - x14*x37 + x16*x37 + x18*x37 + x21*x37 + x22*x37 - x23*x37 - x24*x37 + x25*x37 - x26*x37 + x29*x37 + x30*x37 - x32*x37 + x33*x37 + x37^2 - x0*x38 + x1*x38 - x2*x38 + x4*x38 + x5*x38 - x6*x38 + x7*x38 + x8*x38 - x10*x38 + x11*x38 - x14*x38 + x15*x38 + x16*x38 - x17*x38 + x18*x38 + x19*x38 - x21*x38 + x22*x38 - x25*x38 - x26*x38 + x28*x38 - x31*x38 - x34*x38 - x36*x38 + x37*x38 - x38^2 + x0*x39 - x1*x39 + x2*x39 + x3*x39 - x4*x39 - x5*x39 + x6*x39 - x9*x39 - x12*x39 - x13*x39 + x14*x39 - x16*x39 + x18*x39 - x19*x39 + x20*x39 - x21*x39 + x23*x39 - x24*x39 + x25*x39 + x26*x39 - x28*x39 + x29*x39 + x30*x39 - x31*x39 - x32*x39 - x33*x39 - x34*x39 - x35*x39 + x36*x39 - x37*x39 - x39^2 + x0*x40 - x1*x40 + x2*x40 - x3*x40 - x4*x40 - x5*x40 - x6*x40 + x7*x40 + x9*x40 - x12*x40 + x14*x40 + x15*x40 - x17*x40 - x18*x40 + x20*x40 - x21*x40 - x22*x40 + x23*x40 - x24*x40 - x26*x40 + x28*x40 - x29*x40 - x32*x40 + x33*x40 + x34*x40 - x35*x40 - x37*x40 + x38*x40 - x40^2 + x0*x41 - x1*x41 - x4*x41 + x5*x41 - x6*x41 + x7*x41 + x8*x41 + x9*x41 + x10*x41 - x11*x41 - x12*x41 - x16*x41 + x17*x41 + x20*x41 - x21*x41 + x22*x41 - x23*x41 + x24*x41 + x26*x41 - x27*x41 + x29*x41 - x30*x41 - x31*x41 - x32*x41 - x33*x41 + x34*x41 - x35*x41 - x36*x41 - x37*x41 - x38*x41 + x39*x41 + x40*x41 + x1*x42 + x2*x42 + x4*x42 - x5*x42 - x6*x42 + x8*x42 + x11*x42 - x12*x42 - x14*x42 - x16*x42 - x17*x42 + x18*x42 + x20*x42 + x21*x42 + x22*x42 - x23*x42 - x24*x42 + x27*x42 + x29*x42 - x30*x42 + x33*x42 + x34*x42 + x36*x42 + x37*x42 + x38*x42 - x39*x42 - x40*x42 + x41*x42 - x0*x43 + x2*x43 + x3*x43 - x5*x43 + x6*x43 - x7*x43 + x9*x43 + x10*x43 + x11*x43 + x12*x43 + x13*x43 - x16*x43 - x17*x43 - x18*x43 + x21*x43 - x23*x43 + x24*x43 + x25*x43 + x26*x43 + x28*x43 + x31*x43 - x32*x43 + x33*x43 + x34*x43 + x35*x43 + x36*x43 - x37*x43 - x39*x43 + x40*x43 + x41*x43 - x43^2 + x0*x44 - x1*x44 - x2*x44 - x3*x44 - x4*x44 - x5*x44 + x6*x44 - x7*x44 + x11*x44 - x12*x44 - x13*x44 + x14*x44 + x15*x44 - x16*x44 - x17*x44 - x18*x44 - x19*x44 + x20*x44 - x21*x44 - x22*x44 + x23*x44 - x25*x44 + x27*x44 - x30*x44 + x31*x44 + x32*x44 + x33*x44 + x34*x44 + x35*x44 + x36*x44 - x37*x44 - x38*x44 - x40*x44 + x43*x44 - x44^2 - x2*x45 + x3*x45 - x4*x45 - x5*x45 + x6*x45 - x7*x45 + x9*x45 - x14*x45 + x16*x45 - x17*x45 + x18*x45 - x20*x45 + x21*x45 - x22*x45 + x23*x45 - x24*x45 + x25*x45 - x26*x45 - x27*x45 - x28*x45 + x29*x45 - x31*x45 - x32*x45 + x34*x45 + x35*x45 + x36*x45 - x38*x45 + x39*x45 - x42*x45 - x43*x45 - x44*x45 + x45^2 - x0*x46 - x1*x46 - x2*x46 + x3*x46 + x5*x46 - x7*x46 + x8*x46 + x10*x46 - x11*x46 + x12*x46 - x14*x46 + x17*x46 + x18*x46 + x20*x46 + x23*x46 + x24*x46 + x25*x46 + x26*x46 + x28*x46 - x29*x46 + x30*x46 + x31*x46 + x34*x46 + x35*x46 - x36*x46 - x37*x46 - x38*x46 - x40*x46 - x41*x46 + x43*x46 + x45*x46 - x0*x47 + x2*x47 - x4*x47 - x5*x47 + x7*x47 - x8*x47 - x9*x47 - x11*x47 + x12*x47 + x14*x47 + x15*x47 + x16*x47 - 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x1*x53 + x3*x53 + x4*x53 - x6*x53 + x7*x53 - x9*x53 - x10*x53 + x11*x53 - x12*x53 - x13*x53 - x14*x53 + x15*x53 + x16*x53 + x17*x53 - x18*x53 + x22*x53 - x23*x53 + x25*x53 + x26*x53 - x27*x53 + x28*x53 - x29*x53 - x32*x53 + x33*x53 + x34*x53 + x35*x53 - x37*x53 + x38*x53 - x40*x53 - x43*x53 - x45*x53 + x46*x53 + x47*x53 - x48*x53 + x49*x53 - x50*x53 + x51*x53 - x52*x53 + x53^2 - x1*x54 + x3*x54 - x4*x54 - x9*x54 + x13*x54 - x15*x54 - x17*x54 + x19*x54 - x20*x54 + x21*x54 - x22*x54 + x27*x54 + x28*x54 + x29*x54 + x31*x54 - x33*x54 + x35*x54 + x36*x54 - x37*x54 - x38*x54 + x39*x54 + x40*x54 - x41*x54 + x43*x54 - x44*x54 - x45*x54 + x49*x54 - x50*x54 - x51*x54 + x0*x55 + x2*x55 - x3*x55 + x4*x55 + x6*x55 + x7*x55 + x8*x55 - x9*x55 - x11*x55 - x12*x55 - x16*x55 + x17*x55 - x18*x55 - x19*x55 + x21*x55 - x22*x55 + x24*x55 + x26*x55 - x27*x55 - x28*x55 - x29*x55 - x30*x55 + x31*x55 + x32*x55 + x34*x55 - x37*x55 - x38*x55 - x40*x55 + x41*x55 - x43*x55 - x46*x55 + x47*x55 - x50*x55 - x51*x55 + x53*x55 + x54*x55 - x0*x56 - x2*x56 - x3*x56 + x4*x56 + x6*x56 + x9*x56 + x10*x56 + x11*x56 + x13*x56 + x14*x56 - x15*x56 + x16*x56 + x17*x56 - x18*x56 - x19*x56 + x20*x56 - x22*x56 - x24*x56 + x25*x56 - x26*x56 - x27*x56 + x28*x56 - x29*x56 + x31*x56 - x32*x56 - x35*x56 + x36*x56 - x38*x56 - x39*x56 + x42*x56 - x43*x56 + x44*x56 - x45*x56 + x46*x56 - x47*x56 - x49*x56 + x52*x56 - x53*x56 + x54*x56 - x55*x56 + x0*x57 + x1*x57 - x3*x57 + x4*x57 + x5*x57 + x6*x57 + x7*x57 - x9*x57 - x10*x57 + x11*x57 - x12*x57 - x14*x57 - x15*x57 - x16*x57 + x17*x57 + x19*x57 + x20*x57 - x22*x57 + x23*x57 - x24*x57 + x26*x57 - x32*x57 + x33*x57 + x36*x57 - x42*x57 + x44*x57 - x45*x57 - x46*x57 - x47*x57 - x50*x57 - x51*x57 + x52*x57 - x53*x57 + x56*x57 + x1*x58 + x2*x58 + x3*x58 + x4*x58 - x5*x58 + x7*x58 + x8*x58 - x9*x58 - x12*x58 + x13*x58 - x14*x58 - x15*x58 - x19*x58 - x20*x58 - x21*x58 - x24*x58 - x27*x58 + x28*x58 - x30*x58 + x31*x58 + x32*x58 - x33*x58 - x34*x58 - x35*x58 + x37*x58 + x41*x58 + x42*x58 + x45*x58 + x47*x58 + x48*x58 + x49*x58 - x51*x58 + x55*x58 + x56*x58 - x58^2 - x0 + x1 + x2 - x5 + x6 + x8 - x9 - x10 - x11 + x12 + x13 - x14 - x15 + x17 - x18 - x19 + x20 + x21 + x22 - x25 + x28 + x29 + x30 + x32 - x33 - x34 + x35 - x36 + x37 - x38 + x42 - x44 + x45 - x46 - x50 + x51 + x54 + x56 - x57 + x58 + 1)
[(2, 1, 1, 1, 2, 2, 0, 2, 0, 0, 0, 0, 0, 2, 1, 2, 2, 1, 0, 0, 1, 2, 0, 2, 0, 2, 1, 0, 2, 0, 1, 2, 2, 1, 0, 1, 2, 0, 2, 1, 0, 2, 0, 0, 2, 0, 0, 2, 0, 2, 0, 1, 2, 0, 0, 2, 2, 1, 1, 0, 1, 0, 2, 1, 2, 2, 1, 0, 1, 1, 2, 0, 1, 0, 2), (0, 1, 1, 1, 1, 2, 2, 0, 2, 0, 1, 2, 1, 1, 0, 2, 2, 1, 0, 0, 0, 2, 0, 2, 2, 0, 1, 2, 1, 1, 2, 1, 1, 0, 2, 1, 2, 1, 2, 2, 1, 2, 0, 1, 1, 0, 0, 2, 1, 0, 1, 2, 0, 2, 0, 0, 0, 0, 2, 1, 2, 0, 1, 1, 1, 1, 2, 0, 0, 0, 2, 1, 0, 2, 1), (2, 2, 0, 0, 1, 1, 0, 2, 2, 2, 1, 2, 2, 2, 0, 0, 0, 1, 2, 1, 1, 2, 0, 2, 1, 2, 0, 1, 0, 2, 1, 0, 2, 0, 1, 2, 1, 0, 0, 0, 0, 1, 0, 2, 2, 1, 0, 2, 0, 2, 2, 0, 2, 2, 1, 1, 0, 2, 0, 1, 1, 1, 1, 2, 1, 2, 1, 2, 0, 1, 1, 1, 0, 1, 2), (2, 0, 0, 2, 2, 2, 2, 0, 1, 2, 2, 2, 0, 0, 1, 0, 2, 2, 0, 2, 2, 1, 0, 0, 1, 1, 2, 0, 2, 0, 2, 0, 1, 0, 1, 0, 2, 1, 2, 2, 2, 0, 1, 1, 0, 2, 0, 0, 1, 0, 2, 1, 2, 2, 1, 1, 1, 0, 1, 2, 2, 1, 2, 1, 2, 0, 0, 1, 2, 0, 1, 1, 0, 2, 1), (0, 0, 2, 1, 0, 2, 1, 0, 1, 0, 0, 1, 1, 0, 0, 1, 2, 0, 0, 0, 1, 1, 0, 0, 1, 0, 0, 2, 0, 1, 1, 0, 2, 1, 0, 1, 2, 2, 1, 0, 1, 2, 2, 1, 2, 0, 1, 0, 1, 0, 0, 1, 0, 2, 2, 0, 1, 2, 2, 2, 0, 2, 2, 0, 1, 0, 2, 0, 0, 0, 2, 0, 2, 1, 1), (0, 0, 2, 0, 0, 1, 1, 2, 1, 0, 2, 2, 2, 2, 0, 0, 0, 1, 2, 1, 0, 1, 1, 0, 2, 0, 2, 2, 0, 0, 1, 1, 0, 0, 1, 1, 1, 0, 0, 2, 2, 1, 0, 0, 1, 0, 1, 0, 0, 0, 0, 0, 0, 1, 2, 0, 1, 0, 2, 1, 1, 2, 2, 1, 2, 1, 0, 2, 0, 0, 1, 2, 1, 1, 0)]
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