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# hellman/0_solve.sage

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PlaidCTF 2020 - MPKC (Crypto 350)
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 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
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 # 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))
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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)]