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September 10, 2017 18:42
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ASIS CTF 2017 Finals - Marijuana (Crypto 394)
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| #-*- coding:utf-8 -*- | |
| ''' | |
| In the challenge we are given a recently proposed cryptosystem | |
| based on Mersenne primes ( https://eprint.iacr.org/2017/481 ). | |
| The cryptosystem was broken quickly in https://eprint.iacr.org/2017/522.pdf | |
| using random partitioning and LLL. Here this attack is implemented. | |
| ''' | |
| from sage.all import * | |
| n, p, h, pk = map(int, "289 994646472819573284310764496293641680200912301594695434880927953786318994025066751066111 6 843235888465494832763255804166953659073735483996382764878385572219936486704848907391083".split()) | |
| assert p == 2**n-1 | |
| H = pk | |
| m = 6 | |
| def random_partition(): | |
| bounds = range(n+1) | |
| shuffle(bounds) | |
| bounds = bounds[:2*m] | |
| bounds.sort() | |
| res = [] | |
| for i in xrange(0, len(bounds), 2): | |
| res.append((bounds[i], bounds[i+1] - bounds[i])) | |
| return res | |
| def hw(x): | |
| return sum(map(int, bin(x)[2:])) | |
| def get_sol(): | |
| while 1: | |
| f = random_partition() | |
| g = random_partition() | |
| mat = matrix(ZZ, 2*m+1, 2*m+1) | |
| for i in xrange(m): | |
| ff, lf = f[i] | |
| gg, lg = g[i] | |
| top = max(lf, lg) | |
| mat[i,0] = -2**ff | |
| mat[i,1+i] = 2**(top - lf) | |
| mat[m+i,0] = H*2**gg % p | |
| mat[m+i,1+m+i] = 2**(top - lg) | |
| mat[2*m,0] = p | |
| ml = mat.LLL() | |
| for row in ml: | |
| # print row | |
| if row[0] != 0: | |
| continue | |
| curF = 0 | |
| curG = 0 | |
| for i in xrange(m): | |
| vf = row[1+i] | |
| vg = row[1+m+i] | |
| ff, lf = f[i] | |
| gg, lg = g[i] | |
| top = max(lf, lg) | |
| assert vf % 2**(top - lf) == 0 | |
| vf //= 2**(top - lf) | |
| assert vg % 2**(top - lg) == 0 | |
| vg //= 2**(top - lg) | |
| if vf >= 2**lf: break | |
| if vg >= 2**lg: break | |
| curF += 2**ff * vf | |
| curG += 2**gg * vg | |
| else: | |
| if curF == 0 or curG == 0: | |
| continue | |
| if curF < 0: | |
| curF = -curF | |
| curG = -curG | |
| curF %= p | |
| curG %= p | |
| assert curF % p == H * curG % p | |
| print "Solution" | |
| print hw(curF), curF | |
| print hw(curG), curG | |
| return curG | |
| curG = get_sol() | |
| def b2s(b): | |
| ret = [] | |
| for pos in range(0, len(b), 8): | |
| ret.append(chr(int(b[pos:pos + 8], 2))) | |
| return "".join(ret) | |
| res = "" | |
| for line in open("FLAG.enc"): | |
| v = int(line) | |
| v = (v * curG) % p | |
| # print hw(v) | |
| if hw(v) < n / 2: | |
| res = res + "0" | |
| else: | |
| res = res + "1" | |
| while len(res) % 8: | |
| res = "0" + res | |
| print `b2s(res)` | |
| ''' | |
| $ time sage solve.py | |
| Solution | |
| 6 442143125341270492632539047516324923841771661106084131391532190171922432 | |
| 6 6277102109531099921227501217202764039016212010661702533120 | |
| 'ASIS{a_N3w_PubL1c-kEy_CryP7osYs7eM_viA_M3rs3nne_Numb3rZ!}' | |
| real 0m2.491s | |
| user 0m2.304s | |
| sys 0m0.176s | |
| ''' |
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| #!/usr/bin/python | |
| import random, math | |
| from operator import mul | |
| from fractions import Fraction | |
| from Crypto.Util.number import * | |
| from flag import FLAG | |
| def n_Choose_k(n, k): | |
| return int(reduce(mul, (Fraction(n-i, i+1) for i in range(k)), 1) ) | |
| def hamweight(h, n): | |
| if h <= n: | |
| H = [] | |
| while True: | |
| r = random.randint(0, n-1) | |
| if 2**r not in H: | |
| H.append(2**r) | |
| if len(H) == h: | |
| break | |
| return sum(H) | |
| def ham(n): | |
| nbin = bin(n)[2:].replace('0', '') | |
| return len(nbin) | |
| def keygen(l): | |
| n = 2 * random.randint(4*l, 5*l) + 1 | |
| p = (1 << n) - 1 | |
| while True: | |
| h = random.randint(1, int(math.sqrt(n) / 2) - 1) | |
| if n_Choose_k(n-1, h-1) >= 2**l: | |
| break | |
| F, G = hamweight(h, n), hamweight(h, n) | |
| pk = (F * inverse(G, p)) % p | |
| sk = G | |
| return n, p, h, pk, sk | |
| def encrypt(msg, h, pk): | |
| msg = bin(bytes_to_long(msg))[2:] | |
| enc = [] | |
| for b in msg: | |
| A, B = hamweight(h, n), hamweight(h, n) | |
| enc.append(int(((-1)**int(b) * (A * pk + B)) % p)) | |
| return enc | |
| n, p, h, pk, sk = keygen(32) | |
| f = open('FLAG.enc', 'w') | |
| E = encrypt(FLAG, h, pk) | |
| f.write('\n'.join([str(d) for d in E])) | |
| f.close() |
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| True | |
| (n, p, h, pk) = 289 994646472819573284310764496293641680200912301594695434880927953786318994025066751066111 6 843235888465494832763255804166953659073735483996382764878385572219936486704848907391083 |
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