# Writeup for bfv and PSN in 2021qwb Final

## bfv

bfv加解密过程如下：

\begin{aligned} c[0] &= 0 \newline c[1] &= \delta = \lfloor q/t \rfloor \end{aligned}

\begin{aligned} tmp &= [c[0] + c[1]*s]_q = [ \delta * s]_q \newline m &= [round( \delta * s * t/q)]_t = s \end{aligned}

  1 2 3 4 5 6 7 8 9 10  def mutual(k, c, s): tmp = t * Roundq(c[0] + c[1] * s) TMP = tmp.list() for i in range(len(TMP)): TMP[i] = round(TMP[i] / q) tmp2 = Roundt(R(TMP)) # decryption is over here if tmp2[min(k, d)] == 0: print(True) else: print(False) 

\begin{aligned} tmp &= [c[0] + c[1] * s]_q = [(-\text{guess} + s) * \delta ]_q \newline m &= [round( \delta * (-\text{guess}+s) * t/q)]_t = -\text{guess} + s \end{aligned}

guess依次取-7~7，即可得到满足$s_i=\text{guess}$的所有位数，即私钥s的每一位。

2. （option 4）注册用户，直至用户数达到1024
3. （option 2）发送$c_0 = -\text{guess} * \delta, c_1 = \delta$来获取所有$s_i == \text{guess}$的下标，从而恢复私钥s
6. （option 3）拿flag
  1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65 66 67 68 69 70 71 72 73 74 75 76 77 78 79 80 81 82 83 84 85 86 87 88 89 90 91 92 93 94 95 96 97 98 99 100 101 102 103 104 105 106 107 108 109 110 111 112 113 114 115 116 117 118 119 120 121 122 123 124 125 126 127 128 129 130 131 132 133 134 135 136 137 138 139 140 141 142 143 144 145 146 147 148 149 150 151 152 153 154 155 156 157 158 159 160 161 162 163 164 165 166 167 168 169 170 171 172 173 174 175 176 177 178  from sage.stats.distributions.discrete_gaussian_integer import DiscreteGaussianDistributionIntegerSampler from random import randint, getrandbits import sys from pwn import * q = 2 ^ 54 t = 83 T = 3 d = 1024 delta = int(q / t) sigma = 2 P. = PolynomialRing(ZZ) f = x ^ d + 1 R. = P.quotient(f) D = DiscreteGaussianDistributionIntegerSampler(sigma=sigma) def sample1(): return R([D() for _ in range(d)]) def sample2(): return R([randint(0, q - 1) for _ in range(d)]) def sample3(x): return [randint(0, T - 1) for _ in range(x)] def Roundq(a): A = a.list() for i in range(len(A)): A[i] = A[i] % q if A[i] > (q / 2): A[i] = A[i] - q return R(A) def Roundt(a): A = a.list() for i in range(len(A)): A[i] = A[i] % t if A[i] > (t / 2): A[i] = A[i] - t return R(A) def keygen(): s = sample1() a = Roundq(sample2()) e = Roundq(sample1()) pk = [Roundq(-(a * s + e)), a] return s, pk def encrypt(m): u = sample1() e1 = sample1() e2 = sample1() return (Roundq(pk[0] * u + e1 + delta * m), Roundq(pk[1] * u + e2)) def baseT(n, b=T): v = [] while True: x = n // b y = n % b v.append(y) if x == 0: break n = x v.reverse() return v def rev_baseT(l): s = 0 for i in l: s += i s *= 3 return s def decrypt(c, s): tmp = t * Roundq(c[0] + c[1] * s) TMP = tmp.list() for i in range(len(TMP)): TMP[i] = round(TMP[i] / q) tmp2 = Roundt(R(TMP)) return tmp2.list() # 0. get admin id_num ciphertext # 1. register users, len(users) = 1024 # 2. send c0 = -sk_i*delta, c1 = delta to get index of sk==sk_i # 3. decrypt "admin" ct to get id_num # 4. add friends admin # 5. get flag conn = remote("127.0.0.1", 9999) # conn = remote("106.15.177.94", 8001) DEBUG = False if DEBUG: context.log_level = 'debug' conn.recvuntil(b"sk:\n") real_sk = conn.recvline().strip().decode() log.info(f"sk: {real_sk}") conn.recvuntil(b"4.Regist") # 0. get admin id_num ciphertext conn.sendlineafter(b">", b"2") conn.sendlineafter(b"recv ct?(Y/N)", b"Y") admin_id_num_ct0 = list(map(int, conn.recvuntil(b"]").decode()[1:-1].split(", "))) admin_id_num_ct1 = list(map(int, conn.recvuntil(b"]").decode()[2:-1].split(", "))) # log.info(f"admin_id_num_ct0: {admin_id_num_ct0}\nadmin_id_num_ct1: {admin_id_num_ct1}\n") conn.sendlineafter(b"continue?(Y/N)", b"N") for _ in range(5): conn.sendlineafter(b"c1:", b"0") conn.sendlineafter(b"c2:", b"0") # 1. register users, len(users) = 1024 for i in range(1024-5): conn.sendlineafter(b">", b"4") conn.sendlineafter(b"name:", str(i).encode()) log.info("Register over!") # 2. send c0 = -sk_i*delta, c1 = delta to get index of sk==sk_i delta=floor(q/t) sk = [0]*1024 ct1 = delta ct1 = str(delta).encode() for sk_i in range(-7, 8): # [-7, 7] log.info(f"guess: {sk_i}") conn.sendlineafter(b">", b"2") conn.sendlineafter(b"recv ct?(Y/N)", b"N") ct0 = [-sk_i*delta] * 1024 ct0 = " ".join(str(num) for num in ct0).encode() for index in range(1024): conn.sendlineafter(b"c1:", ct0) conn.sendlineafter(b"c2:", ct1) res = conn.recvline().strip().decode() if res == "True": sk[index] = sk_i print(f"{index:-4d}: {sk_i}") log.info(f"sk: {sk}") if DEBUG: print(str(sk) == real_sk) real_sk = list(map(int, real_sk[1:-1].split(", "))) # 3. decrypt "admin" ct to get id_num pt = decrypt([R(admin_id_num_ct0), R(admin_id_num_ct1)], R(sk)) # 4. add friends admin for i in range(18, 22): id_num = rev_baseT(pt[:i-1]) log.info(f"id_num: {id_num} {int(id_num).bit_length()}\n") conn.sendlineafter(b">", b"1") conn.sendlineafter(b"name:", b"admin") conn.sendlineafter(b"id:", str(id_num).encode()) res = conn.recvline().strip().decode() log.info(f"res: {res}") if res == "failed": continue break # 5. get flag conn.sendlineafter(b">", b"3") conn.sendlineafter(b"name:", b"admin") conn.sendlineafter(b"message:", b"give me the flag") flag = conn.recvline() log.info(f"flag: {flag}") 

## PSN

bias：32bytes

flag：24bytes，且flag格式为flag{.*}，已知6bytes

 1  key = ((self.seed>>(128 if i%2==0 else 0)) + i)&((1<<128)-1) 

bias是给了的，那么实际上偶数位的key我们未知的只有flag的前8bytes，而通过flag格式，我们其实上能够知道这8bytes的前5bytes，未知的就只有末尾的3bytes，爆破一下就有了。

cipher_list根据seed来选择的，是seed的3进制表示。

seed里面包含了flag的所有信息，只要能解出seed，就能拿到flag。

cipher_list又完全与seed相关，所以如果我们能识别出所有的cipher_list，就能算出seed，算出flag。

cipher_list => seed => flag

 1  cipher = self.cipher_list[i%len(self.cipher_list)](key) 

len(cipher_list)是seed的三进制表示，而seed的位数又主要由bias决定，因此当$\log_3{bias}$为奇数时，len(cipher_list)就是奇数。

  1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65 66 67 68 69 70 71 72 73 74 75 76 77 78 79 80 81 82 83 84 85 86 87 88 89 90 91 92 93 94 95 96 97 98 99 100 101 102 103 104 105 106 107 108 109 110 111 112 113 114 115 116 117 118 119 120 121 122 123 124 125 126 127 128 129 130 131 132 133 134 135 136 137 138 139 140 141 142 143 144 145 146 147 148 149 150 151 152 153 154 155 156 157 158 159 160 161 162 163 164 165 166 167 168 169 170 171 172 173 174 175 176 177 178 179 180 181 182 183 184 185 186 187 188 189 190 191 192 193 194 195 196 197 198 199 200 201 202 203 204 205 206 207 208 209 210 211 212 213 214 215 216 217 218 219 220 221 222 223 224 225 226 227 228 229 230 231 232 233 234 235 236 237 238 239 240 241 242 243 244 245 246 247 248 249 250 251 252 253 254 255 256 257 258 259 260 261 262 263 264 265 266  from pwn import * from cryptography.hazmat.primitives.ciphers import Cipher, algorithms, modes import random import math xtime = lambda a: (((a << 1) ^ 0x1B) & 0xFF) if (a & 0x80) else (a << 1) def text2matrix(text): text = int.from_bytes(text,"big") matrix = [] for i in range(16): byte = (text >> (8 * (15 - i))) & 0xFF if i % 4 == 0: matrix.append([byte]) else: matrix[i // 4].append(byte) return matrix def matrix2text(matrix): text = 0 for i in range(4): for j in range(4): text |= (matrix[i][j] << (120 - 8 * (4 * i + j))) text = text.to_bytes(16,"big") return text class backdoorAES: def __init__(self, master_key): self.change_key(master_key) def change_key(self, master_key): self.Sbox = ( 0x63, 0x7C, 0x77, 0x7B, 0xF2, 0x6B, 0x6F, 0xC5, 0x30, 0x01, 0x67, 0x2B, 0xFE, 0xD7, 0xAB, 0x76, 0xCA, 0x82, 0xC9, 0x7D, 0xFA, 0x59, 0x47, 0xF0, 0xAD, 0xD4, 0xA2, 0xAF, 0x9C, 0xA4, 0x72, 0xC0, 0xB7, 0xFD, 0x93, 0x26, 0x36, 0x3F, 0xF7, 0xCC, 0x34, 0xA5, 0xE5, 0xF1, 0x71, 0xD8, 0x31, 0x15, 0x04, 0xC7, 0x23, 0xC3, 0x18, 0x96, 0x05, 0x9A, 0x07, 0x12, 0x80, 0xE2, 0xEB, 0x27, 0xB2, 0x75, 0x09, 0x83, 0x2C, 0x1A, 0x1B, 0x6E, 0x5A, 0xA0, 0x52, 0x3B, 0xD6, 0xB3, 0x29, 0xE3, 0x2F, 0x84, 0x53, 0xD1, 0x00, 0xED, 0x20, 0xFC, 0xB1, 0x5B, 0x6A, 0xCB, 0xBE, 0x39, 0x4A, 0x4C, 0x58, 0xCF, 0xD0, 0xEF, 0xAA, 0xFB, 0x43, 0x4D, 0x33, 0x85, 0x45, 0xF9, 0x02, 0x7F, 0x50, 0x3C, 0x9F, 0xA8, 0x51, 0xA3, 0x40, 0x8F, 0x92, 0x9D, 0x38, 0xF5, 0xBC, 0xB6, 0xDA, 0x21, 0x10, 0xFF, 0xF3, 0xD2, 0xCD, 0x0C, 0x13, 0xEC, 0x5F, 0x97, 0x44, 0x17, 0xC4, 0xA7, 0x7E, 0x3D, 0x64, 0x5D, 0x19, 0x73, 0x60, 0x81, 0x4F, 0xDC, 0x22, 0x2A, 0x90, 0x88, 0x46, 0xEE, 0xB8, 0x14, 0xDE, 0x5E, 0x0B, 0xDB, 0xE0, 0x32, 0x3A, 0x0A, 0x49, 0x06, 0x24, 0x5C, 0xC2, 0xD3, 0xAC, 0x62, 0x91, 0x95, 0xE4, 0x79, 0xE7, 0xC8, 0x37, 0x6D, 0x8D, 0xD5, 0x4E, 0xA9, 0x6C, 0x56, 0xF4, 0xEA, 0x65, 0x7A, 0xAE, 0x08, 0xBA, 0x78, 0x25, 0x2E, 0x1C, 0xA6, 0xB4, 0xC6, 0xE8, 0xDD, 0x74, 0x1F, 0x4B, 0xBD, 0x8B, 0x8A, 0x70, 0x3E, 0xB5, 0x66, 0x48, 0x03, 0xF6, 0x0E, 0x61, 0x35, 0x57, 0xB9, 0x86, 0xC1, 0x1D, 0x9E, 0xE1, 0xF8, 0x98, 0x11, 0x69, 0xD9, 0x8E, 0x94, 0x9B, 0x1E, 0x87, 0xE9, 0xCE, 0x55, 0x28, 0xDF, 0x8C, 0xA1, 0x89, 0x0D, 0xBF, 0xE6, 0x42, 0x68, 0x41, 0x99, 0x2D, 0x0F, 0xB0, 0x54, 0xBB, 0x16, ) self.InvSbox = ( 0x52, 0x09, 0x6A, 0xD5, 0x30, 0x36, 0xA5, 0x38, 0xBF, 0x40, 0xA3, 0x9E, 0x81, 0xF3, 0xD7, 0xFB, 0x7C, 0xE3, 0x39, 0x82, 0x9B, 0x2F, 0xFF, 0x87, 0x34, 0x8E, 0x43, 0x44, 0xC4, 0xDE, 0xE9, 0xCB, 0x54, 0x7B, 0x94, 0x32, 0xA6, 0xC2, 0x23, 0x3D, 0xEE, 0x4C, 0x95, 0x0B, 0x42, 0xFA, 0xC3, 0x4E, 0x08, 0x2E, 0xA1, 0x66, 0x28, 0xD9, 0x24, 0xB2, 0x76, 0x5B, 0xA2, 0x49, 0x6D, 0x8B, 0xD1, 0x25, 0x72, 0xF8, 0xF6, 0x64, 0x86, 0x68, 0x98, 0x16, 0xD4, 0xA4, 0x5C, 0xCC, 0x5D, 0x65, 0xB6, 0x92, 0x6C, 0x70, 0x48, 0x50, 0xFD, 0xED, 0xB9, 0xDA, 0x5E, 0x15, 0x46, 0x57, 0xA7, 0x8D, 0x9D, 0x84, 0x90, 0xD8, 0xAB, 0x00, 0x8C, 0xBC, 0xD3, 0x0A, 0xF7, 0xE4, 0x58, 0x05, 0xB8, 0xB3, 0x45, 0x06, 0xD0, 0x2C, 0x1E, 0x8F, 0xCA, 0x3F, 0x0F, 0x02, 0xC1, 0xAF, 0xBD, 0x03, 0x01, 0x13, 0x8A, 0x6B, 0x3A, 0x91, 0x11, 0x41, 0x4F, 0x67, 0xDC, 0xEA, 0x97, 0xF2, 0xCF, 0xCE, 0xF0, 0xB4, 0xE6, 0x73, 0x96, 0xAC, 0x74, 0x22, 0xE7, 0xAD, 0x35, 0x85, 0xE2, 0xF9, 0x37, 0xE8, 0x1C, 0x75, 0xDF, 0x6E, 0x47, 0xF1, 0x1A, 0x71, 0x1D, 0x29, 0xC5, 0x89, 0x6F, 0xB7, 0x62, 0x0E, 0xAA, 0x18, 0xBE, 0x1B, 0xFC, 0x56, 0x3E, 0x4B, 0xC6, 0xD2, 0x79, 0x20, 0x9A, 0xDB, 0xC0, 0xFE, 0x78, 0xCD, 0x5A, 0xF4, 0x1F, 0xDD, 0xA8, 0x33, 0x88, 0x07, 0xC7, 0x31, 0xB1, 0x12, 0x10, 0x59, 0x27, 0x80, 0xEC, 0x5F, 0x60, 0x51, 0x7F, 0xA9, 0x19, 0xB5, 0x4A, 0x0D, 0x2D, 0xE5, 0x7A, 0x9F, 0x93, 0xC9, 0x9C, 0xEF, 0xA0, 0xE0, 0x3B, 0x4D, 0xAE, 0x2A, 0xF5, 0xB0, 0xC8, 0xEB, 0xBB, 0x3C, 0x83, 0x53, 0x99, 0x61, 0x17, 0x2B, 0x04, 0x7E, 0xBA, 0x77, 0xD6, 0x26, 0xE1, 0x69, 0x14, 0x63, 0x55, 0x21, 0x0C, 0x7D, ) self.round_keys = text2matrix(master_key) Rcon = ( 0x00, 0x01, 0x02, 0x04, 0x08, 0x10, 0x20, 0x40, 0x80, 0x1B, 0x36, 0x6C, 0xD8, 0xAB, 0x4D, 0x9A, 0x2F, 0x5E, 0xBC, 0x63, 0xC6, 0x97, 0x35, 0x6A, 0xD4, 0xB3, 0x7D, 0xFA, 0xEF, 0xC5, 0x91, 0x39, ) for i in range(4, 4 * 5): self.round_keys.append([]) if i % 4 == 0: byte = self.round_keys[i - 4][0] ^ self.Sbox[self.round_keys[i - 1][1]] ^ Rcon[i // 4] self.round_keys[i].append(byte) for j in range(1, 4): byte = self.round_keys[i - 4][j] ^ self.Sbox[self.round_keys[i - 1][(j + 1) % 4]] self.round_keys[i].append(byte) else: for j in range(4): byte = self.round_keys[i - 4][j] ^ self.round_keys[i - 1][j] self.round_keys[i].append(byte) def encrypt(self, plaintext): self.plain_state = text2matrix(plaintext) self.__add_round_key(self.plain_state, self.round_keys[:4]) for i in range(1, 4): self.__round_encrypt(self.plain_state, self.round_keys[4 * i : 4 * (i + 1)]) self.__sub_bytes(self.plain_state) self.__shift_rows(self.plain_state) self.__add_round_key(self.plain_state, self.round_keys[16:]) return matrix2text(self.plain_state) def decrypt(self, ciphertext): self.cipher_state = text2matrix(ciphertext) self.__add_round_key(self.cipher_state, self.round_keys[16:]) self.__inv_shift_rows(self.cipher_state) self.__inv_sub_bytes(self.cipher_state) for i in range(3, 0, -1): self.__round_decrypt(self.cipher_state, self.round_keys[4 * i : 4 * (i + 1)]) self.__add_round_key(self.cipher_state, self.round_keys[:4]) return matrix2text(self.cipher_state) def __add_round_key(self, s, k): for i in range(4): for j in range(4): s[i][j] ^= k[i][j] def __round_encrypt(self, state_matrix, key_matrix): self.__sub_bytes(state_matrix) self.__shift_rows(state_matrix) self.__mix_columns(state_matrix) self.__add_round_key(state_matrix, key_matrix) def __round_decrypt(self, state_matrix, key_matrix): self.__add_round_key(state_matrix, key_matrix) self.__inv_mix_columns(state_matrix) self.__inv_shift_rows(state_matrix) self.__inv_sub_bytes(state_matrix) def __sub_bytes(self, s): for i in range(4): for j in range(4): s[i][j] = self.Sbox[s[i][j]] def __inv_sub_bytes(self, s): for i in range(4): for j in range(4): s[i][j] = self.InvSbox[s[i][j]] def __shift_rows(self, s): s[0][1], s[1][1], s[2][1], s[3][1] = s[1][1], s[2][1], s[3][1], s[0][1] s[0][2], s[1][2], s[2][2], s[3][2] = s[2][2], s[3][2], s[0][2], s[1][2] s[0][3], s[1][3], s[2][3], s[3][3] = s[3][3], s[0][3], s[1][3], s[2][3] def __inv_shift_rows(self, s): s[0][1], s[1][1], s[2][1], s[3][1] = s[3][1], s[0][1], s[1][1], s[2][1] s[0][2], s[1][2], s[2][2], s[3][2] = s[2][2], s[3][2], s[0][2], s[1][2] s[0][3], s[1][3], s[2][3], s[3][3] = s[1][3], s[2][3], s[3][3], s[0][3] def __mix_single_column(self, a): t = a[0] ^ a[1] ^ a[2] ^ a[3] u = a[0] a[0] ^= t ^ xtime(a[0] ^ a[1]) a[1] ^= t ^ xtime(a[1] ^ a[2]) a[2] ^= t ^ xtime(a[2] ^ a[3]) a[3] ^= t ^ xtime(a[3] ^ u) def __mix_columns(self, s): for i in range(4): self.__mix_single_column(s[i]) def __inv_mix_columns(self, s): for i in range(4): u = xtime(xtime(s[i][0] ^ s[i][2])) v = xtime(xtime(s[i][1] ^ s[i][3])) s[i][0] ^= u s[i][1] ^= v s[i][2] ^= u s[i][3] ^= v self.__mix_columns(s) class safeCipher: def __init__(self, key, alg): self.cipher = Cipher(alg(key), modes.ECB()) def encrypt(self, pt): encryptor = self.cipher.encryptor() ct = encryptor.update(pt) assert encryptor.finalize() == b'' return ct def decrypt(self, ct): decryptor = self.cipher.decryptor() pt = decryptor.update(ct) assert decryptor.finalize() == b'' return pt class Camellia(safeCipher): def __init__(self, key): safeCipher.__init__(self, key, algorithms.Camellia) class SEED(safeCipher): def __init__(self, key): safeCipher.__init__(self, key, algorithms.SEED) HOST = "47.96.164.154" PORT = 62344 conn = remote(HOST, PORT) DEBUG = False if DEBUG: context.log_level = "debug" conn.recvuntil(b"Security bias: ") bias = int(conn.recvline().strip().decode(), 16) log.info(f"bias: {bias} {hex(bias)}") bits = math.ceil(math.log(bias, 3)) if bits & 1 == 0: log.info(f"Try again: bias={bits}") exit(-1) conn.sendlineafter(b"choice: ", b"0") conn.sendlineafter(b"Encrypt: ", b"0"*32*bits*2) ciphers = bytes.fromhex(conn.recvline().strip().decode()) cipher_list = [] for i in range(0, len(ciphers), 32): cipher_list.append(ciphers[i:i+16]) seed = int.from_bytes(b"flag{E4s",'big') + (bias >> 128) index = [] for i, ci in enumerate(cipher_list): key = (seed + 2*i) & ((1<<128)-1) key = key.to_bytes(16,"big") for alg in [backdoorAES, Camellia, SEED]: c = alg(key) if c.encrypt(b"\x00"*16) == ci: print(i, alg) if alg == backdoorAES: index.append(0) elif alg == Camellia: index.append(1) elif alg == SEED: index.append(2) if len(index) == 0: log.info("Try again!") exit(-1) print(len(index), index) new_index = [] for i in range(len(index)//2): new_index.append(index[i]) new_index.append(index[len(index)//2+1+i]) new_index.append(index[len(index)//2]) print(len(new_index), new_index) s = int(''.join(str(n) for n in new_index), 3) print(s, bias) print((s-bias).to_bytes(24, 'big'))