RSA 複雜題目¶

2018 Tokyo Western Mixed Cipher¶

題目給的信息如下所示：

每次交互可以維持的時間長度約爲 5 分鐘
每次交互中中n是確定的 1024 bit，但是未知， e 爲 65537
使用 aes 加密了 flag，密鑰和 IV 均不知道
每次密鑰是固定的，但是 IV 每次都會隨機
可以使用 encrypt 功能隨意使用 rsa 和 aes 進行加密，其中每次加密都會對 aes 的 iv 進行隨機
可以使用 decrypt 對隨意的密文進行解密，但是隻能知道最後一個字節是什麼
可以使用 print_flag 獲取 flag 密文
可以使用 print_key 獲取 rsa 加密的 aes 密鑰

本題目看似一個題目，實則是 3 個題目，需要分步驟解決。在此之前，我們準備好交互的函數

def get_enc_key(io):
    io.read_until("4: get encrypted keyn")
    io.writeline("4")
    io.read_until("here is encrypted key :)n")
    c=int(io.readline()[:-1],16)
    return c

def encrypt_io(io,p):
    io.read_until("4: get encrypted keyn")
    io.writeline("1")
    io.read_until("input plain text: ")
    io.writeline(p)
    io.read_until("RSA: ")
    rsa_c=int(io.readline()[:-1],16)
    io.read_until("AES: ")
    aes_c=io.readline()[:-1].decode("hex")
    return rsa_c,aes_c

def decrypt_io(io,c):
    io.read_until("4: get encrypted keyn")
    io.writeline("2")
    io.read_until("input hexencoded cipher text: ")
    io.writeline(long_to_bytes(c).encode("hex"))
    io.read_until("RSA: ")
    return io.read_line()[:-1].decode("hex")

GCD attack n¶

第一步我們需要把沒有給出的 n 算出來，因爲我們可以利用 encrypt 功能對我們輸入的明文 x 進行 rsa 加密，那麼可以利用整除的性質算 n

因爲x ^ e = c mod n
所以 n | x ^ e - c

我們可以構造足夠多的 x，算出最夠多的 x ^ e - c，從而計算最大公約數，得到 n。

def get_n(io):
    rsa_c,aes_c=encrypt_io(io,long_to_bytes(2))
    n=pow(2,65537)-rsa_c
    for i in range(3,6):
        rsa_c, aes_c = encrypt_io(io, long_to_bytes(i))
        n=primefac.gcd(n,pow(i,65537)-rsa_c)
    return n

可以利用加密進行 check

def check_n(io,n):
    rsa_c, aes_c = encrypt_io(io, "123")
    if pow(bytes_to_long("123"), e, n)==rsa_c:
        return True
    else:
        return False

RSA parity oracle¶

利用 leak 的的最後一個字節，我們可以進行選擇密文攻擊，使用 RSA parity oracle 回覆 aes 的祕鑰

def guess_m(io,n,c):
    k=1
    lb=0
    ub=n
    while ub!=lb:
        print lb,ub
        tmp = c * gmpy2.powmod(2, k*e, n) % n
        if ord(decrypt_io(io,tmp)[-1])%2==1:
            lb = (lb + ub) / 2
        else:
            ub = (lb + ub) / 2
        k+=1
    print ub,len(long_to_bytes(ub))
    return ub

PRNG Predict¶

這裏我們可以解密 flag 的16字節之後的內容了，但是前16個字節沒有 IV 是解密不了的。這時我們可以發現，IV 生成使用的隨機數使用了 getrandbits，並且我們可以獲取到足夠多的隨機數量，那麼我們可以進行 PRNG 的 predict，從而直接獲取隨機數

這裏使用了一個現成的的 java 進行 PRNG 的 Predict

public class Main {

   static int[] state;
   static int currentIndex;
40huo
   public static void main(String[] args) {
      state = new int[624];
      currentIndex = 0;

//    initialize(0);

//    for (int i = 0; i < 5; i++) {
//       System.out.println(state[i]);
//    }

      // for (int i = 0; i < 5; i++) {
      // System.out.println(nextNumber());
      // }

      if (args.length != 624) {
         System.err.println("must be 624 args");
         System.exit(1);
      }
      int[] arr = new int[624];
      for (int i = 0; i < args.length; i++) {
         arr[i] = Integer.parseInt(args[i]);
      }


      rev(arr);

      for (int i = 0; i < 6240huo4; i++) {
         System.out.println(state[i]);
      }

//    System.out.println("currentIndex " + currentIndex);
//    System.out.println("state[currentIndex] " + state[currentIndex]);
//    System.out.println("next " + nextNumber());

      // want -2065863258
   }

   static void nextState() {
      // Iterate through the state
      for (int i = 0; i < 624; i++) {
         // y is the first bit of the current number,
         // and the last 31 bits of the next number
         int y = (state[i] & 0x80000000)
               + (state[(i + 1) % 624] & 0x7fffffff);
         // first bitshift y by 1 to the right
         int next = y >>> 1;
         // xor it with the 397th next number
         next ^= state[(i + 397) % 624];
         // if y is odd, xor with magic number
         if ((y & 1L) == 1L) {
            next ^= 0x9908b0df;
         }
         // now we have the result
         state[i] = next;
      }
   }

   static int nextNumber() {
      currentIndex++;
      int tmp = state[currentIndex];
      tmp ^= (tmp >>> 11);
      tmp ^= (tmp << 7) & 0x9d2c5680;
      tmp ^= (tmp << 15) & 0xefc60000;
      tmp ^= (tmp >>> 18);
      return tmp;
   }

   static void initialize(int seed) {

      // http://code.activestate.com/recipes/578056-mersenne-twister/

      // global MT
      // global bitmask_1
      // MT[0] = seed
      // for i in xrange(1,624):
      // MT[i] = ((1812433253 * MT[i-1]) ^ ((MT[i-1] >> 30) + i)) & bitmask_1

      // copied Python 2.7's impl (probably uint problems)
      state[0] = seed;
      for (int i = 1; i < 624; i++) {
         state[i] = ((1812433253 * state[i - 1]) ^ ((state[i - 1] >> 30) + i)) & 0xffffffff;
      }
   }

   static int unBitshiftRightXor(int value, int shift) {
      // we part of the value we are up to (with a width of shift bits)
      int i = 0;
      // we accumulate the result here
      int result = 0;
      // iterate until we've done the full 32 bits
      while (i * shift < 32) {
         // create a mask for this part
         int partMask = (-1 << (32 - shift)) >>> (shift * i);
         // obtain the part
         int part = value & partMask;
         // unapply the xor from the next part of the integer
         value ^= part >>> shift;
         // add the part to the result
         result |= part;
         i++;
      }
      return result;
   }

   static int unBitshiftLeftXor(int value, int shift, int mask) {
      // we part of the value we are up to (with a width of shift bits)
      int i = 0;
      // we accumulate the result here
      int result = 0;
      // iterate until we've done the full 32 bits
      while (i * shift < 32) {
         // create a mask for this part
         int partMask = (-1 >>> (32 - shift)) << (shift * i);
         // obtain the part
         int part = value & partMask;
         // unapply the xor from the next part of the integer
         value ^= (part << shift) & mask;
         // add the part to the result
         result |= part;
         i++;
      }
      return result;
   }

   static void rev(int[] nums) {
      for (int i = 0; i < 624; i++) {

         int value = nums[i];
         value = unBitshiftRightXor(value, 18);
         value = unBitshiftLeftXor(value, 15, 0xefc60000);
         value = unBitshiftLeftXor(value, 7, 0x9d2c5680);
         value = unBitshiftRightXor(value, 11);

         state[i] = value;
      }
   }
}

寫了一個 python 直接調用 java

from Crypto.Util.number import long_to_bytes,bytes_to_long



def encrypt_io(io,p):
    io.read_until("4: get encrypted keyn")
    io.writeline("1")
    io.read_until("input plain text: ")
    io.writeline(p)
    io.read_until("RSA: ")
    rsa_c=int(io.readline()[:-1],16)
    io.read_until("AES: ")
    aes_c=io.readline()[:-1].decode("hex")
    return rsa_c,aes_c
import subprocess
import random
def get_iv(io):
    rsa_c, aes_c=encrypt_io(io,"1")
    return bytes_to_long(aes_c[0:16])
def splitInto32(w128):
    w1 = w128 & (2**32-1)
    w2 = (w128 >> 32) & (2**32-1)
    w3 = (w128 >> 64) & (2**32-1)
    w4 = (w128 >> 96)
    return w1,w2,w3,w4
def sign(iv):
    # converts a 32 bit uint to a 32 bit signed int
    if(iv&0x80000000):
        iv = -0x100000000 + iv
    return iv
def get_state(io):
    numbers=[]
    for i in range(156):
        print i
        numbers.append(get_iv(io))
    observedNums = [sign(w) for n in numbers for w in splitInto32(n)]
    o = subprocess.check_output(["java", "Main"] + map(str, observedNums))
    stateList = [int(s) % (2 ** 32) for s in o.split()]
    r = random.Random()
    state = (3, tuple(stateList + [624]), None)
    r.setstate(state)
    return r.getrandbits(128)

EXP¶

整體攻擊代碼如下：

from zio import *
import primefac
from Crypto.Util.number import long_to_bytes,bytes_to_long
target=("crypto.chal.ctf.westerns.tokyo",5643)
e=65537

def get_enc_key(io):
    io.read_until("4: get encrypted keyn")
    io.writeline("4")
    io.read_until("here is encrypted key :)n")
    c=int(io.readline()[:-1],16)
    return c

def encrypt_io(io,p):
    io.read_until("4: get encrypted keyn")
    io.writeline("1")
    io.read_until("input plain text: ")
    io.writeline(p)
    io.read_until("RSA: ")
    rsa_c=int(io.readline()[:-1],16)
    io.read_until("AES: ")
    aes_c=io.readline()[:-1].decode("hex")
    return rsa_c,aes_c

def decrypt_io(io,c):
    io.read_until("4: get encrypted keyn")
    io.writeline("2")
    io.read_until("input hexencoded cipher text: ")
    io.writeline(long_to_bytes(c).encode("hex"))
    io.read_until("RSA: ")
    return io.read_line()[:-1].decode("hex")

def get_n(io):
    rsa_c,aes_c=encrypt_io(io,long_to_bytes(2))
    n=pow(2,65537)-rsa_c
    for i in range(3,6):
        rsa_c, aes_c = encrypt_io(io, long_to_bytes(i))
        n=primefac.gcd(n,pow(i,65537)-rsa_c)
    return n

def check_n(io,n):
    rsa_c, aes_c = encrypt_io(io, "123")
    if pow(bytes_to_long("123"), e, n)==rsa_c:
        return True
    else:
        return False


import gmpy2
def guess_m(io,n,c):
    k=1
    lb=0
    ub=n
    while ub!=lb:
        print lb,ub
        tmp = c * gmpy2.powmod(2, k*e, n) % n
        if ord(decrypt_io(io,tmp)[-1])%2==1:
            lb = (lb + ub) / 2
        else:
            ub = (lb + ub) / 2
        k+=1
    print ub,len(long_to_bytes(ub))
    return ub


io = zio(target, timeout=10000, print_read=COLORED(NONE, 'red'),print_write=COLORED(NONE, 'green'))
n=get_n(io)
print check_n(io,n)
c=get_enc_key(io)
print len(decrypt_io(io,c))==16


m=guess_m(io,n,c)
for i in range(m - 50000,m+50000):
    if pow(i,e,n)==c:
        aeskey=i
        print long_to_bytes(aeskey)[-1]==decrypt_io(io,c)[-1]
        print "found aes key",hex(aeskey)

import fuck_r
next_iv=fuck_r.get_state(io)
print "##########################################"
print next_iv
print aeskey
io.interact()

2016 ASIS Find the flag¶

這裏我們以 ASIS 2016 線上賽中 Find the flag 爲例進行介紹。

文件解壓出來，有一個密文，一個公鑰，一個 py 腳本。看一下公鑰。

➜  RSA openssl rsa -pubin -in pubkey.pem -text -modulus
Public-Key: (256 bit)
Modulus:
    00:d8:e2:4c:12:b7:b9:9e:fe:0a:9b:c0:4a:6a:3d:
    f5:8a:2a:94:42:69:b4:92:b7:37:6d:f1:29:02:3f:
    20:61:b9
Exponent: 12405943493775545863 (0xac2ac3e0ca0f5607)
Modulus=D8E24C12B7B99EFE0A9BC04A6A3DF58A2A944269B492B7376DF129023F2061B9

這麼小的一個 $N$ ，先分解一下。

p = 311155972145869391293781528370734636009
q = 315274063651866931016337573625089033553

再看給的 py 腳本。

#!/usr/bin/python
import gmpy
from Crypto.Util.number import *
from Crypto.PublicKey import RSA
from Crypto.Cipher import PKCS1_v1_5

flag = open('flag', 'r').read() * 30

def ext_rsa_encrypt(p, q, e, msg):
    m = bytes_to_long(msg)
    while True:
        n = p * q
        try:
            phi = (p - 1)*(q - 1)
            d = gmpy.invert(e, phi)
            pubkey = RSA.construct((long(n), long(e)))
            key = PKCS1_v1_5.new(pubkey)
            enc = key.encrypt(msg).encode('base64')
            return enc
        except:
            p = gmpy.next_prime(p**2 + q**2)
            q = gmpy.next_prime(2*p*q)
            e = gmpy.next_prime(e**2)

p = getPrime(128)
q = getPrime(128)
n = p*q
e = getPrime(64)
pubkey = RSA.construct((long(n), long(e)))
f = open('pubkey.pem', 'w')
f.write(pubkey.exportKey())
g = open('flag.enc', 'w')
g.write(ext_rsa_encrypt(p, q, e, flag))

邏輯很簡單，讀取 flag，重複 30 遍爲密文。隨機取 $p$ 和 $q$ ，生成一個公鑰，寫入 pubkey.pem，再用腳本中的 ext_rsa_encrypt 函數進行加密，最後將密文寫入 flag.enc。

嘗試一下解密，提示密文過長，再看加密函數，原來當加密失敗時，函數會跳到異常處理，以一定算法重新取更大的 $p$ 和 $q$ ，直到加密成功。

那麼我們只要也寫一個相應的解密函數即可。

#!/usr/bin/python
import gmpy
from Crypto.Util.number import *
from Crypto.PublicKey import RSA
from Crypto.Cipher import PKCS1_v1_5

def ext_rsa_decrypt(p, q, e, msg):
    m = bytes_to_long(msg)
    while True:
        n = p * q
        try:
            phi = (p - 1)*(q - 1)
            d = gmpy.invert(e, phi)
            privatekey = RSA.construct((long(n), long(e), long(d), long(p), long(q)))
            key = PKCS1_v1_5.new(privatekey)
            de_error = ''
            enc = key.decrypt(msg.decode('base64'), de_error)
            return enc
        except Exception as error:
            print error
            p = gmpy.next_prime(p**2 + q**2)
            q = gmpy.next_prime(2*p*q)
            e = gmpy.next_prime(e**2)

p = 311155972145869391293781528370734636009
q = 315274063651866931016337573625089033553
n = p*q
e = 12405943493775545863 
# pubkey = RSA.construct((long(n), long(e)))
# f = open('pubkey.pem', 'w')
# f.write(pubkey.exportKey())
g = open('flag.enc', 'r')
msg = g.read()
flag = ext_rsa_decrypt(p, q, e, msg)
print flag

拿到 flag

ASIS{F4ct0R__N_by_it3rat!ng!}

SCTF RSA1¶

這裏我們以 SCTF RSA1 爲例進行介紹，首先解壓壓縮包後，得到如下文件

➜  level0 git:(master) ✗ ls -al
總用量 4
drwxrwxrwx 1 root root    0 7月  30 16:36 .
drwxrwxrwx 1 root root    0 7月  30 16:34 ..
-rwxrwxrwx 1 root root  349 5月   2  2016 level1.passwd.enc
-rwxrwxrwx 1 root root 2337 5月   6  2016 level1.zip
-rwxrwxrwx 1 root root  451 5月   2  2016 public.key

嘗試解壓縮了一下 level1.zip 現需要密碼。然後根據 level1.passwd.enc 可知，應該是我們需要解密這個文件才能得到對應的密碼。查看公鑰

➜  level0 git:(master) ✗ openssl rsa -pubin -in public.key -text -modulus 
Public-Key: (2048 bit)
Modulus:
    00:94:a0:3e:6e:0e:dc:f2:74:10:52:ef:1e:ea:a8:
    89:d6:f9:8d:01:11:51:db:5e:90:92:48:fd:39:0c:
    70:87:24:d8:98:3c:f3:33:1c:ba:c5:61:c2:ce:2c:
    5a:f1:5e:65:b2:b2:46:91:56:b6:19:d5:d3:b2:a6:
    bb:a3:7d:56:93:99:4d:7e:4c:2f:aa:60:7b:3e:c8:
    fc:90:b2:00:62:4b:53:18:5b:a2:30:10:60:a8:21:
    ab:61:57:d7:e7:cc:67:1b:4d:cd:66:4c:7d:f1:1a:
    2a:1d:5e:50:80:c1:5e:45:12:3a:ba:4a:53:64:d8:
    72:1f:84:4a:ae:5c:55:02:e8:8e:56:4d:38:70:a5:
    16:36:d3:bc:14:3e:2f:ae:2f:31:58:ba:00:ab:ac:
    c0:c5:ba:44:3c:29:70:56:01:6b:57:f5:d7:52:d7:
    31:56:0b:ab:0a:e6:8d:ad:08:22:a9:1f:cb:6e:49:
    cc:01:4c:12:d2:ab:a3:a5:97:e5:10:49:19:7f:69:
    d9:3b:c5:53:53:71:00:18:60:cc:69:1a:06:64:3b:
    86:94:70:a9:da:82:fc:54:6b:06:23:43:2d:b0:20:
    eb:b6:1b:91:35:5e:53:a6:e5:d8:9a:84:bb:30:46:
    b8:9f:63:bc:70:06:2d:59:d8:62:a5:fd:5c:ab:06:
    68:81
Exponent: 65537 (0x10001)
Modulus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
writing RSA key
-----BEGIN PUBLIC KEY-----
MIIBIjANBgkqhkiG9w0BAQEFAAOCAQ8AMIIBCgKCAQEAlKA+bg7c8nQQUu8e6qiJ
1vmNARFR216Qkkj9OQxwhyTYmDzzMxy6xWHCzixa8V5lsrJGkVa2GdXTsqa7o31W
k5lNfkwvqmB7Psj8kLIAYktTGFuiMBBgqCGrYVfX58xnG03NZkx98RoqHV5QgMFe
RRI6ukpTZNhyH4RKrlxVAuiOVk04cKUWNtO8FD4vri8xWLoAq6zAxbpEPClwVgFr
V/XXUtcxVgurCuaNrQgiqR/LbknMAUwS0qujpZflEEkZf2nZO8VTU3EAGGDMaRoG
ZDuGlHCp2oL8VGsGI0MtsCDrthuRNV5TpuXYmoS7MEa4n2O8cAYtWdhipf1cqwZo
gQIDAQAB
-----END PUBLIC KEY-----

發現雖然說是 2048 位，但是顯然模數沒有那麼長，嘗試分解下，得到

p=250527704258269
q=74891071972884336452892671945839935839027130680745292701175368094445819328761543101567760612778187287503041052186054409602799660254304070752542327616415127619185118484301676127655806327719998855075907042722072624352495417865982621374198943186383488123852345021090112675763096388320624127451586578874243946255833495297552979177208715296225146999614483257176865867572412311362252398105201644557511678179053171328641678681062496129308882700731534684329411768904920421185529144505494827908706070460177001921614692189821267467546120600239688527687872217881231173729468019623441005792563703237475678063375349

然後就可以構造，並且解密，代碼如下

from Crypto.PublicKey import RSA
import gmpy2
from base64 import b64decode
p = 250527704258269
q = 74891071972884336452892671945839935839027130680745292701175368094445819328761543101567760612778187287503041052186054409602799660254304070752542327616415127619185118484301676127655806327719998855075907042722072624352495417865982621374198943186383488123852345021090112675763096388320624127451586578874243946255833495297552979177208715296225146999614483257176865867572412311362252398105201644557511678179053171328641678681062496129308882700731534684329411768904920421185529144505494827908706070460177001921614692189821267467546120600239688527687872217881231173729468019623441005792563703237475678063375349
e = 65537
n = p * q


def getprivatekey(n, e, p, q):
    phin = (p - 1) * (q - 1)
    d = gmpy2.invert(e, phin)
    priviatekey = RSA.construct((long(n), long(e), long(d)))
    with open('private.pem', 'w') as f:
        f.write(priviatekey.exportKey())


def decrypt():
    with open('./level1.passwd.enc') as f:
        cipher = f.read()
    cipher = b64decode(cipher)
    with open('./private.pem') as f:
        key = RSA.importKey(f)
    print key.decrypt(cipher)


#getprivatekey(n, e, p, q)
decrypt()

發現不對

➜  level0 git:(master) ✗ python exp.py
一堆亂碼。。

這時候就要考慮其他情況了，一般來說現實中實現的 RSA 都不會直接用原生的 RSA，都會加一些填充比如 OAEP，我們這裏試試，修改代碼

def decrypt1():
    with open('./level1.passwd.enc') as f:
        cipher = f.read()
    cipher = b64decode(cipher)
    with open('./private.pem') as f:
        key = RSA.importKey(f)
        key = PKCS1_OAEP.new(key)
    print key.decrypt(cipher)

果然如此，得到

➜  level0 git:(master) ✗ python exp.py
FaC5ori1ati0n_aTTA3k_p_tOO_sma11

得到解壓密碼。繼續，查看 level1 中的公鑰

➜  level1 git:(master) ✗ openssl rsa -pubin -in public.key -text -modulus
Public-Key: (2048 bit)
Modulus:
    00:c3:26:59:69:e1:ed:74:d2:e0:b4:9a:d5:6a:7c:
    2f:2a:9e:c3:71:ff:13:4b:10:37:c0:6f:56:19:34:
    c5:cb:1f:6d:c0:e3:57:3b:47:c4:76:3e:21:a3:b0:
    11:11:78:d4:ee:4f:e8:99:2b:15:cb:cb:d7:73:e4:
    f9:a6:28:20:fd:db:8c:ea:16:ed:67:c2:48:12:6e:
    4b:01:53:4a:67:cb:22:23:3b:34:2e:af:13:ef:93:
    45:16:2b:00:9f:e0:4b:d1:90:c9:2c:27:9a:34:c3:
    3f:d7:ee:40:f5:82:50:39:aa:8c:e9:c2:7b:f4:36:
    e3:38:9d:04:50:db:a9:b7:3f:4b:2a:d6:8a:2a:5c:
    87:2a:eb:74:35:98:6a:9c:e4:52:cb:93:78:d2:da:
    39:83:f3:0c:d1:65:1e:66:9c:40:56:06:0d:58:fc:
    41:64:5e:06:da:83:d0:3b:06:42:70:da:38:53:e0:
    54:35:53:ce:de:79:4a:bf:f5:3b:e5:53:7f:6c:18:
    12:67:a9:de:37:7d:44:65:5e:68:0a:78:39:3d:bb:
    00:22:35:0e:a3:94:e6:94:15:1a:3d:39:c7:50:0e:
    b1:64:a5:29:a3:69:41:40:69:94:b0:0d:1a:ea:9a:
    12:27:50:ee:1e:3a:19:b7:29:70:b4:6d:1e:9d:61:
    3e:7d
Exponent: 65537 (0x10001)
Modulus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
writing RSA key
-----BEGIN PUBLIC KEY-----
MIIBIjANBgkqhkiG9w0BAQEFAAOCAQ8AMIIBCgKCAQEAwyZZaeHtdNLgtJrVanwv
Kp7Dcf8TSxA3wG9WGTTFyx9twONXO0fEdj4ho7AREXjU7k/omSsVy8vXc+T5pigg
/duM6hbtZ8JIEm5LAVNKZ8siIzs0Lq8T75NFFisAn+BL0ZDJLCeaNMM/1+5A9YJQ
OaqM6cJ79DbjOJ0EUNuptz9LKtaKKlyHKut0NZhqnORSy5N40to5g/MM0WUeZpxA
VgYNWPxBZF4G2oPQOwZCcNo4U+BUNVPO3nlKv/U75VN/bBgSZ6neN31EZV5oCng5
PbsAIjUOo5TmlBUaPTnHUA6xZKUpo2lBQGmUsA0a6poSJ1DuHjoZtylwtG0enWE+
fQIDAQAB
-----END PUBLIC KEY-----

似乎還是不是很大，再次分解，然後試了 factordb 不行，試試 yafu。結果分解出來了。

P309 = 156956618844706820397012891168512561016172926274406409351605204875848894134762425857160007206769208250966468865321072899370821460169563046304363342283383730448855887559714662438206600780443071125634394511976108979417302078289773847706397371335621757603520669919857006339473738564640521800108990424511408496383

P309 = 156956618844706820397012891168512561016172926274406409351605204875848894134762425857160007206769208250966468865321072899370821460169563046304363342283383730448855887559714662438206600780443071125634394511976108979417302078289773847706397371335621757603520669919857006339473738564640521800108990424511408496259

可以發現這兩個數非常相近，可能是 factordb 沒有實現這類分解。

繼而下面的操作類似於 level0。只是這次是直接解密就好，沒啥填充，試了填充反而錯

得到密碼 fA35ORI11TLoN_Att1Ck_cL0sE_PrI8e_4acTorS。繼續下一步，查看公鑰

➜  level2 git:(master) ✗ openssl rsa -pubin -in public.key -text -modulus
Public-Key: (1025 bit)
Modulus:
    01:ba:0c:c2:45:b4:5c:e5:b5:f5:6c:d5:ca:a5:90:
    c2:8d:12:3d:8a:6d:7f:b6:47:37:fb:7c:1f:5a:85:
    8c:1e:35:13:8b:57:b2:21:4f:f4:b2:42:24:5f:33:
    f7:2c:2c:0d:21:c2:4a:d4:c5:f5:09:94:c2:39:9d:
    73:e5:04:a2:66:1d:9c:4b:99:d5:38:44:ab:13:d9:
    cd:12:a4:d0:16:79:f0:ac:75:f9:a4:ea:a8:7c:32:
    16:9a:17:d7:7d:80:fd:60:29:64:c7:ea:50:30:63:
    76:59:c7:36:5e:98:d2:ea:5b:b3:3a:47:17:08:2d:
    d5:24:7d:4f:a7:a1:f0:d5:73
Exponent:
    01:00:8e:81:dd:a0:e3:19:28:e8:ee:51:11:08:c7:
    50:5f:61:31:05:d2:e2:ff:9b:83:71:e4:29:c2:dd:
    92:70:65:d4:09:6d:58:c3:76:31:07:f1:d4:fc:cf:
    2d:b3:0a:6d:02:7c:56:61:7c:be:7e:0b:7e:d9:22:
    28:66:9e:fb:3d:2f:2c:20:59:3c:21:ef:ff:31:00:
    6a:fb:a7:68:de:4a:0a:4c:1a:a7:09:d5:48:98:c8:
    1f:cf:fb:dd:f7:9c:ae:ae:0b:15:f4:b2:c7:e0:bc:
    ba:31:4f:5e:07:83:ad:0e:7f:b9:82:a4:d2:01:fa:
    68:29:6d:66:7c:cf:57:b9:4b
Modulus=1BA0CC245B45CE5B5F56CD5CAA590C28D123D8A6D7FB64737FB7C1F5A858C1E35138B57B2214FF4B242245F33F72C2C0D21C24AD4C5F50994C2399D73E504A2661D9C4B99D53844AB13D9CD12A4D01679F0AC75F9A4EAA87C32169A17D77D80FD602964C7EA5030637659C7365E98D2EA5BB33A4717082DD5247D4FA7A1F0D573
writing RSA key
-----BEGIN PUBLIC KEY-----
MIIBIDANBgkqhkiG9w0BAQEFAAOCAQ0AMIIBCAKBgQG6DMJFtFzltfVs1cqlkMKN
Ej2KbX+2Rzf7fB9ahYweNROLV7IhT/SyQiRfM/csLA0hwkrUxfUJlMI5nXPlBKJm
HZxLmdU4RKsT2c0SpNAWefCsdfmk6qh8MhaaF9d9gP1gKWTH6lAwY3ZZxzZemNLq
W7M6RxcILdUkfU+nofDVcwKBgQEAjoHdoOMZKOjuUREIx1BfYTEF0uL/m4Nx5CnC
3ZJwZdQJbVjDdjEH8dT8zy2zCm0CfFZhfL5+C37ZIihmnvs9LywgWTwh7/8xAGr7
p2jeSgpMGqcJ1UiYyB/P+933nK6uCxX0ssfgvLoxT14Hg60Of7mCpNIB+mgpbWZ8
z1e5Sw==
-----END PUBLIC KEY-----

發現私鑰 e 和 n 幾乎一樣大，考慮 d 比較小，使用 Wiener's Attack。得到 d，當然也可以再次驗證一遍。

➜  level2 git:(master) ✗ python RSAwienerHacker.py
Testing Wiener Attack
Hacked!
('hacked_d = ', 29897859398360008828023114464512538800655735360280670512160838259524245332403L)
-------------------------
Hacked!
('hacked_d = ', 29897859398360008828023114464512538800655735360280670512160838259524245332403L)
-------------------------
Hacked!
('hacked_d = ', 29897859398360008828023114464512538800655735360280670512160838259524245332403L)
-------------------------
Hacked!
('hacked_d = ', 29897859398360008828023114464512538800655735360280670512160838259524245332403L)
-------------------------
Hacked!
('hacked_d = ', 29897859398360008828023114464512538800655735360280670512160838259524245332403L)
-------------------------

這時我們解密密文，解密代碼如下

from Crypto.PublicKey import RSA
from Crypto.Cipher import PKCS1_v1_5, PKCS1_OAEP
import gmpy2
from base64 import b64decode
d = 29897859398360008828023114464512538800655735360280670512160838259524245332403L
with open('./public.key') as f:
    key = RSA.importKey(f)
    n = key.n
    e = key.e


def getprivatekey(n, e, d):
    priviatekey = RSA.construct((long(n), long(e), long(d)))
    with open('private.pem', 'w') as f:
        f.write(priviatekey.exportKey())


def decrypt():
    with open('./level3.passwd.enc') as f:
        cipher = f.read()
    with open('./private.pem') as f:
        key = RSA.importKey(f)
    print key.decrypt(cipher)


getprivatekey(n, e, d)
decrypt()

利用末尾的字符串 wIe6ER1s_1TtA3k_e_t00_larg3 解密壓縮包，注意去掉 B。至此全部解密結束，得到 flag。

2018 WCTF RSA¶

題目基本描述爲

Description:
Encrypted message for user "admin":

<<<320881698662242726122152659576060496538921409976895582875089953705144841691963343665651276480485795667557825130432466455684921314043200553005547236066163215094843668681362420498455007509549517213285453773102481574390864574950259479765662844102553652977000035769295606566722752949297781646289262341623549414376262470908749643200171565760656987980763971637167709961003784180963669498213369651680678149962512216448400681654410536708661206594836597126012192813519797526082082969616915806299114666037943718435644796668877715954887614703727461595073689441920573791980162741306838415524808171520369350830683150672985523901>>>

admin public key:

n = 483901264006946269405283937218262944021205510033824140430120406965422208942781742610300462772237450489835092525764447026827915305166372385721345243437217652055280011968958645513779764522873874876168998429546523181404652757474147967518856439439314619402447703345139460317764743055227009595477949315591334102623664616616842043021518775210997349987012692811620258928276654394316710846752732008480088149395145019159397592415637014390713798032125010969597335893399022114906679996982147566245244212524824346645297637425927685406944205604775116409108280942928854694743108774892001745535921521172975113294131711065606768927
e = 65537

Service: http://36.110.234.253

這個題目現在已經沒有辦法在線獲取 binary 了，現在得到的 binary 是之前已經下載好的，我們當時需要登錄用戶的 admin 來下載對應的 generator。

通過簡單逆向這個 generator，我們可以發現這個程序是這麼工作的

利用用戶給定的 license（32 個字節），迭代解密某個固定位置之後的數據，每 32 個字節一組，與密鑰相異或得到結果。
密鑰的生成方法爲
- $k_1=key$
- $k_2 =sha256(k_1)$
- ...
- $k_n=sha256(k_{n-1})$

其中，固定位置就是在找源文件 generator 中第二次出現 ENCRYPTED 的位置，然後再次偏移 32 個字節。

    _ENCRYPT_STR = ENCRYPTED_STR;
    v10 = 0;
    ENCRYPTED_LEN = strlen(ENCRYPTED_STR);
    do
    {
      do
        ++v9;
      while ( strncmp(&file_contents[v9], _ENCRYPT_STR, ENCRYPTED_LEN) );
      ++v10;
    }
    while ( v10 <= 1 );
    v11 = &file_start_off_32[loc2 + ENCRYPTED_LEN];
    v12 = loc2 + ENCRYPTED_LEN;
    len = file_size - (loc2 + ENCRYPTED_LEN) - 32;
    decrypt(&file_start_off_32[v12], &license, len);
    sha256_file_start(v11, len, &output);
    if ( !memcmp(&output, &file_contents[v12], 0x20u) )
    {
      v14 = fopen("out.exe", "wb");
      fwrite(v11, 1u, len, v14);
      fclose(v14);
      sprintf(byte_406020, "out.exe %s", argv[1]);
      system(byte_406020);
    }

同時，我們需要確保生成的文件的校驗對應的哈希值恰好爲指定的值，由於文件最後是一個 exe 文件，所以我們可以認爲最後的文件頭就是標準的 exe 文件，因此就不需要知道原始的 license 文件，進而我們可以編寫 python 腳本生成 exe。

在生成的 exe 中，我們分析出程序的基本流程爲

讀取 license
使用 license 作爲 seed 分別生成 pq
利用 p，q 生成 n，e，d。

其漏洞出現在生成 p，q 的方法上，而且生成 p 和 q 的方法類似。

我們如果仔細分析下生成素數的函數的話，可以看到每個素數都是分爲兩部分生成的

生成左半部分 512 位。
生成右半部分 512 位。
左右構成 1024 比特位，判斷是不是素數，是素數就成功，不是素數，繼續生成。

其中生成每部分的方式相同，方式爲

sha512(const1|const2|const3|const4|const5|const6|const7|const8|v9)
v9=r%1000000007

只有 v9 會有所變化，但是它的範圍卻是固定的。

那麼，如果我們表示 p，q 爲

$p=a*2^{512}+b$

$q=c*2^{512}+d$

那麼

$n=pq=ac*2^{1024}+(ad+bc)*2^{512}+bd$

那麼

$n \equiv bd \bmod 2^{512}$

而且由於 p 和 q 在生成時，a，b，c，d 均只有 1000000007 種可能性。

進而，我們可以枚舉所有的可能性，首先計算出 b 可能的集合爲 S，同時我們使用中間相遇攻擊，計算

$n/d \equiv b \bmod 2^{512}$

這裏由於 b 和 d 都是 p 的尾數，所以一定不會是 2 的倍數，進而必然存在逆元。

這樣做雖然可以，然而，我們可以簡單算一下存儲空間

$64*1000000007 / 1024 / 1024 / 1024=59$

也就是說需要 59 G，太大了，，所以我們仍然需要進一步考慮

$n \equiv bd \bmod 2^{64}$

這樣，我們的內存需求瞬間就降到了 8 G左右。我們仍然使用枚舉的方法進行運算。

其次，我們不能使用 python，，python 佔據空間太大，因此需要使用 c/c++ 編寫。

枚舉所有可能的 d 計算對應的值 $n/d$ 如果對應的值在集合 S 中，那麼我們就可以認爲找到了一對合法的 b 和 d，因此我們就可以恢復 p 和 q 的一半。

之後，我們根據

$n-bd=ac*2^{1024}+(ad+bc)*2^{512}$

可以得到

$\frac{n-bd}{2^{512}} = ac*2^{512}+ad+bc$

$\frac{n-bd}{2^{512}} \equiv ad+bc \bmod 2^{512}$

類似地，我們可以計算出 a 和 c，從而我們就可以完全恢復出 p 和 q。

在具體求解的過程中，在求 p 和 q 的一部分時，可以發現因爲是模 $2^{64}$ ，所以可能存在碰撞（但其實就是一個是 p，另外一個是q，恰好對稱。）。下面我們就求得了 b 對應的 v9。

注意：這裏枚舉出來的空間大約佔用 11 個 G（包括索引），所以請選擇合適的位置。

b64: 9646799660ae61bd idx_b: 683101175 idx_d: 380087137
search 23000000
search 32000000
search 2b000000
search d000000
search 3a000000
search 1c000000
search 6000000
search 24000000
search 15000000
search 33000000
search 2c000000
search e000000
b64: 9c63259ccab14e0b idx_b: 380087137 idx_d: 683101175
search 1d000000
search 3b000000
search 7000000
search 16000000
search 25000000
search 34000000

其實，我們在真正得到 p 或者 q 的一部分後，另外一部分完全可以使用暴力枚舉的方式獲取，因爲計算量幾乎都是一樣的，最後結果爲

...
hash 7000000
hash 30000000
p = 13941980378318401138358022650359689981503197475898780162570451627011086685747898792021456273309867273596062609692135266568225130792940286468658349600244497842007796641075219414527752166184775338649475717002974228067471300475039847366710107240340943353277059789603253261584927112814333110145596444757506023869
q = 34708215825599344705664824520726905882404144201254119866196373178307364907059866991771344831208091628520160602680905288551154065449544826571548266737597974653701384486239432802606526550681745553825993460110874794829496264513592474794632852329487009767217491691507153684439085094523697171206345793871065206283
plain text 13040004482825754828623640066604760502140535607603761856185408344834209443955563791062741885
hash 16000000
hash 25000000
hash b000000
hash 34000000
hash 1a000000
...
➜  2018-WCTF-rsa git:(master) ✗ python
Python 2.7.14 (default, Mar 22 2018, 14:43:05)
[GCC 4.2.1 Compatible Apple LLVM 9.0.0 (clang-900.0.39.2)] on darwin
Type "help", "copyright", "credits" or "license" for more information.
>>> p=13040004482825754828623640066604760502140535607603761856185408344834209443955563791062741885
>>> hex(p)[2:].decode('hex')
Traceback (most recent call last):
  File "<stdin>", line 1, in <module>
  File "/usr/local/Cellar/python@2/2.7.14_3/Frameworks/Python.framework/Versions/2.7/lib/python2.7/encodings/hex_codec.py", line 42, in hex_decode
    output = binascii.a2b_hex(input)
TypeError: Odd-length string
>>> hex(p)[2:-1].decode('hex')
'flag{fa6778724ed740396fc001b198f30313}'

最後我們便拿到 flag 了。

詳細的利用代碼請參見 ctf-challenge 倉庫。

相關編譯指令，需要鏈接相關的庫。

g++  exp2.cpp -std=c++11 -o main2 -lgmp -lcrypto -pthread

參考¶

https://upbhack.de/posts/wctf-2018-writeup-rsa/