# Super Secure Vault
Binary, 400 points
## Description
*Open the Vault to get the treasure.*
## Solution
We were given an ELF, [vault](vault). Let's try it out:
```console╭─face0xff@aniesu-chan ~/ctf/pragyan/vault  ╰─$ ./vault    Enter the key: abcWrong key.```
Let's disassemble it and generate some pseudocode using IDA:
```cint __cdecl main(int argc, const char **argv, const char **envp){  unsigned int v3; // ST0C_4  unsigned int v4; // ST0C_4  unsigned int v5; // ST0C_4  unsigned int v6; // ST0C_4  __int64 v7; // rsi  int v9; // [rsp+14h] [rbp-BCh]  int v10; // [rsp+14h] [rbp-BCh]  int v11; // [rsp+14h] [rbp-BCh]  int v12; // [rsp+20h] [rbp-B0h]  int v13; // [rsp+24h] [rbp-ACh]  int v14; // [rsp+28h] [rbp-A8h]  int v15; // [rsp+2Ch] [rbp-A4h]  int v16; // [rsp+30h] [rbp-A0h]  int v17; // [rsp+34h] [rbp-9Ch]  int v18; // [rsp+38h] [rbp-98h]  int v19; // [rsp+3Ch] [rbp-94h]  int v20; // [rsp+40h] [rbp-90h]  int v21; // [rsp+44h] [rbp-8Ch]  char s; // [rsp+50h] [rbp-80h]  char v23; // [rsp+90h] [rbp-40h]  unsigned __int64 v24; // [rsp+C8h] [rbp-8h]   v24 = __readfsqword(0x28u);  v12 = 213;  v13 = 8;  v14 = 229;  v15 = 5;  v16 = 25;  v17 = 4;  v18 = 83;  v19 = 7;  v20 = 135;  v21 = 5;  printf("Enter the key: ", argv, envp);  __isoc99_scanf("%s", &s);  if ( strlen(&s) > 0x1E )    fail(0LL);  v3 = getNum((__int64)"27644437104591489104652716127", 0, v13);  if ( (unsigned int)mod(&s, v3) != v12 )    fail(0LL);  v9 = v13;  v4 = getNum((__int64)"27644437104591489104652716127", v13, v15);  if ( (unsigned int)mod(&s, v4) != v14 )    fail(0LL);  v10 = v15 + v9;  v5 = getNum((__int64)"27644437104591489104652716127", v10, v17);  if ( (unsigned int)mod(&s, v5) != v16 )    fail(0LL);  v11 = v17 + v10;  v6 = getNum((__int64)"27644437104591489104652716127", v11, v19);  if ( (unsigned int)mod(&s, v6) != v18 )    fail(0LL);  v7 = (unsigned int)getNum((__int64)"27644437104591489104652716127", v19 + v11, v21);  if ( (unsigned int)mod(&s, v7) != v20 )    fail(0LL);  printf("Enter password: ", v7);  __isoc99_scanf("%s", &v23);  func2(&v23, &s, "27644437104591489104652716127");  return 0;}```
This is the main function. It basically asks for a key that should not exceed 30 bytes, and then runs several getNum calls on a certain string "27644437104591489104652716127".
```c__int64 __fastcall getNum(__int64 a1, int a2, int a3){  unsigned int v4; // [rsp+18h] [rbp-8h]  int i; // [rsp+1Ch] [rbp-4h]   v4 = 0;  for ( i = a2; i < a2 + a3; ++i )    v4 = 10 * v4 + *(char *)(i + a1) - 48;  return v4;}```
getNum(s, i, j) simply seems to take the argument string s and cut it starting from the i-th byte and keeping j bytes. It then converts it into a integer.
Here is the function mod :
```c__int64 __fastcall mod(const char *a1, int a2){  unsigned int v3; // [rsp+18h] [rbp-18h]  int i; // [rsp+1Ch] [rbp-14h]   v3 = 0;  for ( i = 0; i < strlen(a1); ++i )    v3 = (signed int)(10 * v3 + a1[i] - 48) % a2;  return v3;}```
It just seems to compute a1 mod a2.
Let's put everything together ; the big number is divided into 5 parts: `27644437, 10459, 1489, 1046527, 16127`, and for each of these, the program calculates our input, which has to be a number less than 30 digits, modulus the part. It then tests if it is equal to a certain hardcoded value. Here are the conditions that need to be reunited:
```s = 213 mod 27644437s = 229 mod 10459s = 25 mod 1489s = 83 mod 1046527s = 135 mod 16127```
So it happens that all these moduli are co-prime, so we can use the Chinese Remainder Theorem to compute s. You can check the full script to see how it is computed.
We find that the lowest solution for s is 3087629750608333480917556.
Once we entered the key, we are asked for a password, and there is a call to func2(password, s, "27644437104591489104652716127").
Here are the contents of func2:
```cint __fastcall func2(__int64 a1, char *a2, const char *a3){  unsigned __int64 v3; // rax  int v4; // ST30_4  int v5; // ST34_4  int v7; // [rsp+24h] [rbp-3Ch]  int v8; // [rsp+28h] [rbp-38h]  int v9; // [rsp+28h] [rbp-38h]  int v10; // [rsp+2Ch] [rbp-34h]  int v11; // [rsp+2Ch] [rbp-34h]  char *v12; // [rsp+40h] [rbp-20h]   v12 = strcat(a2, a3);  v3 = (unsigned __int64)&v12[strlen(v12)];  *(_WORD *)v3 = 12344;  *(_BYTE *)(v3 + 2) = 0;  v7 = 0;  v8 = 0;  v10 = strlen(v12) >> 1;  while ( v8 < strlen(v12) >> 1 )  {    if ( *(_BYTE *)(v7 + a1) != matrix[100 * (10 * (v12[v8] - 48) + v12[v8 + 1] - 48)                                     - 48                                     + 10 * (v12[v10] - 48)                                     + v12[v10 + 1]] )      fail(1LL);    ++v7;    v8 += 2;    v10 += 2;  }  v9 = 0;  v11 = strlen(v12) >> 1;  while ( v9 < strlen(v12) >> 1 )  {    v4 = 10 * (v12[v9] - 48) + v12[v9 + 1] - 48;    v5 = 10 * (v12[v11] - 48) + v12[v11 + 1] - 48;    if ( *(_BYTE *)(v7 + a1) != matrix[100 * (v4 * v4 % 97) + v5 * v5 % 97] )      fail(1LL);    ++v7;    v9 += 2;    v11 += 2;  }  puts("Your Skills are really great. Flag is:");  return printf("pctf{%s}\n", a1);}```
So basically what this does is, we concat s with 27644437104591489104652716127 and then we append "80" (the 12344 decimal). We obtain a string v3 = "30876297506083334809175562764443710459148910465271612780".
Then some loops will compare each character of our password to a certain value in "matrix". Looking it up on IDA, matrix is a 10000-byte chunk of ascii characters, from which I dumped the contents in [matrix.txt](matrix.txt).
The only thing that is left for us to do is to calculate all the indexes that will be read in the matrix to figure out the password. Here's the final keygen:
```pythonfrom functools import reduceimport binascii
def chinese_remainder(n, a):    s = 0    prod = reduce(lambda a, b: a*b, n)    for n_i, a_i in zip(n, a):        p = prod // n_i        s += a_i * mul_inv(p, n_i) * p    return s % prod def mul_inv(a, b):    b0 = b    x0, x1 = 0, 1    if b == 1: return 1    while a > 1:        q = a // b        a, b = b, a % b        x0, x1 = x1 - q * x0, x0    if x1 < 0:        x1 += b0    return x1
a = [213, 229, 25, 83, 135]n = [27644437, 10459, 1489, 1046527, 16127]N = 27644437104591489104652716127
s = chinese_remainder(n, a)
matrix = open('matrix.txt', 'r').read()matrix = matrix.replace(' ', '').replace('\r', '').replace('\n', '')matrix = binascii.unhexlify(matrix)
v12 = str(s) + str(N) + "80"v12 = list(map(int, list(v12)))
v8 = 0v10 = len(v12) // 2
password = b""while v8 < len(v12) // 2:    q_ = 100*(10*v12[v8]+v12[v8+1])+10*v12[v10]+v12[v10+1]    password += bytes([matrix[q_]])    v8 += 2    v10 += 2
v9 = 0v11 = len(v12) // 2
while v9 < len(v12) // 2:    v4 = 10 * v12[v9] + v12[v9 + 1]    v5 = 10 * v12[v11] + v12[v11 + 1]    password += bytes([matrix[100*(v4**2%97)+v5**2%97]])    v9 += 2    v11 += 2
print(s, password)```
and its output:
```console╭─face0xff@aniesu-chan ~/ctf/pragyan/vault  ╰─$ python vault.py3087629750608333480917556 b'R3v3rS1Ng_#s_h311_L0t_Of_Fun'```
Let's try it out.
```console╭─face0xff@aniesu-chan ~/ctf/pragyan/vault  ╰─$ ./vaultEnter the key: 3087629750608333480917556Enter password: R3v3rS1Ng_#s_h311_L0t_Of_FunYour Skills are really great. Flag is:pctf{R3v3rS1Ng_#s_h311_L0t_Of_Fun}```
Enjoy!
Note: actually, there are more than 100000 correct (key, password) couples. Indeed, the solutions to the modular system of equations are all congruent modulo the product of the five integers. I lost a lot of time because I was looking for a 30-digit key instead of simply choosing the lowest solution, which yields a "readable" "flag-looking" password flag.
Some examples of other valid flags...
```10353650500772965893596379 b'XbGeQsfL#soYFTr$Byze@PIFPiRf'17619671250937598306275202 b'F$ihT}L(nF$IqTGpajfB{hNgi@wf'24885692001102230718954025 b'@XvtR)LWe&v(mTeWiT(j!Yhp{gzf'32151712751266863131632848 b'VPgdRtre{hQXVTyiy*)WQjoEz@Zf'39417733501431495544311671 b'jayNQrSOeiABITafV_zKQVEoH!hf'46683754251596127956990494 b'edUeTTxtvxa)MTE^QjfpQWTp{cvf'```