**The challenge was:**

*“We've found the source to the Arstotzka spies rendevous server, we must find out their new vault key.”*You are also provided with a slurp.py python script and the ip:port.

Trivia: Arstotzka is the place of a indie game called “Papers, Please”. The game has a very unusual gameplay. In the game you play a border control guard who is checking the the passports of persons wanting to enter Arstotzka. The game plays in a fictional Eastern Block state but the setting could also portrait modern day USA.

slurp.py is the server listening on 128.238.66.222:7788. The goal is to pass it’s authentication scheme to get the flag.

The authentication protocol is as follows:

**Phase 1, sha1 challenge**

base64_encoded_24bit_urandom

Client -> Server:

The client has to find a value so that the hash $$sha1(base64\_encoded\_24bit\_urandom\ +\ client\_choosen\_chars)$$ ends with "\xFF\xFF\xFF". If the client is able to generate such a hash, the server continues to phase 2.

We solved this phase with a simple brute force of the sha1 hash. After a few seconds you can find a sha1 hash which suffices the

server-supplied challenge.

**Phase 2, authentication**

**Remark 1:**

The values sent by the client are transfer in a way that would potentially allow negative values. The formatting uses a unsigned short as packed length, concatenated with a string of the value in hexadecimal notation. However we couldn’t find any use for this during the exploitation.

**Remark 2:**

The random number generation used by the server is not perfect. It generates a 2048 bit number from 320 bit output of urandom, however it’s hashed in such a way that a portion of the resulting number will be contain zero bit. We couldn’t find a way to use this during the exploitation.

def cryptrand(self,n=2048): p1=self.hashToInt(os.urandom(40))<<1600 p1+=self.hashToInt(p1)<<1000 p1+=self.hashToInt(p1)<<500 p1+=self.hashToInt(p1) bitmask=((2<<(n+1))-1) p1=(p1&bitmask) return (p1% self.N)Server -> Client:

“Welcome to Arstotzka's check in server, please provide the agent number”

Client -> Server:

The client chooses the values of index and cEphemeral and send them to the server.

Index has to be at least 2, but not greater than N/2 (N is constant, known prime used as modulus in all operations).

The only restriction on cEphemeral is that cEphemeral % N != 0.

Server -> Client:

The server sends the values of sEphermeral, salt.

The client already knows salt because it’s sha512(index).

Client -> Server:

The client calculates the gennedKey and sends it to the server.

Server -> Client:

The server checks whether the server-calculated gennedKey is the same as the one provided by the client. Then it tells the client whether authentication was successful or not. If the authentication was successful, the server sends:

“Well done comrade, the key to the vault is $flag_value”.

If you’re able to crack the authentication, you get the flag and solve the challenge.

So how does the server test authentication?

The client-supplied values are: index, cEphemeral

The server settings are: password, 2048 bit modulus N

The password is actually empty in the provided sources which let us to a dead end; more on this later.

Then the server calculates:

$$salt = sha512(index)$$ $$storedKey = index^{sha512(salt, password)}\ mod\ N$$ $$sEphemeralPriv = cryptrand()\ \ \#a\ 2048\ bit\ value,\ random\ in\ every\ connection$$ $$sEphemeral = index^{sEphemeralPriv} + 3 * storedKey\ mod\ N$$ $$sEphemeral = index^{sEphemeralPriv} + 3 * index^{sha512(salt + password)}\ mod\ N$$ $$\Longleftrightarrow$$ $$index^{sEphemeralPriv} = sEphemeral - 3 * index^{sha512(salt + password)}\ mod\ N$$ Note that sEphemeral and salt are now sent to the client $$slush = sha512(cEphemeral, sEphemeral)$$ $$agreedKey = sha512( (cEphemeral * storedKey^{slush})^{sEphemeralPriv}\ mod\ N )$$ $$gennedKey = sha512(sha512(N) \oplus sha512(index), sha512(index), salt, cEphemeral, sEphemeral, agreedKey)$$ The only unknown value is agreedKey.

Then, the compare on the client-supplied gennedKey is done. So, the client needs to find a known agreedKey.

**First attempt**

$$agreedKey = sha512( (cEphemeral * storedKey^{slush})^{sEphemeralPriv}\ mod\ N )$$ $$agreedKey = sha512( agreedKey\_withouthash )$$ $$agreedKey\_withouthash = (cEphemeral * storedKey^{slush})^{sEphemeralPriv}\ mod\ N$$ $$agreedKey\_withouthash = (cEphemeral * index^{sha512(salt, password) * slush})^{sEphemeralPriv}\ mod\ N$$ If we choose $$cEphemeral = index^{xxx}$$ we get

$$agreedKey\_withouthash = (index^{xxx} * index^{sha512(salt, password) * slush})^{sEphemeralPriv}\ mod\ N$$ $$agreedKey\_withouthash = (index^{xxx + sha512(salt, password) * slush})^{sEphemeralPriv}\ mod\ N$$ $$agreedKey\_withouthash = (index^{sEphemeralPriv})^{xxx + sha512(salt, password) * slush}\ mod\ N$$ With the following value from above:

$$index^{sEphemeralPriv} = sEphemeral - 3 * index^{sha512(salt + password)}\ mod\ N$$ We get the following:

$$agreedKey\_withouthash = (sEphemeral - 3 * index^{sha512(salt + password)})^{xxx + sha512(salt, password) * slush}\ mod\ N$$ So, we had a formula for calculating agreedKey and knew all variables of the formula. By calculating agreedKey we were able to calculate gennedKey and send it to the server. This all worked well for our test environment. However it did not work on the live flag server. Our guess was that the

*password*is different on the live flag server, thus we were not able to calculate the correct agreedKey.

**Second attempt**

$$agreedKey\_withouthash = index^{sha512(salt, password) * slush * sEphemeralPriv}\ mod\ N$$ Because the exponential is changing for each connection on random, we can assume that it’s divisible by some small number, e.g. 3 or 4 (if not, we can retry until it is). So, let’s now find the index, such that:

$$index^3 = 1\ (mod\ N)$$

*(index is cubic root of 1 modulo prime N)*

If we manage to find such index and (sha512(salt, password) * slush * sEphemeralPriv) will be divisible by 3, agreedKey_withouthash will equals to 1. How to find it? One line in mathematica:

Reduce[x^3 == 1, x, Modulus->5924486056224…(the value of N)]

Result: http://wklej.org/hash/3ca3fcbd545/txt/

There are 3 solutions, but only the second one satisfies constraints on the index value. Now we have to set the index to this number, calculate agreedKey = SHA512(1) and send the data, until it succeeds ;)

**Remarks:**

Instead of 3 you could use 4 and then find the root using formula: $$a^{\frac{N-1}{4}}$$ (

*a*can be, for example, 2).

You couldn’t use 2 instead of 3, because the only quadratic roots of 1 are 1 and -1 (refused by server’s constraints).

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