# PicoCTF 2014 Write-ups

## Low Entropy - 110 (Cryptography)

#### Writeup by Oksisane

Created: 2014-11-07 21:04:24

### Problem

Daedalus Corp's spy in Thyrin Labs seems to sometimes use an encrypted drop box for their messages. We intercepted one of their messages, but we don't seem to be able to decrypt it. Fortunately, we have the source and the address of their key generation server: maybe there's a way to use that to decrypt their message? Unfortunately, we don't have their list of cached primes...

Their source, and our intercepted message, are here. The key generation service is running at vuln2014.picoctf.com:51818.

### Hint

We're pretty sure that 30 total possible primes is low enough that we should be able to correlate different keys somehow...

### Overview

Factor the given publickey since we have only 30 primes, then use the factors to compute the private key and decrypt the message.

### Details

For the purposes of this solution, I am going to assume you are familiar with the basics of the RSA encryption/decryption

In this problem we are given a server which generates RSA moduli and are told that there are only 30 possible primes that the server uses to generate them. First, we copy the given modulus 0xc20a1d8b3903e1864d14a4d1f32ce57e4665fc5683960d2f7c0f30d5d247f5fa264fa66b49e801943ab68be3d9a4b393ae22963888bf145f07101616e62e0db2b04644524516c966d8923acf12af049a1d9d6fe3e786763613ee9b8f541291dcf8f0ac9dccc5d47565ef332d466bc80dc5763f1b1139f14d3c0bae072725815f and the cipher text 0x49f573321bdb3ad0a78f0e0c7cd4f4aa2a6d5911c90540ddbbaf067c6aabaccde78c8ff70c5a4abe7d4efa19074a5249b2e6525a0168c0c49535bc993efb7e2c221f4f349a014477d4134f03413fd7241303e634499313034dbb4ac96606faed5de01e784f2706e85bf3e814f5f88027b8aeccf18c928821c9d2d830b5050a1e out of the intercepted_message.pcap using Wireshark. We can glean two key observations from this:

1. This modulus must be a product of two of the 30 primes being used.
2. The server lets us generate as many moduli as we want.

If we repeatedly request moduli it wont be too long before one of the keys shares a prime with our modulus (See a python sample of this below). If this occurs, the GCD of the two moduli will be the p component of the modulus and dividing the modulus, N by p will yeild the other prime q. Now that we have the two primes and know the public exponent, e equals 65537, we can easily compute the private exponent d as 70118266743531770742541821902419779511239599511770182985184154839839018008193467076473425663009520539545369270962530218486192141886710801892227602961998928213537654150767790772889562458972990736497832327701547730106443165511208494243221689127374975558908570745982766653293349491204750802344491680794276628473 and the message as 41494331723717004087840363038436379137678883897083290482504777727264502253603851759718429929339251785330232810578894088434140357577326859552100822540305154760883449774383987636028167175968532867374560076986174085246917601557997651752239365247146390720857679075750704377311507373191982 Finally, converting the decrypted number to plaintext reveals the message:

Good thing no one can read this! I'd hate for them to know that the flag is make_sure_your_rng_generates_lotsa_primes. and the flag!

Python Sample Code for finding the gcd of the modulus:

from fractions import gcd
import socket

server = "vuln2014.picoctf.com"
c =  int(0xc20a1d8b3903e1864d14a4d1f32ce57e4665fc5683960d2f7c0f30d5d247f5fa264fa66b49e801943ab68be3d9a4b393ae22963888bf145f07101616e62e0db2b04644524516c966d8923acf12af049a1d9d6fe3e786763613ee9b8f541291dcf8f0ac9dccc5d47565ef332d466bc80dc5763f1b1139f14d3c0bae072725815f)
test_num = 1
while gcd(c,test_num) == 1:
sock = socket.socket(socket.AF_INET, socket.SOCK_STREAM)
sock.connect((server, 51818))
sock.recv(2048)
string = "0x" + sock.recv(2048)
test_num =  int(string,16)
print gcd(test_num,c)
print c/gcd(test_num,c)


### Flag

make_sure_your_rng_generates_lotsa_primes