Breaking Random Number Generators with Chosen Seed

June 20 2017

# # # #

Find the flag.

Source is as follows

#!/usr/bin/env python3

import random
import time
import string
import signal

# use secure seed
random.seed(int(time.time()))

with open('flag.txt') as f:
	flag = f.read()

# large constant prime
p = 174807157365465092731323561678522236549173502913317875393564963123330281052524687450754910240009920154525635325209526987433833785499384204819179549544106498491589834195860008906875039418684191252537604123129659746721614402346449135195832955793815709136053198207712511838753919608894095907732099313139446299843

# large constant generator
g = 41899070570517490692126143234857256603477072005476801644745865627893958675820606802876173648371028044404957307185876963051595214534530501331532626624926034521316281025445575243636197258111995884364277423716373007329751928366973332463469104730271236078593527144954324116802080620822212777139186990364810367977

# large secret key because safety is number one priority
a = random.randrange(1, p - 2)

print("A = ", pow(g, a, p))

# In 60 seconds, deliver a SIGALRM and terminate                  
signal.alarm(30)

_a = int(input("Now give me \"a\" back!\n"))

if a == _a:
	print("Nice!")
	print(flag)
else:
	print("Boo!")

Solution

The prime, p, generator g, are given and an a is closen from the interval 1 to p - 2. The user is asked to break the discrete log problem and find a number a, with a given constant g, p, and A. Where

A = pow(g, a, p)

Normally, this sort of a problem would involve a weakness in the prime or a weakness in the choice of numbers used. However in this case, the numbers chosen are relatively secure, and there was no brute-force attack which could solve the problem in such a small time-frame.

A decent suggestion that I heard was to randomly compute a, and the corresponding A for a computable set of numbers, and run the script until we get lucky. This could perhaps yeild results.

However the primary flaw here was the usage of a bad seed.

The seed is

# use secure seed
random.seed(int(time.time()))

This is very, very bad to use for security. The reason being that you can now find out exactly what the secret pseudo-random number generator will generate next and you can do this deterministically. So as long as you are able to match the seed.

From the top of my memory, the time.time() returns a float, which is seconds from unix epoch in a UTC timezone. This remains constant for all timezones too.

What's more is that I make it even easier… the seed is cast to int. So you don't even need to match the floating point seed generated, you just need to match it to the closest integer second.

Let's break it by re-using the same code itself.

#!/usr/bin/env python3

import random
import time
import string

# use secure seed
random.seed(int(time.time()))

# large constant prime
p = 174807157365465092731323561678522236549173502913317875393564963123330281052524687450754910240009920154525635325209526987433833785499384204819179549544106498491589834195860008906875039418684191252537604123129659746721614402346449135195832955793815709136053198207712511838753919608894095907732099313139446299843

a = random.randrange(1, p - 2)

print(a)

Now we run this, and connect to the service at the same time…. Something interesting happens. We are able to find out what the randomly generated a is. It takes a few tries to get the timing exact, but withing a few tries, I got my answers.

$ python solution.py # run in a different terminal
58549141448034825724470421456086046263783009030305765315029214732103597725629643584640519564993686952460147424224499824237208320256710402644454885977843807235722331321468491704215402565594336151254611301385405568079284208008877673066349902012892511931129094269278697396028322758428218881094326448915019094908

$ nc yuno.parthkolekar.me 1143
A =  162845821762659164979308029502757300883569801448030678296035478772826604789837097465089145966790720159023736096828352235603744416863113902483466847601771758735899269752092426466070440647296490845779612629770248165661953834419624646583136868830812897675991836545617055134899212084666355607002046548367853130352
Now give me "a" back!
58549141448034825724470421456086046263783009030305765315029214732103597725629643584640519564993686952460147424224499824237208320256710402644454885977843807235722331321468491704215402565594336151254611301385405568079284208008877673066349902012892511931129094269278697396028322758428218881094326448915019094908
Nice!
flag{random number generators? Or did you break discrete log?}

Flag

flag{random number generators? Or did you break discrete log?}

Additional Notes

An interesting observation was to be made when the above source is run as python2 instead of python3.

The input() in python2 can be considered to be equivalent to eval(raw_input())

Which meant that the problem would be solved quite easily by giving the input "a".

This would convert the

_a = int(input("Now give me \"a\" back!\n"))

to become

_a = int(eval(raw_input("Now give me \"a\" back!\n")))

This would make _a = int(a). Which would pass the test. I was intending to give this as the follow-up question. But I was unable to find a decent sandboxer. Because the eval() can be used to do a lot of un-intended things, the lack of a decent sandbox prevented me from giving this follow-up problem.

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