Untempering the Mersenne Twister RNG

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I've been working on the Matasano Crypto Challenges. They are worth an article on their own1. In them you learn crypto as deployed in web applications. They teach you crypto systems by breaking them. This is an excellent way to learn crypto. It teaches you to assess systems based on potential weaknesses rather than their abstract properties.

A crypto implementation is usually built out of smaller units called crypto primitives. It is important that these smaller units be secure against leaking information. If not the integrity of the entire system is compromised. An important cryptographic primative is the pseudo-random number generator (PRNG). While cryptographically secure PRNGs (CSPRNGs) exist2, programmers may not realize that they should use them. Instead they reach for their language's native PRNG.

This this is in all likelihood the Mersenne Twister PRNG, based on a Mersenne prime number. I still haven't grokked why this is important to the algorithm. I do know that it makes the period of the PRNG (the amount of time before the PRNG repeats an output) huge (~106000). The algorithm is explained in technical detail on the wikipedia page linked above. What interests me the most is the final step of the algorithm. This is the "tempering step" when the state array output is transformed. Reversing the tempering step is covered in many places as well. The mathematical description of the algorithm calls for the tempering matrix to be invertible, meaning: $$ x_{i} = T s_{i} $$ $$ s_{i} = T^{-1}x_{i} $$ Where \(T\) is the tempering matrix and \(T^{-1}\) must exist. On paper, it is possible to take an output from the Mersenne RNG and multiply the bit-vector by \(T^{-1}\). This recovers one of the internal states \(s_{i}\). Repeat this 624 times and you have discovered the entire internal state of the RNG. You can now predict every future output of the RNG for all time. If your crypto system is based on the Mersenne RNG once you've cloned it you can create the same stream of bytes. This may allow you to decrypt all future encrypted messages (and with a little more work, all past messages).

I say on paper, because it turns out that inverting \(T\) as it is stated isn't as clear as inverting a matrix. The algorithm specifies \(T\) as: $$ x := s\oplus ((s >> u) \otimes d)\\ x := x\oplus ((x << s) \otimes b)\\ x := x\oplus ((x << t) \otimes c)\\ x := x\oplus ((x >> t))\\ $$ Where \(s\) is an input from the internal state, \(x\) is the final result, \(\oplus\) is addition mod 2 (bitwise xor), >> and << are the right and left bitshift operators respectively, \(\otimes\) is multiplication mod 2 (bitwise and). Finally \(u\), \(d\), \(s\), \(b\), \(t\), and \(c\) are parameters for the tempering transform. The specific values can be referenced on the wiki page or in the reference implementation provided by the author of the algorithm.

As attackers we're after the internal state of the RNG. This means inverting the tempering transformation outlined above. How can we do this? If we wanted we could construct a matrix representation of \(T\) and invert this matrix. This seems infeasible on the face of it. Let's instead look at some interesting properties of each transformation step. I don't know if this combination of operations has a name in the literature. I've taken to calling them diffuse operations.3 If you are trying to wrap your head around these transformations, I recommend playing with them on paper.

Let's look at one in detail: $$ x := x\oplus ((x >> a) \otimes b) $$ A source of confusion is the bitwise and since we think of this operation as "throwing away" information. It seems that we're loosing information when a 0 in \(b\)'s bit vector is propagated through multiplication by (x >> a). For the moment lets ignore this and set the \(b\) vector to all 1's (0xFFFFFFFF for a 32 bit Mersenne RNG). Let's also work with 8 bit numbers. You can write them on a sheet of paper without cramping your hand or making too many mistakes.

In the following, \(|x|\) is the number of bits represented by \(x\) (32 in the full case, 8 in our toy). First note that if \(a > |x|/2\) we have the following interesting property. Say \(a=5\), if we put each operation on its own row4 we have:

       x: 1  0  1  1  1  0  1  1 
    x>>a: 0  0  0  0  0  1  0  1 
x^(x>>a): 1  0  1  1  1  1  1  0

Notice that the first \(a=5\) bits of the output is the same as the input. This is remarkable. It provides an immediate answer for how we recover the original \(x\), inverting the transformation:

       x: 1  0  1  1  1  1  1  0 
    x>>a: 0  0  0  0  0  1  0  1 
x^(x>>a): 1  0  1  1  1  0  1  1

When we shift over 5 places, we put the same three bits in the lower part that we xor, this recovers the input. Those zeros also mean that if you multiply by a mask the same pattern will hold:

           x: 1  0  1  1  1  0  1  1 
    (x>>a)&b: 0  0  0  0  0  1  0  0 
x^((x>>a)&b): 1  0  1  1  1  1  1  1

We now know that the first \(a\) bits are untouched by this transformation. This is the key to reversing it. The left shift case is symmetric.

Let us now turn to the more complicated case when \(a < |x|/2\). In this case, when we go to recover the bits, some of the scrambled bits will be "in the way" and we may not get a clean recovery the way we do when \(a > |x|/2\). How do we proceed? We still have \(a\) bits at the front that are untouched, so we can get \(2a\) bytes with one diffuse step. If we continue to feed the output of the diffuse step as the input to the next, eventually we'll recover the original integer.

The only trouble with this is that it is not clear how to calculate the number of times you need to apply the diffuse step in order to recover the (unknown) integer. Furthermore, the number of steps required depends on \(x\). While it seems we're at an impasse, it is possible to write down a modified diffuse procedure that takes the original \(x\) value. Then, it is simply a matter of applying the modified procedure \(\lceil |x|/a \rceil\) times. The modified procedure is: $$ r := x r := x \oplus ((r >> a) \otimes b) $$

There you have it, a prescription for undoing each step of the tempering function. When implemented, you apply the inverse of each diffusion step in reverse order. This inverts the tempering of RNG output. From here its a short step to getting the internal state of the RNG and thus break the whole thing wide open.

There are some unanswered questions though. The diffuse steps are cyclic, meaning that if you let: $$ d(x,c,b) = x \oplus (x >> c) \otimes b $$ and you repeatedly feed the output of one to the input of the next: $$ d^{(n)}(x,c,b)=d(d^{(n-1)}(x,c,b),c,b)\\ d^{0}(x,c,b)=d(x,c,b) $$ then there exists a finite \(n\) such that $$ x = d^{(n)}(x,c,b) $$

In this case, reversing the tempering is just a matter of figuring out how to derive \(n\) for values of \(x\), \(c\) and \(b\). I suspect that the answer is independent of \(b\), but I haven't been able to get anywhere on actually constructing an argument that proves any real relationship. Furthermore, my small explorations show that while \(n\) is finite, it depends on \(x\), this is undesirable since it requires knowledge that is initially inaccessible.

One final point before I close: This entire attack hinges on the tempering step being invertible. If the tempering step weren't invertible, the attack wouldn't work and the internal state would remain hidden from attackers, securing this PRNG. The most obvious way of achieving this is to hash the output and then use whatever bits of the hash are required for the application. The issue with this is that hashes may skew the statistical properties of the PRNG, or worse, provide side-channel leaks.

In closing, crypto is a lot of fun. The math involved while simple, has subtleties that are fun to play around with. I hope that if you are also wandering the path towards crypto enlightenment that you find this explanation useful. If you have any comments, you can tweet me @dbjergaard.

Footnotes:

1

The folks on #cryptopals in freenode have been extraordinarily helpful and are always patient with me when I'm trying to figure out how something works. If you get stuck there are regulars there who have completed all of the exercises.

2

I pronounce it "pring", and the cryptographically-secure PRNG (CSPRNG) as "see-es-pring". I don't know if this is wide spread as I've never had a verbal conversation about PRNGs with someone.

3

Since the tempering step is meant to "diffuse" the state bits into the final output and are there to improve the statistical properties of the random numbers

4

Remember, we set \(b:=\) 0xFFFFFFFF, so the bitwise and step just "copies" the bits from \(x>>a\)