How (if at all) does a predictable random number generator get more secure after SHA-1ing its output?

Question

This article states that

Despite the fact that the Mersenne Twister is an extremely good pseudo-random number generator, it is not cryptographically secure by itself for a very simple reason. It is possible to determine all future states of the generator from the state the generator has at any given time, and either 624 32-bit outputs, or 19,937 one-bit outputs are sufficient to provide that state. Using a cryptographically-secure hash function, such as SHA-1, on the output of the Mersenne Twister has been recommended as one way of obtaining a keystream useful in cryptography.

But there are no references on why digesting the output would make it any more secure. And honestly, I don't see why this should be the case. The Mersenne Twister has a period of 2^19937-1, but I think my reasoning would also apply to any periodic PRNG, e.g. Linear Congruential Generators as well. Due to the properties of a secure one-way function h, one could think of h as an injective function (otherwise we could produce collisions), thus simply mapping the values from its domain into its range in a one-to-one manner.

With this thought in mind I would argue that the hashed values will produce exactly the same periodical behaviour as the original Mersenne Twister did. This means if you observe all values of one period and the values start to recur, then you are perfectly able to predict all future values.

I assume this to be related to the same principle that is applied in password-based encryption (PKCS#5) - because the domain of passwords does not provide enough entropy, simply hashing passwords doesn't add any additional entropy - that's why you need to salt passwords before you hash them. I think that exactly the same principle applies here.

One simple example that finally convinced me: Suppose you have a very bad PRNG that will always produce a "random number" of 1. Then even if SHA-1 would be a perfect one-way function, applying SHA-1 to the output will always yield the same value, thus making the output no less predictable than previously.

Still, I'd like to believe there is some truth to that article, so surely I must have overlooked something. Can you help me out? To a large part, I have left out the seed value from my arguments - maybe this is where the magic happens?

Answer 1

The state of the mersenne twister is defined by the previous n outputs, where n is the degree of recurrence (a constant). As such, if you give the attacker n outputs straight from a mersenne twister, they will immediately be able to predict all future values.

Passing the values through SHA-1 makes it more difficult, as now the attacker must try to reverse the RNG. However, for a 32-bit word size, this is unlikely to be a severe impediment to a determined attacker; they can build a rainbow table or use some other standard approach for reversing SHA-1s, and in the event of collisions, filter candidates by whether they produce the RNG stream observed. As such, the mersenne twister should not be used for cryptographically sensitive applications, SHA-1 masking or no. There are a number of standard CSPRNGs that may be used instead.