Do the limitations of PRNGs apply to CSPRNGs when shuffling a list?

Question

Suppose, as an example, we want to use the Fisher-Yates algorithm along with a pseudorandom number generator to shuffle a deck of cards. We know that there are limitations with respect to the internal state of the PRNG and its ability to reach the distinct permutations of the deck. If, for example, the PRNG can only be seeded with $2^{32}$ bits, then it cannot possibly produce all the $52!$ distinct permutations. In addition, even if we could meet the 226 bits needed for a 52-card deck, the distribution may not be uniform.

Now, suppose we switch and swap the PRNG with a CSPRNG. How do these limitations apply? In practice, they're seeded and reseeded frequently, depending on implementation, but this is generally done behind the scenes. Since CSPRNGs are PRNGs, my initial thoughts would lead me to believe that they will suffer the same limitations. Yet, if we used a PRNG with the same seed, but continuously draw numbers from it, we could reach additional permutations.

Over yonder in response to this question, one answer suggests that a cryptographically strong source of unbiased numbers will reach the desired permutations. Would you agree?

Finally, I used a deck of cards as an example. I'm curious about list sizes with respect to this question. I know that with PRNGs, the seed bits must increase to accommodate larger lists.

Answer 1

If, for example, the PRNG can only be seeded with $2^{32}$ bits, then it cannot possibly produce all the $52!$ distinct permutations. In addition, even if we could meet the 226 bits needed for a 52-card deck, the distribution may not be uniform.

This is true, but a good PRNG (even non-cryptographic) will give you a random sample from a uniform distribution. That means that until you get close to the square root of the seed size, your sampling will still look random. With a 32-bit seed you can easily go over that limit, but with larger seeds you will not.

Yet, if we used a PRNG with the same seed, but continuously draw numbers from it, we could reach additional permutations.

Yes, but even then the state size will limit which outputs are possible.

Now, suppose we switch and swap the PRNG with a CSPRNG. How do these limitations apply? In practice, they're seeded and reseeded frequently, depending on implementation, but this is generally done behind the scenes.

If you take a sequence from a CSPRNG that is longer than the state size, without reseeding in between, it cannot produce every possible output value. However, it is guaranteed to be indistinguishable from random.

For example, suppose you have a CSPRNG based on AES CTR. A single AES key will produce $2^{128}$ different 128-bit outputs, but as long as you see (significantly) fewer than $2^{64}$ you do not notice this. As long as the RNG is reseeded often enough and you do not use it enough that both keys and inputs would repeat (impossible in practice), you are fine.

So yes, there are $52!$ different ways to shuffle cards, but since there is no chance that you will look at them all, it is sufficient to generate deck orders from a set of e.g. $2^{128}$ that are themselves a random selection of all possible ones.