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This follows on from How do I produce a stream of secure random numbers from AES-Counter mode?

Consider then the generator:-

$$ \operatorname{AES_{k1}}(n) \oplus \operatorname{AES_{k2}}(n) $$

where $ k1 \ne k2 $, and $n$ is the single counter variable. And we take care to ensure that there is no relationship between $k1$ and $k2$. If this really does create a pseudo random function, will the birthday boundary for AES-CTR output be overcome?

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    $\begingroup$ Related and helpful: crypto.stackexchange.com/q/5447/32035 $\endgroup$ – hardyrama Jun 24 at 13:53
  • $\begingroup$ @hardyrama Indeed very. Apologies for the repetition then. I haven't used the PRF acronym so the related ones didn't pop up when I authored it... $\endgroup$ – Paul Uszak Jun 24 at 20:41
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Yes. The number of $n$-bit blocks you can encrypt with this construction is proportional to $2^n$ instead of just $2^{n/2}$. So for AES this shouldn't be a problem because you can't process anything near $2^{128}$ blocks.

A secure stream cipher can be constructed from a PRF using something similar to AES-CTR.

AES is more accurately described as a PRP than a PRF, but this distinction is unimportant if the number of queries an attacker is allowed to make under a given key is much much less than $2^{64}$.

(See PRF switching lemma. If the number of queries the attacker can make is much less than $2^{n/2}$, then it is safe to substitute an $n$-bit PRP for a PRF. This is where AES-CTR's limit comes from.)

However it has been proven that the XOR of two PRPs is a secure PRF instead for up to $\mathcal{O}(2^n)$ queries. 1 2

Readers should note that this isn't the same as saying that you can use up to precisely $2^n$ blocks. For this reason, I strongly discourage using a block cipher algorithm with smaller blocks (like DES, Simon, or Speck) in this kind of construct. (Or any other mode, I suppose.)

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    $\begingroup$ Are you sure that 2 instance of AES-CTR xor'd together gives $2^n$ blocks for the bound? Unless I am misreading something, that appears to directly contradict with this answer, which claims security up to $2^{2b/3}$ queries for the construction $\endgroup$ – Ella Rose Jun 24 at 16:54
  • $\begingroup$ @EllaRose One of the papers I linked to cites Lucks for prior work. (As well as another paper that improved on Lucks' bounds.) Each of these sources, I think, give a minimum number of queries an attacker needs to make. Lucks provided a bound on how few q the best possible attack must require. Subsequent work provided tighter bounds. None were numbers from actual discovered attacks. "At least this difficult" in other words. $\endgroup$ – Future Security Jun 24 at 17:40

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