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For example AES-128 starting with a 128-bit message $m_0$ and static 128 key $k$


$AES128(m_0,k)\rightarrow c_0$
$c_0\rightarrow m_1$
$AES128(m_1,k)\rightarrow c_1$
$c_1\rightarrow m_2$
...
continue until $m_i$ is equal to any $m_j, j<i$


The period length would be $l = i-j$

Any theory about how big that $l$ will be?
Will it be equal for any possible $m_0$?
Is $l=j$ for every $m_0$?

(edit: in AES $j$ is always 0 because symmetric algorithm. Each cipher value has only one possible plain text)

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    $\begingroup$ Yes, there is a theory about this. By fixing the key randomly you selected a random permutation among the permutations of AES. Actually, you are asking about the distribution of the cycles of a permutation. See this answer of Cycles in SHA256 They i.e. Squeamish Ossifrage $\endgroup$ – kelalaka Oct 27 '19 at 13:42
  • $\begingroup$ @kelalaka sure about this? Sha256 is a hash algorithm. AES a symmetric block cipher. SHA256 can have many inverse results (or many values can give one SHA256 value). At AES only one inverse value. AES not a normal permutation. $\endgroup$ – J. Doe Oct 27 '19 at 23:13
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Since AES under any fixed key is a permutation, we necessarily have $j = 0$ and $i = l$—iterating a permutation enough times will always return you to the starting point.

From Harris 1960 (paywall-free), if we model AES as a uniform random permutation, every period length $l$ has equal probability $1/n$ (Eq. 5.2) for any particular starting point, where $n = 2^{128}$ is the size of the domain, so the expected cycle length is $\sum_{i=1}^n i/n = (n + 1)/2 \approx 2^{127}$.

(Any substantial deviation from this would imply an attack on AES.)

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  • $\begingroup$ So it is same as a real uniform random permutation? "if we model AES as a uniform random permutation" <-- Is it proven that this can be done. With this an assumption is made there is no kind of inner structure. $\endgroup$ – J. Doe Oct 27 '19 at 15:49
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    $\begingroup$ @J.Doe Any substantial deviation from this would imply an attack on AES. $\endgroup$ – Squeamish Ossifrage Oct 27 '19 at 15:52
  • $\begingroup$ Not sure why that is that case (the attack thing). But this might be the topic of a new question. $\endgroup$ – J. Doe Oct 27 '19 at 16:39
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    $\begingroup$ It would imply a PRP distinguisher: Given an oracle $\mathcal O$ (which may be either $\operatorname{AES}_k$ for uniform random $k$, or a uniform random permutation), pick an arbitrary input $x$, query the oracle for $\mathcal O(x)$, $\mathcal O(\mathcal O(x))$, $\dotsc$, $\mathcal O^q(x)$, and check for a duplicate (maybe use a constant-memory cycle-detection algorithm). If there's a duplicate substantially more often, or substantially less often, for AES than for a uniform random permutation, then that's a distinguishing attack on the central security conjecture of AES. $\endgroup$ – Squeamish Ossifrage Oct 27 '19 at 16:43
  • $\begingroup$ @SqueamishOssifrage so not an answer? But your answer makes more sense than fgrieu comment oO. $\endgroup$ – J. Doe Oct 27 '19 at 17:18

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