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Suppose having an arbitrary $GF(2^n)$ element $x$. Its distribution is unknown.

The task is to derive two $GF(2^n)$ elements $y$ and $z$, that have uniform distribution and are independent from each other.

Let $x$ be known to a possible adversary.

The obvious solutions are:

  1. Use some KDF, but it takes a lot of time to evaluate. This operation will be used often.
  2. $y = E_k(x), z = E_k(\overline{x})$ ($E$ is a block cipher), but knowing $y$ and $z$ for some $x$ we can easily find out $y' = z, z' = y$ for $x' = \overline{x}$.
  3. $y = E_k(x), z = E_{k'}(x)$, but this approach uses two potentially distinct PRP's and makes more difficult the rest of one proof, so I want to avoid using distinct PRP's.
  4. Let $x$ to be an element of $GF(2 ^ {2n})$ and put $y = E_k(x[0..n]), z = E_k(x[n..2n])$, but $x$ usually will be a small number, so the upper half will likely be zero.
  5. Saw this question, but solutions are to use KDF's and hashes, that are too expensive in terms of performance.

I have several ideas, but I'm not sure if such $y$ and $z$ are independent.

  1. Let $y = E_k(x), z = E_k(y \oplus k')$, where $k'$ is uniform, random and independent.
  2. Let $y = E_k(x), z = E_k(x \oplus k')$, where $k'$ meets the same condition as for the previous option. This solution, however, has the following flaw: $y = z \implies k' = 0$. It potentially affects some practical security, if an adversary can intercept these values. In theory this adversary's opportunity is omitted (he or she interacts with a cryptosystem as a black-box), but if 1-st or 3-rd case produces independent values I'd prefer one of them.
  3. Let $y = E_k(x), z = y \oplus k'$, where $k'$ is again uniformly random and independent.

The question is: are $y$ and $z$ independent from each other in such cases. If not, is there any "lightweight" method to derive such values.

Let $k, k'$ to be master-keys and $y, z$ to be some concrete keys I want to derive from master keys and that should differ from each other respectively for different $x$'s. Also they should be uniformly random and independent from each other.

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If $k$ is uniform and independent then for fixed $x,$ $\{E_k(x): k \in GF(2)^n\}$ (assume keylength=blocklength) is in general a PRF not a PRP as $k$ ranges over the keyspace. Therefore it has the random balls incident in bins distribution and for two different $k\neq k'$ $E_k(x)$ may equal $E_{k'}(x)$ with probability approximately $e^{-1}$ (derangements). Therefore option 2 may have $y$ and $z$ be the same even if $k'\neq 0.$

You can use the PRP property directly if you treat $x$ as the key [fixed, with nonuniform distribution but that won't matter] for example you can define $$ y=E_x(k),\quad z=E_x(k'), $$ and now these are two "one-point" samples from the same PRP that you are using and are uniformly distributed and independent.

I wonder if this or something like this would meet your requirements. If $x$ needs to be hidden, I suppose you could do $x_0=H(x)$ for some hash function and use $E_{x_0}(\cdot)$ instead.

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  • $\begingroup$ That's what I was looking for :) I forgot, that key space actually differs from message space in the second option (ah those sleepless nights spent on studying cryptography). However, the idea to use $x$ as a key looks like a better solution for my purposes $\endgroup$ Commented Jul 24, 2022 at 20:23

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