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Consider a stream cipher $E(k)$ which takes a key $k$ to produce a pseudo random keystream.

It should have the property that you can easily generate a set of at least 3 different keys $\{k_m, k_0, k_1, \cdots\}$ such that the key stream generated by the master key $E(k_m)$ is the same as XOR-ing all key streams of the other keys $E(k_0) \oplus E(k_1) \oplus \cdots$

Is this property a security risk by itself, or could such a cipher be used to speed up an encryption scheme with nested encryption, like onion routing?

If it could be secure, are there already any ciphers with such a property?

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  • $\begingroup$ What is your actual problem? $\endgroup$
    – kelalaka
    Jul 28, 2019 at 18:54
  • $\begingroup$ I'm just curious $\endgroup$
    – Aemyl
    Jul 28, 2019 at 18:57
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    $\begingroup$ If you consider symmetric encryption schemes instead of just stream ciphers, you're looking for "key-homomorphic encryption". There has been some work on lattice-based fully key homomorphic encryption schemes (see here). $\endgroup$
    – Mark
    Jul 28, 2019 at 23:21
  • $\begingroup$ @Mark thank you, I'd accept your comment as an answer $\endgroup$
    – Aemyl
    Jul 29, 2019 at 23:09

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If you consider symmetric encryption schemes instead of just stream ciphers, you're looking for "key-homomorphic encryption". There has been some work on lattice-based fully key homomorphic encryption schemes (see here).

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