# secure “related-key” stream cipher

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?

• What is your actual problem? – kelalaka Jul 28 '19 at 18:54
• I'm just curious – Aemyl Jul 28 '19 at 18:57
• 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). – Mark Jul 28 '19 at 23:21
• @Mark thank you, I'd accept your comment as an answer – Aemyl Jul 29 '19 at 23:09