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We have cryptography based on hard problems from number theory, like DDH.

When we speak about symmetric cryptography with AES, a mode of encryption (CBC, ...), what is the type of problem ? What is the term used ?

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You could say breaking symmetric encryption is a combinatorial problem. – otus Jul 1 '14 at 10:12
The underlying problem is called AES. Typical symmetrical schemes don't reduce to an elegant mathematical problem. They rely on the complexity emerging out of the interaction of many simple operations. In a way RSA/DH etc. are structured whereas AES is unstructured. – CodesInChaos Jul 1 '14 at 10:28
@CodesInChaos That looks interesting. Thank you. – Dingo13 Jul 1 '14 at 14:28
@CodesInChaos The usual comment :) – Maarten Bodewes Jul 3 '14 at 13:44

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