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\item Alice uses a ring-LWE key exchange protocol to establish secure connections  \item Alice uses the same key share to communicate with both Bob and the attacker Eve  \item Eve's goal is to recover the value $s$ the corresponds to Alice's public key share (and thus be able to decrypt Alice's traffic)  \item Eve can perform the ring-LWE exchange protocol with Alice multiple times (with Eve providing a fresh  key share) share each time)  \item Each time after Alice and Eve has performed the key exchange protocol, Alice will derive her shared secret; Eve when then be able to generate one guess to that shared secret, and Alice will indicate whether that guess matches what she has or not.  \end{itemize}   This last step can be implemented by continuing on with the protocol that used the key establishment; Alice and Eve may derive keys based on the shared secret. What Eve can do is generate her keys based on her guess; if Alice is able to decrypt (and respond) based on those keys, then (with high probability) her guess was correct; if Alice rejects the exchange, then Eve's guess was not correct.  \subsection{Phase 1 of the attack}  Write something here  ...