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Why it is highly recommended to use a nonce once and only once in SSL/TLS? Same question with key streams in stream ciphers? What sort of attack(s) could take place if ever a nonce/key stream is used more than once? During the SSL/TLS handshake, is the client nonce/server nonce session related or connection related ?

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Well, in general, reusing a nonce enables replay attacks. This is a fundamental concept in authentication protocols.

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