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What would happen if the random bytes provided by the client and the server in the first two messages would be reused by one peer/by both peers?

I think it would highly reduce the provided security, like for example if you use a nonce multiple times, but i am not quite sure.

Does using the same random bytes multiple times lead to a less secure TLS connection. If so, why and what would happen?

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If the server uses the same server_random in the 'server hello' message, then this could enable an attacker to launch a replay attack. See SSL replay attack when client/server random is missing for more info.

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