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The TLSSLed tool found that the host that I use has SSL connection that accepts 8 different TLSv1 and SSLv3 ciphers that are weak by their length (40 bits).

How can an attacker exploit this fact?

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There is a few "popular" attacks on SSL. They were discussed i.e. in How SSL works, section "How to crack SSL". In addition, you can read more about BREACH attack, which was discovered in 2013.

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