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If Alice and bob have a secure channel for key-exchange and mallory don't man-in-the-middle attack them but in the future eavesdrop connection and see the key exchanged, can mallory break it like RSA with a quantum computer?

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While the title (after moving is at the beginning) makes a good question, the body has problems: it is assumed "a secure channel for key-exchange", when I think it should be assumed "an insecure channel over which it is performed Diffie-Hellman key exchange"; also "in the future eavesdrop connection and see the key exchanged" likely should be "eavesdrops the messages in a key exchange, and saves that for attack in the future". Ah and only Alice gets caps :-) –  fgrieu Dec 7 '12 at 11:28

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A quantum computer solves the discrete logarithm problem for both finite fields and elliptic curves. Being able to efficiently calculate discrete logarithms implies being able to break Diffie-Hellman, so Diffie-Hellman on either of them is not secure against an adversary who owns a large quantum computer.

There might be other groups in which DL problem is hard even against a quantum computer, but I've never heard of such a group.

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what is best alternative for public use today –  mary Dec 7 '12 at 11:19
    
@mary Leave that to your other question, and don't ask new questions in comments. –  CodesInChaos Dec 7 '12 at 11:32

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