Exit Zen
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../../principles/security.rst:951
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Alice chooses a secret integer : :math:`a=8` and sends :math:`A= g^{a} \mod p= 5^{8} \mod 23=16` to Bob
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../../principles/security.rst:953
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Bob chooses a secret integer : :math:`b=13` and sends :math:`B= g^{b} \mod p=5^{13} \mod 23=21` to Alice
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../../principles/security.rst:955
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Alice computes :math:`S_{A}=B^{a} \mod p= 21^{8} \mod 23=3`
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../../principles/security.rst:956
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Bob computes :math:`S_{B}=A^{b} \mod p= 16^{13} \mod 23=3`
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../../principles/security.rst:958
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Alice and Bob have agreed on the secret information :math:`3` without having sent it explicitly through the network. If the integers used are large enough and have good properties, then even Eve who can capture all the messages sent by Alice and Bob cannot recover the secret key that they have exchanged. There is no formal proof of the security of the algorithm, but mathematicians have tried to solve similar problems with integers during centuries without finding an efficient algorithm. As long as the integers that are used are random and large enough, the only possible attack for Eve is to test all possible integers that could have been chosen by Alice and Bob. This is computationally very expensive. This algorithm is widely used in security protocols to agree on a secret key.
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../../principles/security.rst:970
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Unfortunately, the Diffie-Hellman key exchange alone cannot cope with man-in-the middle attacks. Consider Mallory who sits in the middle between Alice and Bob and can easily capture and modify their messages. The modulus and the base are public. They are thus known by Mallory as well. He could then operate as follows :
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../../principles/security.rst:976
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Alice chooses a secret integer and sends :math:`A= g^{a} \mod p` to Mallory
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../../principles/security.rst:977
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Mallory generates a secret integer, :math:`m` and sends :math:`M=g^{m} \mod p` to Bob
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../../principles/security.rst:978
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Bob chooses a secret integer and sends :math:`B=g^{b} \mod p` to Mallory
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../../principles/security.rst:979
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Mallory computes :math:`S_{A}=A^{m} \mod p` and :math:`S_{B}=B^{m} \mod p`
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../../principles/security.rst:980
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Alice computes :math:`S_{A}=M^{a} \mod p` and uses this key to communicate with Mallory (acting as Bob)
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../../principles/security.rst:981
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Bob computes :math:`S_{B}=M^{b} \mod p` and uses this key to communicate with Mallory (acting as Alice)
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../../principles/security.rst:983
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When Alice sends a message, she encrypts it with :math:`S_{A}`. Mallory decrypts it with :math:`S_{A}` and encrypts the plaintext with :math:`S_{B}`. When Bob receives the message, he can decrypt it by using :math:`S_{B}`.
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../../principles/security.rst:988
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To safely use the Diffie-Hellman key exchange, Alice and Bob must use an `authenticated` exchange. Some of the information sent by Alice or Bob must be signed with a public key known by the other user. In practice, it is often important for Alice to authenticate Bob. If Bob has a certificated signed by Ted, the authenticated key exchange could be organized as follows.
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../../principles/security.rst:995
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Alice chooses a secret integer : :math:`a` and sends :math:`A= g^{a} \mod p` to Bob
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../../principles/security.rst:997
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Bob chooses a secret integer : :math:`b`, computes :math:`B= g^{b} \mod p` and sends :math:`Cert(Bob,Bob_{pub},Ted), E_p(Bob_{priv},B)` to Alice
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../../principles/security.rst:1000
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Alice checks the signature (with :math:`Bob_{pub}`) and the certificate and computes :math:`S_{A}=B^{a} \mod p`
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../../principles/security.rst:1002
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Bob computes :math:`S_{B}=A^{b} \mod p`
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../../principles/security.rst:1006
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This prevents the attack mentioned above since Mallory cannot create a fake certificate and cannot sign a value by using Bob's private key. Given the risk of man-in-the-middle attacks, the Diffie-Hellman key exchange mechanism should never be used without authentication.
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../../principles/security.rst:1016
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The wikipedia page on passwords provides many of these references : https://en.wikipedia.org/wiki/Password_strength
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