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../../principles/security.rst:600
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Public key cryptography provides another possibility to allow Alice to authenticate herself on Bob's computer. Assume again that Alice and Bob know each other from previous encounters. Alice knows Bob's public key (:math:`Bob_{pub}`) and Bob also knows Alice's key (:math:`Alice_{pub}`). To authenticate herself, Alice could send her user identifier. Bob would reply with a random number encrypted with Alice's public key : :math:`E_p(Alice_{pub},R)`. Alice can decrypt this message to recover `R` and sends :math:`E_p(Bob_{pub},R)`. Bob decrypts the nonce and confirms that Alice knows :math:`Alice_{priv}`. If an eavesdropper captures the messages exchanged, he cannot recover the value `R` which could be used as a key to encrypt the information with a secret key algorithm. This is illustrated in the time sequence diagram below.
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../../principles/security.rst:634
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A drawback of this approach is that Bob is forced to perform two public key computations : one encryption to send the random nonce to Alice and one decryption to recover the nonce encrypted by Alice. If these computations are costly from a CPU viewpoint, this creates a risk of Denial of Service Attacks were attackers could try to access Bob's computer and force it to perform such costly computations. Bob is more at risk than Alice in this situation and he should not perform complex operations before being sure that he is talking with Alice. An alternative is shown in the time sequence diagram below.
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../../principles/security.rst:664
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Here, Bob simply sends a random nonce to Alice and verifies her signature. Since the random nonce and the signature could be captured by an eavesdropper, they cannot be used as a secret key to encrypt further data. However Bob could propose a secret key and send it encrypted with Alice's public key in response to the signed nonce that he received.
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../../principles/security.rst:671
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The solution described above works provided that Bob and Alice know their respective public keys before communicating. Otherwise, the protocol is not secure against man-in-the-middle attackers. Consider Mallory sitting in the middle between Alice and Bob and assume that neither Alice nor Bob knows the other's public key.
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../../principles/security.rst:705
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In the above example, Alice sends her public key, (:math:`Alice_{pub}`), in her first message together with her identity. Mallory intercepts the message and replaces Alice's key with his own key, (:math:`Mallory_{pub}`). Bob replies with a nonce, `R`. Alice then signs the random nonce to prove that she knows :math:`Alice_{priv}`. Mallory discards the information and instead computes :math:`E_p(Mallory_{priv},R)`. Bob now thinks that he is discussing with Alice while Mallory sits in the middle.
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../../principles/security.rst:714
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There are situations where symmetric authentication is required. In this case, each user must perform some computation with his/her private key. A possible exchange is the following. Alice sends her certificate to Bob. Bob replies with a nonce, :math:`R1`, and provides his certificate. Alice encrypts :math:`R1` with her private key and generates a nonce, :math:`R2`. Bob verifies Alice's computation and encrypts :math:`R2` with his private key. Alice verifies the computation and both have been authenticated.
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../../principles/security.rst:747
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The protocol described above works, but it takes a long time for Bob to authenticate Alice and for Alice to authenticate Bob. A faster authentication could be the following.
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../../principles/security.rst:771
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Alice sends her random nonce, :math:`R2`. Bob signs :math:`R2` and sends his nonce : :math:`R1`. Alice signs :math:`R1` and both are authenticated.
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../../principles/security.rst:775
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Now consider that Mallory wants to be authenticated as Alice. The above protocol has a subtle flaw that could be exploited by Mallory. This flaw can be exploited if Alice and Bob can act as both client and server. Knowing this, Mallory could operate as follows. Mallory starts an authentication with Bob faking himself as Alice. He sends a first message to Bob including Alice's identity.
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../../principles/security.rst:799
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In this exchange, Bob authenticates himself by signing the :math:`RA` nonce that was sent by Mallory. Now, to authenticate as Alice, Mallory needs to compute the signature of nonce :math:`RB` with Alice's private key. Mallory does not know Alice's key, but he could exploit the protocol to force Alice to perform the required computation. For this, Mallory can start an authentication to Alice as shown below.
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../../principles/security.rst:822
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In this example, Mallory has forced Alice to compute :math:`E_p(Alice_{priv},RB)` which is the information required to finalize the first exchange and be authenticated as Alice. This illustrates a common problem with authentication schemes when the same information can be used for different purposes. The problem comes from the fact that Alice agrees to compute her signature on a nonce chosen by Bob (and relayed by Mallory). This problem occurs if the nonce is a simple integer without any structure. If the nonce includes some structure such as some information about Alice and Bob's identities or even a single bit indicating whether the nonce was chosen by a user acting as a client (i.e. starting the authentication) or as a server, then the protocol is not vulnerable anymore.
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../../principles/security.rst:837
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To cope with some of the above mentioned problems, public-key cryptography is usually combined with certificates. A `certificate` is a data structure that includes a signature from a trusted third party. A simple explanation of the utilization of certificates is to consider that Alice and Bob both know Ted. Ted is trusted by these two users and both have stored Ted's public key : :math:`Ted_{pub}`. Since they both know Ted's key, he can issue certificates. A certificate is mainly a cryptographic link between the identity of a user and his/her public key. Such a certificate can be computed in different ways. A simple solution is for Ted to generate a file that contains the following information for each certified user :
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../../principles/security.rst:849
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his/her identity
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../../principles/security.rst:850
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his/her public key
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../../principles/security.rst:851
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a hash of the entire file signed with Ted's private key
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../../principles/security.rst:853
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Then, knowing Ted's public key, anyone can verify the validity of a certificate. When a user sends his/her public key, he/she must also attach the certificate to prove the link between his/her identity and the public key. In practice, certificates are more complex than this. Certificates will often be used to authenticate the server and sometimes to authenticate the client.
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../../principles/security.rst:860
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A possible protocol could then be the following. Alice sends :math:`Cert(Alice_{pub},Ted)`. Bob replies with a random nonce.
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../../principles/security.rst:885
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Until now, we have only discussed the authentication problem. This is an important but not sufficient step to have a secure communication between two users through an insecure network. To securely exchange information, Alice and Bob need to both :
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../../principles/security.rst:890
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mutually authenticate each other
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../../principles/security.rst:891
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agree on a way to encrypt the messages that they will exchange
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