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the server generates the random number :math:`b`. It then computes :math:`B=g^{b} \mod p`, :math:`K=B^{a} \mod p` and signs with its private key :math:`hash(V_{Client} || V_{Server} || KEX\_INIT_{Client} || KEX\_INIT_{Server} || Server_{pub} || A || B || K )` where :math:`V_{Server}` (resp. :math:`V_{Client}`) is the initial messages sent by the client (resp. server), :math:`KEX\_INIT_{Client}` (resp. :math:`KEX\_INIT_{Server}`) is the key exchange message sent by the client (resp. server) and :math:`A`, :math:`B` and :math:`K` are the messages of the Diffie Hellman key exchange
the client can recompute :math:`K=A^{b} \mod p` and verify the signature provided by the server
This is a slightly modified authenticated Diffie Hellman key exchange with two interesting points. The first point is that when the server authenticates the key exchange it does not provide a certificate. This is because ``ssh`` assumes that the client will store inside its cache the public key of the servers that it uses on a regular basis. This assumption is valid for a protocol like ``ssh`` because users typically use it to interact with a small number of servers, typically a few or a few tens. Storing this information does not require a lot of storage. In practice, most ``ssh`` clients will accept to connect to remote servers without knowing their public key before the connection. In this case, the client issues a warning to the user who can decide to accept or reject the key. This warning can be associated with a fingerprint of the key, either as a sequence of letters or as an ASCII art which can be posted on the web or elsewhere [#fdnsssh]_ by the system administrator of the server. If a client connects to a server whose public key does not match the stored one, a stronger warning is issued because this could indicate a man-in-the-middle attack or that the remote server has been compromised. It can also indicate that the server has been upgraded and that a new key has been generated during this upgrade.
The second point is that the server authenticates not only the result of the Diffie Hellman exchange but also a hash of all the information sent and received during the exchange. This is important to prevent `downgrade attacks`. A `downgrade attack` is an attack where an active attacker modifies the messages sent by the communicating hosts (typically the client) to request the utilization of weaker encryption algorithms. Consider a client that supports two encryption schemes. The preferred one uses 128 bits secret keys and the second one is an old encryption scheme that uses 48 bits keys. This second algorithm is kept for backward compatibility with older implementations. If an attacker can remove the preferred algorithm from the list of encryption algorithms supported by the client, he can force the server to use a weaker encryption scheme that will be easier to break. Thanks to the hash that covers all the messages exchanged by the server, the downgrade attack cannot occur against ``ssh``. Algorithm agility is a key requirement for security protocols that need to evolve when encryption algorithms are broken by researchers. This agility cannot be used without care and signing a hash of all the messages exchanged is a technique that is frequently used to prevent downgrade attacks.
Single use keys
Thanks to the Diffie Hellman key exchange, the client and the servers share key :math:`K`. A naive implementation would probably directly use this key for all the cryptographic algorithms that have been negotiated for this session. Like most security protocols, ``ssh`` does not directly use key :math:`K`. Instead, it uses the negotiated hash function with different parameters [#fsshkeys]_ to allow the client and the servers to compute six keys from :math:`K` :
a key used by the client (resp. server) to encrypt the data that it sends
a key used by the client (resp. server) to authenticate the data that it sends
a key used by the client (resp. server) to initialize the negotiated encryption scheme (if required by this scheme)
It is common practice among designers of security protocols to never use the same key for different purposes. For example, allowing the client and the server to use the same key to encrypt data could enable an attacker to launch a replay attack by sending to the client data that it has itself encrypted.
At this point, all the messages sent over the TCP connection will be encrypted with the negotiated keys. The ``ssh`` protocol uses messages that are encoded according to the Binary Packet Protocol defined in :rfc:`4253`. Each of these messages contains the following information :
``length`` : this is the length of the message in bytes, excluding the MAC and length fields
``padding length`` : this is the number of random bytes that have been added at the end of the message.
``payload`` : the data (after optional compression) passed by the user
``padding`` : random bytes added in each message (at least four) to ensure that the message length is a multiple of the block size used by the negotiated encryption algorithm
``MAC`` : this field is present if a Message Authentication Code has been negotiated for the session (in practice, using ``ssh`` without authentication is risky and this field should always be present). Note that to compute the MAC, an ``ssh`` implementation must maintain a message counter. This counter is incremented by one every time a message is sent and the MAC is computed with the negotiated authentication algorithm using the MAC key over the concatenation of the message counter and the cleartext message. The message counter is not transmitted, but the recipient can easily recover its value. The ``MAC`` is computed as :math:`mac = MAC(key, sequence\_number || unencrypted\_message)` where the key is the negotiated authentication key.
Authenticating messages with HMAC
`ssh` is one example of a protocol that uses Message Authentication Codes (MAC) to authenticates the messages that are sent. A naive implementation of such a MAC would be to simply use a hash function like SHA-1. However, such a construction would not be safe from a security viewpoint. Internet protocols usually rely on the HMAC construction defined in :rfc:`2104`. It works with any hash function (`H`) and a key (`K`). As an example, let us consider HMAC with the SHA-1 hash function. SHA-1 uses 20 bytes blocks and the block size will play an important role in the operation of HMAC. We first require the key to be as long as the block size. Since this key is the output of the key generation algorithm, this is one parameter of this algorithm.
HMAC uses two padding strings : `ipad` (resp. `opad`) which is a string containing 20 times byte ``0x36`` (resp. byte ``0x5C``). The HMAC is then computed as :math:`H[K \oplus opad, H(K \oplus ipad, data) ]` where :math:`\oplus` denotes the bitwise XOR operation. This computation has been shown to be stronger than the naive :math:`H(K,data)` against some types of cryptographic attacks.
Among the various features of the ``ssh`` protocol, it is interesting to mention how users are authenticated by the server. The ``ssh`` protocol supports the classical username/password authentication (but both the username and the password are transmitted over the secure encrypted channel). In addition, ``ssh`` supports two authentication mechanisms that rely on public keys. To use the first one, each user needs to generate his/her own public/private key pair and store the public key on the server. To be authenticated, the user needs to sign a message containing his/her public key by using his/her private key. The server can easily verify the validity of the signature since it already knows the user's public key. The second authentication scheme is designed for hosts that trust each other. Each host has a public/private key pair and stores the public keys of the other hosts that it trusts. This is typically used in environments such as university labs where each user could access any of the available computers. If Alice has logged on ``computer1`` and wants to execute a command on ``computer2``, she can create an ``ssh`` session on this computer and type (again) her password. With the host-based authentication scheme, ``computer1`` signs a message with its private key to confirm that it has already authenticated Alice. ``computer2`` would then accept Alice's session without asking her credentials.
The ``ssh`` protocol includes other features that are beyond the scope of this book. Additional details may be found in [BS2005]_.
Footnotes
For some of the algorithms, it is possible to negotiate the utilization of no algorithm. This happens frequently for the compression algorithm that is not always used. For this, both the client and the server must announce ``null`` in their ordered list of supported algorithms.
For example, :rfc:`4255` describes a DNS record that can be used to associate an ``ssh`` fingerprint to a DNS name.
The exact algorithms used for the computation of these keys are defined in :rfc:`4253`

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../../protocols/ssh.rst:231
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locale/pot/protocols/ssh.pot, string 32