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Olm: A Crytographic Ratchet
===========================

An implementation of the cryptographic ratchet described by
https://github.com/trevp/axolotl/wiki.


The Olm Algorithm
-----------------

.. figure:: Axolotl.svg


Initial setup
~~~~~~~~~~~~~

The setup takes four Curve25519 inputs: Identity keys for Alice and Bob,
:math:`I_A` and :math:`I_B`, and emphemeral keys for Alice and Bob,
:math:`E_A` and :math:`E_B`. A shared secret, :math:`S`, is generated using
Triple Diffie-Hellman. The initial 256 bit root key, :math:`R_0`, and 256 bit
chain key, :math:`C_{0,0}`, are derived from the shared secret using an
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HMAC-based Key Derivation Function (HKDF) with default salt.
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.. math::
    \begin{align}
        S&=ECDH\left(I_A,\,E_B\right)\;\parallel\;ECDH\left(E_A,\,I_B\right)\;
            \parallel\;ECDH\left(E_A,\,E_B\right)\\
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        R_0\;\parallel\;C_{0,0}&=HKDF\left(S,\,\text{"OLM\_ROOT"}\right)
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    \end{align}

Advancing the root key
~~~~~~~~~~~~~~~~~~~~~~

Advancing a root key takes the previous root key, :math:`R_{i-1}`, and two
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Curve25519 inputs: the previous ratchet key, :math:`T_{i-1}`, and the current
ratchet key :math:`T_i`. The even ratchet keys are generated by Alice.
The odd ratchet keys are generated by Bob. A shared secret is generated
using Diffie-Hellman on the ratchet keys. The next root key, :math:`R_i`, and
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chain key, :math:`C_{i,0}`, are derived from the shared secret using an
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HMAC-based Key Derivation Function (HKDF) using :math:`R_{i-1}` as the salt.
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.. math::
    \begin{align}
        R_i\;\parallel\;C_{i,0}&=HKDF\left(
            ECDH\left(T_{i-1},\,T_i\right),\,
            R_{i-1},\,
            \text{"OLM\_RATCHET"}
        \right)
    \end{align}


Advancing the chain key
~~~~~~~~~~~~~~~~~~~~~~~

Advancing a root key takes the previous chain key, :math:`C_{i,j-i}`. The next
chain key, :math:`C_{i,j}`, is the HMAC of ``"\x02"`` using the previous chain
key as the key.

.. math::
     \begin{align}
        C_{i,j}&=HMAC\left(C_{i,j-1},\,\text{"\textbackslash x02"}\right)
    \end{align}

Creating a message key
~~~~~~~~~~~~~~~~~~~~~~

Creating a message key takes the current chain key, :math:`C_{i,j}`. The
message key, :math:`M_{i,j}`, is the HMAC of ``"\x01"`` using the current
chain key as the key. The message keys where :math:`i` is even are used by
Alice to encrypt messages. The message keys where :math:`i` is odd are used
by Bob to encrypt messages.

.. math::
    \begin{align}
        M_{i,j}&=HMAC\left(C_{i,j},\,\text{"\textbackslash x01"}\right)
    \end{align}


The Olm Protocol
----------------

Creating an outbound session
~~~~~~~~~~~~~~~~~~~~~~~~~~~~

Bob publishes his identity key, :math:`I_B`, and some single-use one-time
keys :math:`E_B`.

Alice downloads Bob's identity key, :math:`I_B`, and a one-time key,
:math:`E_B`. Alice takes her identity key, :math:`I_A`, and generates a new
single-use key, :math:`E_A`. Alice computes a root key, :math:`R_0`, and a
chain key :math:`C_{0,0}`. Alice generates a new ratchet key :math:`T_0`.

Sending the first pre-key messages
~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~

Alice computes a message key, :math:`M_{0,j}`, using the current chain key,
:math:`C_{0,j}`. Alice replaces the current chain key with :math:`C_{0,j+1}`.
Alice encrypts her plain-text with the message key, :math:`M_{0,j}`, using an
authenticated encryption scheme to get a cipher-text, :math:`X_{0,j}`. Alice
sends her identity key, :math:`I_A`, her single-use key, :math:`E_A`, Bob's
single-use key, :math:`E_B`, the current chain index, :math:`j`, her ratchet
key, :math:`T_0`, and the cipher-text, :math:`X_{0,j}`, to Bob.

Alice will continue to send pre-key messages until she receives a message from
Bob.

Creating an inbound session from a pre-key message
~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~

Bob receives a pre-key message with Alice's identity key, :math:`I_A`,
Alice's single-use key, :math:`E_A`, the public part of his single-use key,
:math:`E_B`, the current chain index, :math:`j`, Alice's ratchet key,
:math:`T_0`, and the cipher-text, :math:`X_{0,j}`. Bob looks up the private
part of the single-use key, :math:`E_B`. Bob computes the root key :math:`R_0`,
and the chain key :math:`C_{0,0}`. Bob then advances the chain key to compute
the chain key used by the message, :math:`C_{0,j}`. Bob then creates the
message key, :math:`M_{0,j}`, and attempts to decrypt the ciphertext,
:math:`X_{0,j}`. If the cipher-text's authentication is correct then Bob can
discard private part of his single-use one-time key, :math:`E_B`.

Sending messages
~~~~~~~~~~~~~~~~

To send a message the user checks if they have a sender chain key,
:math:`C_{i,j}`. Alice use chain keys where :math:`i` is even. Bob uses chain
keys where :math:`i` is odd. If the chain key doesn't exist then a new ratchet
key :math:`T_i` is generated and a the chain key, :math:`C_{i,0}`, is computed
using :math:`R_{i-1}`, :math:`T_{i-1}` and :math:`T_i`. A message key,
:math:`M_{i,j}` is computed from the current chain key, :math:`C_{i,j}`, and
the chain key is replaced with the next chain key, :math:`C_{i,j+1}`. The
plain-text is encrypted with :math:`M_{i,j}`, using an authenticated encryption
scheme to get a cipher-text, :math:`X_{i,j}`. Then user sends the current
chain index, :math:`j`, the ratchet key, :math:`T_i`, and the cipher-text,
:math:`X_{i,j}`, to the other user.

Receiving messages
~~~~~~~~~~~~~~~~~~

The user receives a message with the current chain index, :math:`j`, the
ratchet key, :math:`T_i`, and the cipher-text, :math:`X_{i,j}`, from the
other user. The user checks if they have a receiver chain with the correct
:math:`i` by comparing the ratchet key, :math:`T_i`. If the chain doesn't exist
then they compute a new receiver chain, :math:`C_{i,0}`, using :math:`R_{i-1}`,
:math:`T_{i-1}` and :math:`T_i`. If the :math:`j` of the message is less than
the current chain index on the receiver then the message may only be decrypted
if the receiver has stored a copy of the message key :math:`M_{i,j}`. Otherwise
the receiver computes the chain key, :math:`C_{i,j}`. The receiver computes the
message key, :math:`M_{i,j}`, from the chain key and attempts to decrypt the
cipher-text, :math:`X_{i,j}`.
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If the decryption succeeds the reciever updates the chain key for :math:`T_i`
with :math:`C_{i,j+1}` and stores the message keys that were skipped in the
process so that they can decode out of order messages. If the receiver created
a new receiver chain then they discard their current sender chain so that
they will create a new chain when they next send a message.