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Olm: A Cryptographic Ratchet
============================
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An implementation of the cryptographic ratchet described by
https://github.com/trevp/axolotl/wiki.

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Notation
--------

This document uses :math:`\parallel` to represent string concatenation. When
:math:`\parallel` appears on the right hand side of an :math:`=` it means that
the inputs are concatenated. When :math:`\parallel` appears on the left hand
side of an :math:`=` it means that the output is split.

When this document uses :math:`ECDH\left(K_A,\,K_B\right)` it means that each
party computes a Diffie-Hellman agreement using their private key and the
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remote party's public key.
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So party :math:`A` computes :math:`ECDH\left(K_B_public,\,K_A_private\right)`
and party :math:`B` computes :math:`ECDH\left(K_A_public,\,K_B_private\right)`
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The Olm Algorithm
-----------------

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

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The setup takes four Curve25519_ inputs: Identity keys for Alice and Bob,
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:math:`I_A` and :math:`I_B`, and ephemeral keys for Alice and Bob,
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:math:`E_A` and :math:`E_B`. A shared secret, :math:`S`, is generated using
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`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
HMAC-based Key Derivation Function using SHA-256_ as the hash function
(HKDF-SHA-256_) with default salt and ``"OLM_ROOT"`` as the info.
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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
HKDF-SHA-256_ using :math:`R_{i-1}` as the salt and ``"OLM_RATCHET"`` as the
info.
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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
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chain key, :math:`C_{i,j}`, is the HMAC-SHA-256_ of ``"\x02"`` using the
previous chain key as the key.
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.. 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
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message key, :math:`M_{i,j}`, is the HMAC-SHA-256_ 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
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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
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authenticated encryption scheme (see below) 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.
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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
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message key, :math:`M_{0,j}`, and attempts to decrypt the cipher-text,
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:math:`X_{0,j}`. If the cipher-text's authentication is correct then Bob can
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discard the private part of his single-use one-time key, :math:`E_B`.
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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
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scheme (see below) 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.
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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 receiver updates the chain key for :math:`T_i`
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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.
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The Olm Message Format
----------------------

Normal Messages
~~~~~~~~~~~~~~~

Olm messages start with a one byte version followed by a variable length
payload followed by a fixed length message authentication code.

.. code::

   +--------------+------------------------------------+-----------+
   | Version Byte | Payload Bytes                      | MAC Bytes |
   +--------------+------------------------------------+-----------+

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The version byte is ``"\x01"``.

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The payload consists of key-value pairs where the keys are integers and the
values are integers and strings. The keys are encoded as a variable length
integer tag where the 3 lowest bits indicates the type of the value:
0 for integers, 2 for strings. If the value is an integer then the tag is
followed by the value encoded as a variable length integer. If the value is
a string then the tag is followed by the length of the string encoded as
a variable length integer followed by the string itself.

Olm uses a variable length encoding for integers. Each integer is encoded as a
sequence of bytes with the high bit set followed by a byte with the high bit
clear. The seven low bits of each byte store the bits of the integer. The least
significant bits are stored in the first byte.

=========== ===== ======== ================================================
    Name     Tag    Type                     Meaning
=========== ===== ======== ================================================
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Ratchet-Key  0x0A String   The public part of the ratchet key, :math:`T_{i}`,
                           of the message
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Chain-Index  0x10 Integer  The chain index, :math:`j`, of the message
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Cipher-Text  0x22 String   The cipher-text, :math:`X_{i,j}`, of the message
=========== ===== ======== ================================================

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The length of the MAC is determined by the authenticated encryption algorithm
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being used. The MAC protects all of the bytes preceding the MAC.
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Pre-Key Messages
~~~~~~~~~~~~~~~~

Olm pre-key messages start with a one byte version followed by a variable
length payload.

.. code::

   +--------------+------------------------------------+
   | Version Byte | Payload Bytes                      |
   +--------------+------------------------------------+

The version byte is ``"\x01"``.

The payload uses the same key-value format as for normal messages.

============ ===== ======== ================================================
    Name      Tag    Type                     Meaning
============ ===== ======== ================================================
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One-Time-Key  0x0A String   The public part of Bob's single-use key,
                            :math:`E_b`.
Base-Key      0x12 String   The public part of Alice's single-use key,
                            :math:`E_a`.
Identity-Key  0x1A String   The public part of Alice's identity key,
                            :math:`I_a`.
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Message       0x22 String   An embedded Olm message with its own version and
                            MAC.
============ ===== ======== ================================================

Olm Authenticated Encryption
----------------------------
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Version 1
~~~~~~~~~
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Version 1 of Olm uses AES-256_ in CBC_ mode with `PCKS#7`_ padding for
encryption and HMAC-SHA-256_ for authentication. The 256 bit AES key, 256 bit
HMAC key, and 128 bit AES IV are derived from the message key using
HKDF-SHA-256_ using the default salt and an info of ``"OLM_KEYS"``.
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First the plain-text is encrypted to get the cipher-text, :math:`X_{i,j}`.
Then the entire message, both the headers and cipher-text, are HMAC'd and the
MAC is appended to the message.
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.. math::

    \begin{align}
    AES\_KEY_{i,j}\;\parallel\;HMAC\_KEY_{i,j}\;\parallel\;AES\_IV_{i,j}
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        &= HKDF\left(M_{i,j},\,\text{"OLM\_KEYS"}\right) \\
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    \end{align}
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.. _`Curve25519`: http://cr.yp.to/ecdh.html
.. _`Triple Diffie-Hellman`: https://whispersystems.org/blog/simplifying-otr-deniability/
.. _`HKDF-SHA-256`: https://tools.ietf.org/html/rfc5869
.. _`HMAC-SHA-256`: https://tools.ietf.org/html/rfc2104
.. _`SHA-256`: https://tools.ietf.org/html/rfc6234
.. _`AES-256`: http://csrc.nist.gov/publications/fips/fips197/fips-197.pdf
.. _`CBC`: http://csrc.nist.gov/publications/nistpubs/800-38a/sp800-38a.pdf
.. _`PCKS#7`: https://tools.ietf.org/html/rfc2315