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.. Copyright 2016 OpenMarket Ltd
..
.. Licensed under the Apache License, Version 2.0 (the "License");
.. you may not use this file except in compliance with the License.
.. You may obtain a copy of the License at
..
..     http://www.apache.org/licenses/LICENSE-2.0
..
.. Unless required by applicable law or agreed to in writing, software
.. distributed under the License is distributed on an "AS IS" BASIS,
.. WITHOUT WARRANTIES OR CONDITIONS OF ANY KIND, either express or implied.
.. See the License for the specific language governing permissions and
.. limitations under the License.


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Megolm group ratchet
====================

An AES-based cryptographic ratchet intended for group communications.

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.. contents::

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Background
----------

The Megolm ratchet is intended for encrypted messaging applications where there
may be a large number of recipients of each message, thus precluding the use of
peer-to-peer encryption systems such as `Olm`_.

It also allows a receipient to decrypt received messages multiple times. For
instance, in client/server applications, a copy of the ciphertext can be stored
on the (untrusted) server, while the client need only store the session keys.

Overview
--------

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Each participant in a conversation uses their own outbound session for
encrypting messages. A session consists of a ratchet and an `Ed25519`_ keypair.
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Secrecy is provided by the ratchet, which can be wound forwards but not
backwards, and is used to derive a distinct message key for each message.
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Authenticity is provided via Ed25519 signatures.
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The value of the ratchet, and the public part of the Ed25519 key, are shared
with other participants in the conversation via secure peer-to-peer
channels. Provided that peer-to-peer channel provides authenticity of the
messages to the participants and deniability of the messages to third parties,
the Megolm session will inherit those properties.

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The Megolm ratchet algorithm
----------------------------

The Megolm ratchet :math:`R_i` consists of four parts, :math:`R_{i,j}` for
:math:`j \in {0,1,2,3}`. The length of each part depends on the hash function
in use (256 bits for this version of Megolm).

The ratchet is initialised with cryptographically-secure random data, and
advanced as follows:

.. math::
    \begin{align}
    R_{i,0} &=
      \begin{cases}
        H_0\left(R_{2^24(n-1),0}\right) &\text{if }\exists n | i = 2^24n\\
        R_{i-1,0} &\text{otherwise}
      \end{cases}\\
    R_{i,1} &=
      \begin{cases}
        H_1\left(R_{2^24(n-1),0}\right) &\text{if }\exists n | i = 2^24n\\
        H_1\left(R_{2^16(m-1),1}\right) &\text{if }\exists m | i = 2^16m\\
        R_{i-1,1} &\text{otherwise}
      \end{cases}\\
    R_{i,2} &=
      \begin{cases}
        H_2\left(R_{2^24(n-1),0}\right) &\text{if }\exists n | i = 2^24n\\
        H_2\left(R_{2^16(m-1),1}\right) &\text{if }\exists m | i = 2^16m\\
        H_2\left(R_{2^8(p-1),2}\right) &\text{if }\exists p | i = 2^8p\\
        R_{i-1,2} &\text{otherwise}
      \end{cases}\\
    R_{i,3} &=
      \begin{cases}
        H_3\left(R_{2^24(n-1),0}\right) &\text{if }\exists n | i = 2^24n\\
        H_3\left(R_{2^16(m-1),1}\right) &\text{if }\exists m | i = 2^16m\\
        H_3\left(R_{2^8(p-1),2}\right) &\text{if }\exists p | i = 2^8p\\
        H_3\left(R_{i-1,3}\right) &\text{otherwise}
      \end{cases}
    \end{align}
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where :math:`H_0`, :math:`H_1`, :math:`H_2`, and :math:`H_3` are different hash
functions. In summary: every :math:`2^8` iterations, :math:`R_{i,3}` is
reseeded from :math:`R_{i,2}`. Every :math:`2^16` iterations, :math:`R_{i,2}`
and :math:`R_{i,3}` are reseeded from :math:`R_{i,1}`. Every :math:`2^24`
iterations, :math:`R_{i,1}`, :math:`R_{i,2}` and :math:`R_{i,3}` are reseeded
from :math:`R_{i,0}`.

The complete ratchet value, :math:`R_{i}`, is hashed to generate the keys used
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to encrypt each message.  This scheme allows the ratchet to be advanced an
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arbitrary amount forwards while needing at most 1023 hash computations.  A
client can decrypt chat history onwards from the earliest value of the ratchet
it is aware of, but cannot decrypt history from before that point without
reversing the hash function.

This allows a participant to share its ability to decrypt chat history with
another from a point in the conversation onwards by giving a copy of the
ratchet at that point in the conversation.


The Megolm protocol
-------------------

Session setup
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~~~~~~~~~~~~~

Each participant in a conversation generates their own Megolm session. A
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session consists of three parts:

* a 32 bit counter, :math:`i`.
* an `Ed25519`_ keypair, :math:`K`.
* a ratchet, :math:`R_i`, which consists of four 256-bit values,
  :math:`R_{i,j}` for :math:`j \in {0,1,2,3}`.
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The counter :math:`i` is initialised to :math:`0`. A new Ed25519 keypair is
generated for :math:`K`. The ratchet is simply initialised with 1024 bits of
cryptographically-secure random data.

A single participant may use multiple sessions over the lifetime of a
conversation. The public part of :math:`K` is used as an identifier to
discriminate between sessions.

Sharing session data
~~~~~~~~~~~~~~~~~~~~

To allow other participants in the conversation to decrypt messages, the
session data is formatted as described in `Session-sharing format`_. It is then
shared with other participants in the conversation via a secure peer-to-peer
channel (such as that provided by `Olm`_).

When the session data is received from other participants, the recipient first
checks that the signature matches the public key. They then store their own
copy of the counter, ratchet, and public key.

Message encryption
~~~~~~~~~~~~~~~~~~

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This version of Megolm uses AES-256_ in CBC_ mode with `PCKS#7`_ padding and
HMAC-SHA-256_ (truncated to 64 bits). The 256 bit AES key, 256 bit HMAC key,
and 128 bit AES IV are derived from the megolm ratchet :math:`R_i`:
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.. math::

    \begin{align}
    AES\_KEY_{i}\;\parallel\;HMAC\_KEY_{i}\;\parallel\;AES\_IV_{i}
        &= HKDF\left(0,\,R_{i},\text{"MEGOLM\_KEYS"},\,80\right) \\
    \end{align}

where :math:`\parallel` represents string splitting, and
:math:`HKDF\left(salt,\,IKM,\,info,\,L\right)` refers to the `HMAC-based key
derivation function`_ using using `SHA-256`_ as the hash function
(`HKDF-SHA-256`_) with a salt value of :math:`salt`, input key material of
:math:`IKM`, context string :math:`info`, and output keying material length of
:math:`L` bytes.

The plain-text is encrypted with AES-256, using the key :math:`AES\_KEY_{i}`
and the IV :math:`AES\_IV_{i}` to give the cipher-text, :math:`X_{i}`.

The ratchet index :math:`i`, and the cipher-text :math:`X_{i}`, are then packed
into a message as described in `Message format`_. Then the entire message
(including the version bytes and all payload bytes) are passed through
HMAC-SHA-256. The first 8 bytes of the MAC are appended to the message.

Finally, the authenticated message is signed using the Ed25519 keypair; the 64
byte signature is appended to the message.

The complete signed message, together with the public part of :math:`K` (acting
as a session identifier), can then be sent over an insecure channel. The
message can then be authenticated and decrypted only by recipients who have
received the session data.

Advancing the ratchet
~~~~~~~~~~~~~~~~~~~~~

After each message is encrypted, the ratchet is advanced. This is done as
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described in `The Megolm ratchet algorithm`_, using the following definitions:
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.. math::
    \begin{align}
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        H_0(A) &\equiv HMAC(A,\text{"\textbackslash x00"}) \\
        H_1(A) &\equiv HMAC(A,\text{"\textbackslash x01"}) \\
        H_2(A) &\equiv HMAC(A,\text{"\textbackslash x02"}) \\
        H_3(A) &\equiv HMAC(A,\text{"\textbackslash x03"}) \\
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    \end{align}

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where :math:`HMAC(A, T)` is the HMAC-SHA-256_ of ``T``, using ``A`` as the
key.
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For outbound sessions, the updated ratchet and counter are stored in the
session.

In order to maintain the ability to decrypt conversation history, inbound
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sessions should store a copy of their earliest known ratchet value (unless they
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explicitly want to drop the ability to decrypt that history). They may also
choose to cache calculated ratchet values, but the decision of which ratchet
states to cache is left to the application.

Data exchange formats
---------------------

Session-sharing format
~~~~~~~~~~~~~~~~~~~~~~

The Megolm key-sharing format is as follows:

.. code::

    +---+----+--------+--------+--------+--------+------+-----------+
    | V | i  | R(i,0) | R(i,1) | R(i,2) | R(i,3) | Kpub | Signature |
    +---+----+--------+--------+--------+--------+------+-----------+
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    0   1    5        37       69      101      133    165         229   bytes
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The version byte, ``V``, is ``"\x02"``.

This is followed by the ratchet index, :math:`i`, which is encoded as a
big-endian 32-bit integer; the ratchet values :math:`R_{i,j}`; and the public
part of the Ed25519 keypair :math:`K`.

The data is then signed using the Ed25519 keypair, and the 64-byte signature is
appended.

Message format
~~~~~~~~~~~~~~

Megolm messages consist of a one byte version, followed by a variable length
payload, a fixed length message authentication code, and a fixed length
signature.

.. code::

   +---+------------------------------------+-----------+------------------+
   | V | Payload Bytes                      | MAC Bytes | Signature Bytes  |
   +---+------------------------------------+-----------+------------------+
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   0   1                                    N          N+8                N+72   bytes
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The version byte, ``V``, is ``"\x03"``.

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The payload consists of the following key-value pairs:
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============= ===== ======== ================================================
    Name       Tag    Type                     Meaning
============= ===== ======== ================================================
Message-Index  0x08 Integer  The index of the ratchet, :math:`i`
Cipher-Text    0x12 String   The cipher-text, :math:`X_{i}`, of the message
============= ===== ======== ================================================

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Within the payload, integers are encoded using a variable length encoding. 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.

Strings are encoded as a variable-length integer followed by the string itself.

Each key-value pair is encoded as a variable-length integer giving the tag,
followed by a string or variable-length integer giving the value.

The payload is followed by the MAC. The length of the MAC is determined by the
authenticated encryption algorithm being used (8 bytes in this version of the
protocol). The MAC protects all of the bytes preceding the MAC.
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The length of the signature is determined by the signing algorithm being used
(64 bytes in this version of the protocol). The signature covers all of the
bytes preceding the signaure.

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License
-------
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The Megolm specification (this document) is licensed under the `Apache License,
Version 2.0 <http://www.apache.org/licenses/LICENSE-2.0>`_.
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.. _`Ed25519`: http://ed25519.cr.yp.to/
.. _`HMAC-based key derivation function`: https://tools.ietf.org/html/rfc5869
.. _`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
.. _`Olm`: ./olm.html