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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.
+
+
+Megolm group ratchet
+====================
+
+An AES-based cryptographic ratchet intended for group communications.
+
+.. contents::
+
+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
+--------
+
+Each participant in a conversation uses their own outbound session for
+encrypting messages. A session consists of a ratchet and an `Ed25519`_ keypair.
+
+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.
+
+Authenticity is provided via Ed25519 signatures.
+
+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.
+
+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}
+
+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
+to encrypt each message.  This scheme allows the ratchet to be advanced an
+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
+~~~~~~~~~~~~~
+
+Each participant in a conversation generates their own Megolm session. A
+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}`.
+
+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
+~~~~~~~~~~~~~~~~~~
+
+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`:
+
+.. 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
+described in `The Megolm ratchet algorithm`_, using the following definitions:
+
+.. math::
+    \begin{align}
+        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"}) \\
+    \end{align}
+
+where :math:`HMAC(A, T)` is the HMAC-SHA-256_ of ``T``, using ``A`` as the
+key.
+
+For outbound sessions, the updated ratchet and counter are stored in the
+session.
+
+In order to maintain the ability to decrypt conversation history, inbound
+sessions should store a copy of their earliest known ratchet value (unless they
+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 |
+    +---+----+--------+--------+--------+--------+------+-----------+
+    0   1    5        37       69      101      133    165         229   bytes
+
+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  |
+   +---+------------------------------------+-----------+------------------+
+   0   1                                    N          N+8                N+72   bytes
+
+The version byte, ``V``, is ``"\x03"``.
+
+The payload uses a format based on the `Protocol Buffers encoding`_. It
+consists of the following key-value pairs:
+
+============= ===== ======== ================================================
+    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
+============= ===== ======== ================================================
+
+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.
+
+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.
+
+License
+-------
+
+The Megolm specification (this document) is licensed under the `Apache License,
+Version 2.0 <http://www.apache.org/licenses/LICENSE-2.0>`_.
+
+
+.. _`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
+.. _`Protocol Buffers encoding`: https://developers.google.com/protocol-buffers/docs/encoding