# Megolm group ratchet

An AES-based cryptographic ratchet intended for group communications.

## 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 recipient 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 $`R_i`$ consists of four parts, $`R_{i,j}`$ for
$`j \in {0,1,2,3}`$. 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{aligned}
R_{i,0} &=
  \begin{cases}
  H_0\left(R_{2^{24}(n-1),0}\right) &\text{if }\exists n | i = 2^{24}n\\
  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^{24}n\\
  H_1\left(R_{2^{16}(m-1),1}\right) &\text{if }\exists m | i = 2^{16}m\\
  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^{24}n\\
  H_2\left(R_{2^{16}(m-1),1}\right) &\text{if }\exists m | i = 2^{16}m\\
  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^{24}n\\
  H_3\left(R_{2^{16}(m-1),1}\right) &\text{if }\exists m | i = 2^{16}m\\
  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{aligned}
```

where $`H_0`$, $`H_1`$, $`H_2`$, and $`H_3`$ are different hash
functions. In summary: every $`2^8`$ iterations, $`R_{i,3}`$ is
reseeded from $`R_{i,2}`$. Every $`2^{16}`$ iterations, $`R_{i,2}`$
and $`R_{i,3}`$ are reseeded from $`R_{i,1}`$. Every $`2^{24}`$
iterations, $`R_{i,1}`$, $`R_{i,2}`$ and $`R_{i,3}`$ are reseeded
from $`R_{i,0}`$.

The complete ratchet value, $`R_{i}`$, 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 1020 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, $`i`$.
* an [Ed25519][] keypair, $`K`$.
* a ratchet, $`R_i`$, which consists of four 256-bit values,
  $`R_{i,j}`$ for $`j \in {0,1,2,3}`$.

The counter $`i`$ is initialised to $`0`$. A new Ed25519 keypair is
generated for $`K`$. 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 $`K`$ 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](#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 [PKCS#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 $`R_i`$:

```math
\begin{aligned}
 \mathit{AES\_KEY}_{i}\;\parallel\;\mathit{HMAC\_KEY}_{i}\;\parallel\;\mathit{AES\_IV}_{i}
    &= \operatorname{HKDF}\left(0,\,R_{i},\text{"MEGOLM\_KEYS"},\,80\right) \\
\end{aligned}
```

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

The plain-text is encrypted with AES-256, using the key $`\mathit{AES\_KEY}_{i}`$
and the IV $`\mathit{AES\_IV}_{i}`$ to give the cipher-text, $`X_{i}`$.

The ratchet index $`i`$, and the cipher-text $`X_{i}`$, are then packed
into a message as described in [Message format](#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 $`K`$ (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](#the-megolm-ratchet-algorithm), using the following definitions:

```math
\begin{aligned}
    H_0(A) &\equiv \operatorname{HMAC}(A,\text{``\char`\\x00"}) \\
    H_1(A) &\equiv \operatorname{HMAC}(A,\text{``\char`\\x01"}) \\
    H_2(A) &\equiv \operatorname{HMAC}(A,\text{``\char`\\x02"}) \\
    H_3(A) &\equiv \operatorname{HMAC}(A,\text{``\char`\\x03"}) \\
\end{aligned}
```

where $`\operatorname{HMAC}(A, T)`$ 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 - see [Partial
Forward Secrecy](#partial-forward-secrecy)). 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

This format is used for the initial sharing of a Megolm session with other
group participants who need to be able to read messages encrypted by this
session.

The session sharing format is as follows:

```
+---+----+--------+--------+--------+--------+------+-----------+
| 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, $`i`$, which is encoded as a
big-endian 32-bit integer; the ratchet values $`R_{i,j}`$; and the public
part of the Ed25519 keypair $`K`$.

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

### Session export format

Once the session is initially shared with the group participants, each
participant needs to retain a copy of the session if they want to maintain
their ability to decrypt messages encrypted with that session.

For forward-secrecy purposes, a participant may choose to store a ratcheted
version of the session. But since the ratchet index is covered by the
signature, this would invalidate the signature. So we define a similar format,
called the *session export format*, which is identical to the [session sharing
format](#session-sharing-format) except for dropping the signature.

The Megolm session export format is thus as follows:

```
+---+----+--------+--------+--------+--------+------+
| V | i  | R(i,0) | R(i,1) | R(i,2) | R(i,3) | Kpub |
+---+----+--------+--------+--------+--------+------+
0   1    5        37       69      101      133    165   bytes
```

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

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

### 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.

```
+---+------------------------------------+-----------+------------------+
| 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, i
Cipher-Text|0x12|String|The cipher-text, Xi, 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 signature.

## Limitations

### Message Replays

A message can be decrypted successfully multiple times. This means that an
attacker can re-send a copy of an old message, and the recipient will treat it
as a new message.

To mitigate this it is recommended that applications track the ratchet indices
they have received and that they reject messages with a ratchet index that
they have already decrypted.

### Lack of Transcript Consistency

In a group conversation, there is no guarantee that all recipients have
received the same messages. For example, if Alice is in a conversation with Bob
and Charlie, she could send different messages to Bob and Charlie, or could
send some messages to Bob but not Charlie, or vice versa.

Solving this is, in general, a hard problem, particularly in a protocol which
does not guarantee in-order message delivery. For now it remains the subject of
future research.

### Lack of Backward Secrecy

[Backward secrecy](https://intensecrypto.org/public/lec_08_hash_functions_part2.html#sec-forward-and-backward-secrecy)
(also called 'future secrecy' or 'post-compromise security') is the property
that if current private keys are compromised, an attacker cannot decrypt
future messages in a given session.  In other words, when looking
**backwards** in time at a compromise which has already happened, **current**
messages are still secret.

By itself, Megolm does not possess this property: once the key to a Megolm
session is compromised, the attacker can decrypt any message that was
encrypted using a key derived from the compromised or subsequent ratchet
values.

In order to mitigate this, the application should ensure that Megolm sessions
are not used indefinitely. Instead it should periodically start a new session,
with new keys shared over a secure channel.

<!-- TODO: Can we recommend sensible lifetimes for Megolm sessions? Probably
   depends how paranoid we're feeling, but some guidelines might be useful. -->

### Partial Forward Secrecy

[Forward secrecy](https://intensecrypto.org/public/lec_08_hash_functions_part2.html#sec-forward-and-backward-secrecy)
(also called 'perfect forward secrecy') is the property that if the current
private keys are compromised, an attacker cannot decrypt *past* messages in
a given session. In other words, when looking **forwards** in time towards a
potential future compromise, **current** messages will be secret.

In Megolm, each recipient maintains a record of the ratchet value which allows
them to decrypt any messages sent in the session after the corresponding point
in the conversation. If this value is compromised, an attacker can similarly
decrypt past messages which were encrypted by a key derived from the
compromised or subsequent ratchet values. This gives 'partial' forward
secrecy.

To mitigate this issue, the application should offer the user the option to
discard historical conversations, by winding forward any stored ratchet values,
or discarding sessions altogether.

### Dependency on secure channel for key exchange

The design of the Megolm ratchet relies on the availability of a secure
peer-to-peer channel for the exchange of session keys. Any vulnerabilities in
the underlying channel are likely to be amplified when applied to Megolm
session setup.

For example, if the peer-to-peer channel is vulnerable to an unknown key-share
attack, the entire Megolm session become similarly vulnerable. For example:
Alice starts a group chat with Eve, and shares the session keys with Eve. Eve
uses the unknown key-share attack to forward the session keys to Bob, who
believes Alice is starting the session with him. Eve then forwards messages
from the Megolm session to Bob, who again believes they are coming from
Alice. Provided the peer-to-peer channel is not vulnerable to this attack, Bob
will realise that the key-sharing message was forwarded by Eve, and can treat
the Megolm session as a forgery.

A second example: if the peer-to-peer channel is vulnerable to a replay
attack, this can be extended to entire Megolm sessions.

## 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
[PKCS#7]: https://tools.ietf.org/html/rfc2315
[Olm]: https://gitlab.matrix.org/matrix-org/olm/blob/master/docs/olm.md
[Protocol Buffers encoding]: https://developers.google.com/protocol-buffers/docs/encoding
