Convert olm.rst to markdown

# Olm: A Cryptographic Ratchet

An implementation of the double cryptographic ratchet described by

https://whispersystems.org/docs/specifications/doubleratchet/.

## Notation

This document uses $`\parallel`$ to represent string concatenation. When

$`\parallel`$ appears on the right hand side of an $`=`$ it means that

the inputs are concatenated. When $`\parallel`$ appears on the left hand

side of an $`=`$ it means that the output is split.

When this document uses $`ECDH\left(K_A,\,K_B\right)`$ it means that each

party computes a Diffie-Hellman agreement using their private key and the

remote party's public key.

So party $`A`$ computes $`ECDH\left(K_B^{public},\,K_A^{private}\right)`$

and party $`B`$ computes $`ECDH\left(K_A^{public},\,K_B^{private}\right)`$.

Where this document uses $`HKDF\left(salt,\,IKM,\,info,\,L\right)`$ it

refers to the [HMAC-based key derivation function][] with a salt value of

$`salt`$, input key material of $`IKM`$, context string $`info`$,

and output keying material length of $`L`$ bytes.

## The Olm Algorithm

### Initial setup

The setup takes four [Curve25519][] inputs: Identity keys for Alice and Bob,

$`I_A`$ and $`I_B`$, and one-time keys for Alice and Bob,

$`E_A`$ and $`E_B`$. A shared secret, $`S`$, is generated using

[Triple Diffie-Hellman][]. The initial 256 bit root key, $`R_0`$, and 256

bit chain key, $`C_{0,0}`$, 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.

```math

\begin{aligned}

    S&=ECDH\left(I_A,\,E_B\right)\;\parallel\;ECDH\left(E_A,\,I_B\right)\;

        \parallel\;ECDH\left(E_A,\,E_B\right)\\

    R_0\;\parallel\;C_{0,0}&=

            HKDF\left(0,\,S,\,\text{"OLM\_ROOT"},\,64\right)

\end{aligned}

```

### Advancing the root key

Advancing a root key takes the previous root key, $`R_{i-1}`$, and two

Curve25519 inputs: the previous ratchet key, $`T_{i-1}`$, and the current

ratchet key $`T_i`$. 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, $`R_i`$, and

chain key, $`C_{i,0}`$, are derived from the shared secret using

[HKDF-SHA-256][] using $`R_{i-1}`$ as the salt and ``"OLM_RATCHET"`` as the

info.

```math

\begin{aligned}

    R_i\;\parallel\;C_{i,0}&=HKDF\left(

        R_{i-1},\,

        ECDH\left(T_{i-1},\,T_i\right),\,

        \text{"OLM\_RATCHET"},\,

        64

    \right)

\end{aligned}

```

### Advancing the chain key

Advancing a chain key takes the previous chain key, $`C_{i,j-1}`$. The next

chain key, $`C_{i,j}`$, is the [HMAC-SHA-256][] of ``"\x02"`` using the

previous chain key as the key.

```math

\begin{aligned}

    C_{i,j}&=HMAC\left(C_{i,j-1},\,\text{"\x02"}\right)

\end{aligned}

```

### Creating a message key

Creating a message key takes the current chain key, $`C_{i,j}`$. The

message key, $`M_{i,j}`$, is the [HMAC-SHA-256][] of ``"\x01"`` using the

current chain key as the key. The message keys where $`i`$ is even are used

by Alice to encrypt messages. The message keys where $`i`$ is odd are used

by Bob to encrypt messages.

```math

\begin{aligned}

    M_{i,j}&=HMAC\left(C_{i,j},\,\text{"\x01"}\right)

\end{aligned}

```

## The Olm Protocol

### Creating an outbound session

Bob publishes the public parts of his identity key, $`I_B`$, and some

single-use one-time keys $`E_B`$.

Alice downloads Bob's identity key, $`I_B`$, and a one-time key,

$`E_B`$. She generates a new single-use key, $`E_A`$, and computes a

root key, $`R_0`$, and a chain key $`C_{0,0}`$. She also generates a

new ratchet key $`T_0`$.

### Sending the first pre-key messages

Alice computes a message key, $`M_{0,j}`$, and a new chain key,

$`C_{0,j+1}`$, using the current chain key. She replaces the current chain

key with the new one.

Alice encrypts her plain-text with the message key, $`M_{0,j}`$, using an

authenticated encryption scheme (see below) to get a cipher-text,

$`X_{0,j}`$.

She then sends the following to Bob:

 * The public part of her identity key, $`I_A`$

 * The public part of her single-use key, $`E_A`$

 * The public part of Bob's single-use key, $`E_B`$

 * The current chain index, $`j`$

 * The public part of her ratchet key, $`T_0`$

 * The cipher-text, $`X_{0,j}`$

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 as above.

Bob looks up the private part of his single-use key, $`E_B`$. He can now

compute the root key, $`R_0`$, and the chain key, $`C_{0,0}`$, from

$`I_A`$, $`E_A`$, $`I_B`$, and $`E_B`$.

Bob then advances the chain key $`j`$ times, to compute the chain key used

by the message, $`C_{0,j}`$. He now creates the

message key, $`M_{0,j}`$, and attempts to decrypt the cipher-text,

$`X_{0,j}`$. If the cipher-text's authentication is correct then Bob can

discard the private part of his single-use one-time key, $`E_B`$.

Bob stores Alice's initial ratchet key, $`T_0`$, until he wants to

send a message.

### Sending normal messages

Once a message has been received from the other side, a session is considered

established, and a more compact form is used.

To send a message, the user checks if they have a sender chain key,

$`C_{i,j}`$. Alice uses chain keys where $`i`$ is even. Bob uses chain

keys where $`i`$ is odd. If the chain key doesn't exist then a new ratchet

key $`T_i`$ is generated and a new root key $`R_i`$ and chain key

$`C_{i,0}`$ are computed using $`R_{i-1}`$, $`T_{i-1}`$ and

$`T_i`$.

A message key,

$`M_{i,j}`$ is computed from the current chain key, $`C_{i,j}`$, and

the chain key is replaced with the next chain key, $`C_{i,j+1}`$. The

plain-text is encrypted with $`M_{i,j}`$, using an authenticated encryption

scheme (see below) to get a cipher-text, $`X_{i,j}`$.

The user then sends the following to the recipient:

 * The current chain index, $`j`$

 * The public part of the current ratchet key, $`T_i`$

 * The cipher-text, $`X_{i,j}`$

### Receiving messages

The user receives a message as above with the sender's current chain index, $`j`$,

the sender's ratchet key, $`T_i`$, and the cipher-text, $`X_{i,j}`$.

The user checks if they have a receiver chain with the correct

$`i`$ by comparing the ratchet key, $`T_i`$. If the chain doesn't exist

then they compute a new root key, $`R_i`$, and a new receiver chain, with

chain key $`C_{i,0}`$, using $`R_{i-1}`$, $`T_{i-1}`$ and

$`T_i`$.

If the $`j`$ 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 $`M_{i,j}`$. Otherwise

the receiver computes the chain key, $`C_{i,j}`$. The receiver computes the

message key, $`M_{i,j}`$, from the chain key and attempts to decrypt the

cipher-text, $`X_{i,j}`$.

If the decryption succeeds the receiver updates the chain key for $`T_i`$

with $`C_{i,j+1}`$ 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.

## The Olm Message Format

Olm uses two types of messages. The underlying transport protocol must provide

a means for recipients to distinguish between them.

### Normal Messages

Olm messages start with a one byte version followed by a variable length

payload followed by a fixed length message authentication code.

```

 +--------------+------------------------------------+-----------+

 | Version Byte | Payload Bytes                      | MAC Bytes |

 +--------------+------------------------------------+-----------+

```

The version byte is ``"\x03"``.

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**

:-----:|:-----:|:-----:|:-----:

Ratchet-Key|0x0A|String|The public part of the ratchet key, Ti, of the message

Chain-Index|0x10|Integer|The chain index, j, of the message

Cipher-Text|0x22|String|The cipher-text, Xi, j, of the message

The length of the MAC is determined by the authenticated encryption algorithm

being used. (Olm version 1 uses [HMAC-SHA-256][], truncated to 8 bytes). The

MAC protects all of the bytes preceding the MAC.

### Pre-Key Messages

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

length payload.

```

 +--------------+------------------------------------+

 | Version Byte | Payload Bytes                      |

 +--------------+------------------------------------+

```

The version byte is ``"\x03"``.

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

**Name**|**Tag**|**Type**|**Meaning**

:-----:|:-----:|:-----:|:-----:

One-Time-Key|0x0A|String|The public part of Bob's single-use key, Eb.

Base-Key|0x12|String|The public part of Alice's single-use key, Ea.

Identity-Key|0x1A|String|The public part of Alice's identity key, Ia.

Message|0x22|String|An embedded Olm message with its own version and MAC.

## Olm Authenticated Encryption

### Version 1

Version 1 of Olm uses [AES-256][] in [CBC][] mode with [PKCS#7][] padding for

encryption and [HMAC-SHA-256][] (truncated to 64 bits) 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"``.

```math

\begin{aligned}

    AES\_KEY_{i,j}\;\parallel\;HMAC\_KEY_{i,j}\;\parallel\;AES\_IV_{i,j}

    &= HKDF\left(0,\,M_{i,j},\text{"OLM\_KEYS"},\,80\right) \\

\end{aligned}

```

The plain-text is encrypted with AES-256, using the key $`AES\_KEY_{i,j}`$

and the IV $`AES\_IV_{i,j}`$ to give the cipher-text, $`X_{i,j}`$.

Then the entire message (including the Version Byte and all Payload Bytes) are

passed through [HMAC-SHA-256][]. The first 8 bytes of the MAC are appended to the message.

## Message authentication concerns

To avoid unknown key-share attacks, the application must include identifying

data for the sending and receiving user in the plain-text of (at least) the

pre-key messages. Such data could be a user ID, a telephone number;

alternatively it could be the public part of a keypair which the relevant user

has proven ownership of.

### Example attacks

1. Alice publishes her public [Curve25519][] identity key, $`I_A`$. Eve

   publishes the same identity key, claiming it as her own. Bob downloads

   Eve's keys, and associates $`I_A`$ with Eve. Alice sends a message to

   Bob; Eve intercepts it before forwarding it to Bob. Bob believes the

   message came from Eve rather than Alice.

   This is prevented if Alice includes her user ID in the plain-text of the

   pre-key message, so that Bob can see that the message was sent by Alice

   originally.

2. Bob publishes his public [Curve25519][] identity key, $`I_B`$. Eve

   publishes the same identity key, claiming it as her own. Alice downloads

   Eve's keys, and associates $`I_B`$ with Eve. Alice sends a message to

   Eve; Eve cannot decrypt it, but forwards it to Bob. Bob believes the

   Alice sent the message to him, wheras Alice intended it to go to Eve.

   This is prevented by Alice including the user ID of the intended recpient

   (Eve) in the plain-text of the pre-key message. Bob can now tell that the

   message was meant for Eve rather than him.

## IPR

The Olm specification (this document) is hereby placed in the public domain.

## Feedback

Can be sent to olm at matrix.org.

## Acknowledgements

The ratchet that Olm implements was designed by Trevor Perrin and Moxie

Marlinspike - details at https://whispersystems.org/docs/specifications/doubleratchet/. Olm is

an entirely new implementation written by the Matrix.org team.

[Curve25519]: http://cr.yp.to/ecdh.html

[Triple Diffie-Hellman]: https://whispersystems.org/blog/simplifying-otr-deniability/

[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