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# 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