Network Working Group R. Barnes
Internet-Draft Cisco
Intended status: Informational J. Millican
Expires: November 3, 2019 Facebook
E. Omara
K. Cohn-Gordon
University of Oxford
R. Robert
May 02, 2019

The Messaging Layer Security (MLS) Protocol


Messaging applications are increasingly making use of end-to-end security mechanisms to ensure that messages are only accessible to the communicating endpoints, and not to any servers involved in delivering messages. Establishing keys to provide such protections is challenging for group chat settings, in which more than two clients need to agree on a key but may not be online at the same time. In this document, we specify a key establishment protocol that provides efficient asynchronous group key establishment with forward secrecy and post-compromise security for groups in size ranging from two to thousands.

Status of This Memo

This Internet-Draft is submitted in full conformance with the provisions of BCP 78 and BCP 79.

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This Internet-Draft will expire on November 3, 2019.

Copyright Notice

Copyright (c) 2019 IETF Trust and the persons identified as the document authors. All rights reserved.

This document is subject to BCP 78 and the IETF Trust's Legal Provisions Relating to IETF Documents ( in effect on the date of publication of this document. Please review these documents carefully, as they describe your rights and restrictions with respect to this document. Code Components extracted from this document must include Simplified BSD License text as described in Section 4.e of the Trust Legal Provisions and are provided without warranty as described in the Simplified BSD License.

Table of Contents

1. Introduction

DISCLAIMER: This is a work-in-progress draft of MLS and has not yet seen significant security analysis. It should not be used as a basis for building production systems.

RFC EDITOR: PLEASE REMOVE THE FOLLOWING PARAGRAPH The source for this draft is maintained in GitHub. Suggested changes should be submitted as pull requests at Instructions are on that page as well. Editorial changes can be managed in GitHub, but any substantive change should be discussed on the MLS mailing list.

A group of users who want to send each other encrypted messages needs a way to derive shared symmetric encryption keys. For two parties, this problem has been studied thoroughly, with the Double Ratchet emerging as a common solution [doubleratchet] [signal]. Channels implementing the Double Ratchet enjoy fine-grained forward secrecy as well as post-compromise security, but are nonetheless efficient enough for heavy use over low-bandwidth networks.

For a group of size greater than two, a common strategy is to unilaterally broadcast symmetric “sender” keys over existing shared symmetric channels, and then for each member to send messages to the group encrypted with their own sender key. Unfortunately, while this improves efficiency over pairwise broadcast of individual messages and provides forward secrecy (with the addition of a hash ratchet), it is difficult to achieve post-compromise security with sender keys. An adversary who learns a sender key can often indefinitely and passively eavesdrop on that member’s messages. Generating and distributing a new sender key provides a form of post-compromise security with regard to that sender. However, it requires computation and communications resources that scale linearly with the size of the group.

In this document, we describe a protocol based on tree structures that enable asynchronous group keying with forward secrecy and post-compromise security. Based on earlier work on “asynchronous ratcheting trees” [art], the mechanism presented here use a asynchronous key-encapsulation mechanism for tree structures. This mechanism allows the members of the group to derive and update shared keys with costs that scale as the log of the group size.

1.1. Change Log








2. Terminology

The key words “MUST”, “MUST NOT”, “REQUIRED”, “SHALL”, “SHALL NOT”, “SHOULD”, “SHOULD NOT”, “RECOMMENDED”, “NOT RECOMMENDED”, “MAY”, and “OPTIONAL” in this document are to be interpreted as described in BCP 14 [RFC2119] [RFC8174] when, and only when, they appear in all capitals, as shown here.

An agent that uses this protocol to establish shared cryptographic state with other clients. A client is defined by the cryptographic keys it holds. An application or user may use one client per device (keeping keys local to each device) or sync keys among a user’s devices so that each user appears as a single client.
A collection of clients with shared cryptographic state.
A client that is included in the shared state of a group, hence has access to the group’s secrets.
Initialization Key:
A short-lived HPKE key pair used to introduce a new client to a group. Initialization keys are published for each client (UserInitKey).
Leaf Key:
A secret that represent a member’s contribution to the group secret (so called because the members’ leaf keys are the leaves in the group’s ratchet tree).
Identity Key:
A long-lived signing key pair used to authenticate the sender of a message.

Terminology specific to tree computations is described in Section 5.

We use the TLS presentation language [RFC8446] to describe the structure of protocol messages.

3. Basic Assumptions

This protocol is designed to execute in the context of a Messaging Service (MS) as described in [I-D.ietf-mls-architecture]. In particular, we assume the MS provides the following services:

4. Protocol Overview

The goal of this protocol is to allow a group of clients to exchange confidential and authenticated messages. It does so by deriving a sequence of secrets and keys known only to members. Those should be secret against an active network adversary and should have both forward and post-compromise secrecy with respect to compromise of a participant.

We describe the information stored by each client as a state, which includes both public and private data. An initial state, including an initial set of clients, is set up by a group creator using the Init algorithm and based on information pre-published by clients. The creator sends the Init message to the clients, who can then set up their own group state and derive the same shared secret. Clients then exchange messages to produce new shared states which are causally linked to their predecessors, forming a logical Directed Acyclic Graph (DAG) of states. Members can send Update messages for post-compromise secrecy and new clients can be added or existing members removed from the group.

The protocol algorithms we specify here follow. Each algorithm specifies both (i) how a client performs the operation and (ii) how other clients update their state based on it.

There are four major operations in the lifecycle of a group:

Before the initialization of a group, clients publish UserInitKey objects to a directory provided to the Messaging Service.

A              B              C          Directory       Channel
|              |              |              |              |
| UserInitKeyA |              |              |              |
|------------------------------------------->|              |
|              |              |              |              |
|              | UserInitKeyB |              |              |
|              |---------------------------->|              |
|              |              |              |              |
|              |              | UserInitKeyC |              |
|              |              |------------->|              |
|              |              |              |              |

When a client A wants to establish a group with B and C, it first downloads UserInitKeys for B and C. It then initializes a group state containing only itself and uses the UserInitKeys to compute Welcome and Add messages to add B and C, in a sequence chosen by A. The Welcome messages are sent directly to the new members (there is no need to send them to the group). The Add messages are broadcasted to the group, and processed in sequence by B and C. Messages received before a client has joined the group are ignored. Only after A has received its Add messages back from the server does it update its state to reflect their addition.

A              B              C          Directory            Channel
|              |              |              |                   |
|         UserInitKeyB, UserInitKeyC         |                   |
|<-------------------------------------------|                   |
|state.init()  |              |              |                   |
|              |              |              |                   |
|              |              |              | Add(A->AB)        |
|              |              |              |                   |
|  Welcome(B)  |              |              |                   |
|------------->|state.init()  |              |                   |
|              |              |              |                   |
|              |              |              | Add(A->AB)        |
|state.add(B)  |<------------------------------------------------|
|              |state.join()  |              |                   |
|              |              |              |                   |
|              |              |              | Add(AB->ABC)      |
|              |              |              |                   |
|              |  Welcome(C)  |              |                   |
|---------------------------->|state.init()  |                   |
|              |              |              |                   |
|              |              |              | Add(AB->ABC)      |
|state.add(C)  |<------------------------------------------------|
|              |state.add(C)  |<---------------------------------|
|              |              |state.join()  |                   |

Subsequent additions of group members proceed in the same way. Any member of the group can download an UserInitKey for a new client and broadcast an Add message that the current group can use to update their state and the new client can use to initialize its state.

To enforce forward secrecy and post-compromise security of messages, each member periodically updates its leaf secret which represents its contribution to the group secret. Any member of the group can send an Update at any time by generating a fresh leaf secret and sending an Update message that describes how to update the group secret with that new information. Once all members have processed this message, the group’s secrets will be unknown to an attacker that had compromised the sender’s prior leaf secret.

It is left to the application to determine the interval of time between Update messages. This policy could require a change for each message, or it could require sending an update every week or more.

A              B     ...      Z          Directory        Channel
|              |              |              |              |
| Update(A)    |              |              |              |
|              |              |              |              |
|              |              |              | Update(A)    |
|state.upd(A)  |<-------------------------------------------|
|              |state.upd(A)  |<----------------------------|
|              |              |state.upd(A)  |              |
|              |              |              |              |

Members are removed from the group in a similar way, as an update is effectively removing the old leaf from the group. Any member of the group can generate a Remove message that adds new entropy to the group state that is known to all members except the removed member. After other participants have processed this message, the group’s secrets will be unknown to the removed participant. Note that this does not necessarily imply that any member is actually allowed to evict other members; groups can layer authentication-based access control policies on top of these basic mechanism.

A              B     ...      Z          Directory       Channel
|              |              |              |              |
|              |              | Remove(B)    |              |
|              |              |---------------------------->|
|              |              |              |              |
|              |              |              | Remove(B)    |
|state.del(B)  |              |<----------------------------|
|              |              |state.del(B)  |              |
|              |              |              |              |
|              |              |              |              |

5. Ratchet Trees

The protocol uses “ratchet trees” for deriving shared secrets among a group of clients.

5.1. Tree Computation Terminology

Trees consist of nodes. A node is a leaf if it has no children, and a parent otherwise; note that all parents in our trees have precisely two children, a left child and a right child. A node is the root of a tree if it has no parents, and intermediate if it has both children and parents. The descendants of a node are that node, its children, and the descendants of its children, and we say a tree contains a node if that node is a descendant of the root of the tree. Nodes are siblings if they share the same parent.

A subtree of a tree is the tree given by the descendants of any node, the head of the subtree. The size of a tree or subtree is the number of leaf nodes it contains. For a given parent node, its left subtree is the subtree with its left child as head (respectively right subtree).

All trees used in this protocol are left-balanced binary trees. A binary tree is full (and balanced) if its size is a power of two and for any parent node in the tree, its left and right subtrees have the same size. If a subtree is full and it is not a subset of any other full subtree, then it is maximal.

A binary tree is left-balanced if for every parent, either the parent is balanced, or the left subtree of that parent is the largest full subtree that could be constructed from the leaves present in the parent’s own subtree. Note that given a list of n items, there is a unique left-balanced binary tree structure with these elements as leaves. In such a left-balanced tree, the k-th leaf node refers to the k-th leaf node in the tree when counting from the left, starting from 0.

The direct path of a root is the empty list, and of any other node is the concatenation of that node with the direct path of its parent. The copath of a node is the list of siblings of nodes in its direct path. The frontier of a tree is the list of heads of the maximal full subtrees of the tree, ordered from left to right.

For example, in the below tree:

           /      \
          /        \
         /          \
     ABCD            EFG
    /    \          /  \
   /      \        /    \
  AB      CD      EF    |
 / \     / \     / \    |
A   B   C   D   E   F   G

                    1 1 1
0 1 2 3 4 5 6 7 8 9 0 1 2

Each node in the tree is assigned an node index, starting at zero and running from left to right. A node is a leaf node if and only if it has an even index. The node indices for the nodes in the above tree are as follows:

(Note that left-balanced binary trees are the same structure that is used for the Merkle trees in the Certificate Transparency protocol [I-D.ietf-trans-rfc6962-bis].)

The leaves of the tree are indexed separately, using a leaf index, since the protocol messages only need to refer to leaves in the tree. Like nodes, leaves are numbered left to right. Note that given the above numbering, a node is a leaf node if and only if it has an even node index, and a leaf node’s leaf index is half its node index. The leaf indices in the above tree are as follows:

5.2. Ratchet Tree Nodes

A particular instance of a ratchet tree is based on the following cryptographic primitives, defined by the ciphersuite in use:

Each node in a ratchet tree contains up to three values:

The conditions under which each of these values must or must not be present are laid out in Section 5.3.

A node in the tree may also be blank, indicating that no value is present at that node. The resolution of a node is an ordered list of non-blank nodes that collectively cover all non-blank descendants of the node. The nodes in a resolution are ordered according to their indices.

For example, consider the following tree, where the “_” character represents a blank node:

    /   \
   /     \
  _       CD
 / \     / \
A   _   C   D

0 1 2 3 4 5 6

In this tree, we can see all three of the above rules in play:

Every node, regardless of whether a node is blank or populated, has a corresponding hash that summarizes the contents of the subtree below that node. The rules for computing these hashes are described in Section 6.3.

5.3. Views of a Ratchet Tree

We generally assume that each participant maintains a complete and up-to-date view of the public state of the group’s ratchet tree, including the public keys for all nodes and the credentials associated with the leaf nodes.

No participant in an MLS group has full knowledge of the secret state of the tree, i.e., private keys associated to the nodes. Instead, each member is assigned to a leaf of the tree, which determines the set of secret state known to the member. The credential stored at that leaf is one provided by the member.

In particular, MLS maintains the members’ views of the tree in such a way as to maintain the tree invariant:

The private key for a node in the tree is known to a member of
the group if and only if that member's leaf is a descendant of
the node or equal to it.

In other words, each member holds the private keys for nodes in its direct path, and no others.

5.4. Ratchet Tree Updates

Nodes in a tree are always updated along the direct path from a leaf to the root. The generator of the update chooses a random secret value “path_secret[0]”, and generates a sequence of “path secrets”, one for each node from the leaf to the root. That is, path_secret[0] is used for the leaf, path_secret[1] for its parent, and so on. At each step, the path secret is used to derive a new secret value for the corresponding node, from which the node’s key pair is derived.

path_secret[n] = HKDF-Expand-Label(path_secret[n-1],
                                   "path", "", Hash.Length)
node_secret[n] = HKDF-Expand-Label(path_secret[n],
                                   "node", "", Hash.Length)
node_priv[n], node_pub[n] = Derive-Key-Pair(node_secret[n])

For example, suppose there is a group with four members:

     / \
    /   \
   /     \
  E       F
 / \     / \
A   B   C   D

If the second participant (B) subsequently generates an update based on a secret X, then the sender would generate the following sequence of path secrets and node secrets:

    path_secret[2] ---> node_secret[2]
    path_secret[1] ---> node_secret[1]
X = path_secret[0] ---> node_secret[0]

After the update, the tree will have the following structure, where “ns[i]” represents the node_secret values generated as described above:

         /     \
     ns[1]      F
     /  \      / \
    A   ns[0] C   D

5.5. Synchronizing Views of the Tree

The members of the group need to keep their views of the tree in sync and up to date. When a client proposes a change to the tree (e.g., to add or remove a member), it transmits a handshake message containing a set of public values for intermediate nodes in the direct path of a leaf. The other members of the group can use these public values to update their view of the tree, aligning their copy of the tree to the sender’s.

To perform an update for a leaf, the sender broadcasts to the group the following information for each node in the direct path of the leaf, as well as the root:

The path secret value for a given node is encrypted for the subtree corresponding to the parent’s non-updated child, i.e., the child on the copath of the leaf node. There is one encrypted path secret for each public key in the resolution of the non-updated child. In particular, for the leaf node, there are no encrypted secrets, since a leaf node has no children.

The recipient of an update processes it with the following steps:

  1. Compute the updated path secrets * Identify a node in the direct path for which the local member is in the subtree of the non-updated child * Identify a node in the resolution of the copath node for which this node has a private key * Decrypt the path secret for the parent of the copath node using the private key from the resolution node * Derive path secrets for ancestors of that node using the algorithm described above * The recipient SHOULD verify that the received public keys agree with the public keys derived from the new node_secret values
  2. Merge the updated path secrets into the tree * Replace the public keys for nodes on the direct path with the received public keys * For nodes where an updated path secret was computed in step 1, compute the corresponding node secret and node key pair and replace the values stored at the node with the computed values.

For example, in order to communicate the example update described in the previous section, the sender would transmit the following values:

Public Key Ciphertext(s)
pk(ns[2]) E(pk(C), ps[2]), E(pk(D), ps[2])
pk(ns[1]) E(pk(A), ps[1])

In this table, the value pk(X) represents the public key corresponding derived from the node secret X. The value E(K, S) represents the public-key encryption of the path secret S to the public key K.

6. Cryptographic Objects

Each MLS session uses a single ciphersuite that specifies the following primitives to be used in group key computations:

The ciphersuite must also specify an algorithm Derive-Key-Pair that maps octet strings with the same length as the output of the hash function to key pairs for the asymmetric encryption scheme.

Public keys used in the protocol are opaque values in a format defined by the ciphersuite, using the following types:

opaque HPKEPublicKey<1..2^16-1>;
opaque SignaturePublicKey<1..2^16-1>;

Cryptographic algorithms are indicated using the following types:

enum {
} SignatureScheme;

enum {
} CipherSuite;

6.1. Curve25519, SHA-256, and AES-128-GCM

This ciphersuite uses the following primitives:

Given an octet string X, the private key produced by the Derive-Key-Pair operation is SHA-256(X). (Recall that any 32-octet string is a valid Curve25519 private key.) The corresponding public key is X25519(SHA-256(X), 9).

Implementations SHOULD use the approach specified in [RFC7748] to calculate the Diffie-Hellman shared secret. Implementations MUST check whether the computed Diffie-Hellman shared secret is the all-zero value and abort if so, as described in Section 6 of [RFC7748]. If implementers use an alternative implementation of these elliptic curves, they SHOULD perform the additional checks specified in Section 7 of [RFC7748]

6.1.1. P-256, SHA-256, and AES-128-GCM

This ciphersuite uses the following primitives:

Given an octet string X, the private key produced by the Derive-Key-Pair operation is SHA-256(X), interpreted as a big-endian integer. The corresponding public key is the result of multiplying the standard P-256 base point by this integer.

P-256 ECDH calculations (including parameter and key generation as well as the shared secret calculation) are performed according to [IEEE1363] using the ECKAS-DH1 scheme with the identity map as key derivation function (KDF), so that the shared secret is the x-coordinate of the ECDH shared secret elliptic curve point represented as an octet string. Note that this octet string (Z in IEEE 1363 terminology) as output by FE2OSP, the Field Element to Octet String Conversion Primitive, has constant length for any given field; leading zeros found in this octet string MUST NOT be truncated.

(Note that this use of the identity KDF is a technicality. The complete picture is that ECDH is employed with a non-trivial KDF because MLS does not directly use this secret for anything other than for computing other secrets.)

Clients MUST validate remote public values by ensuring that the point is a valid point on the elliptic curve. The appropriate validation procedures are defined in Section 4.3.7 of [X962] and alternatively in Section of [keyagreement]. This process consists of three steps: (1) verify that the value is not the point at infinity (O), (2) verify that for Y = (x, y) both integers are in the correct interval, (3) ensure that (x, y) is a correct solution to the elliptic curve equation. For these curves, implementers do not need to verify membership in the correct subgroup.

6.2. Credentials

A member of a group authenticates the identities of other participants by means of credentials issued by some authentication system, e.g., a PKI. Each type of credential MUST express the following data:

Credentials MAY also include information that allows a relying party to verify the identity / signing key binding.

enum {
} CredentialType;

struct {
    opaque identity<0..2^16-1>;
    SignatureScheme algorithm;
    SignaturePublicKey public_key;
} BasicCredential;

struct {
    CredentialType credential_type;
    select (credential_type) {
        case basic:

        case x509:
            opaque cert_data<1..2^24-1>;
} Credential;

6.3. Tree Hashes

To allow group members to verify that they agree on the cryptographic state of the group, this section defines a scheme for generating a hash value that represents the contents of the group’s ratchet tree and the members’ credentials.

The hash of a tree is the hash of its root node, which we define recursively, starting with the leaves. The hash of a leaf node is the hash of a LeafNodeHashInput object:

struct {
    uint8 present;
    switch (present) {
        case 0: struct{};
        case 1: T value;
} optional<T>;

struct {
    HPKEPublicKey public_key;
    Credential credential;
} LeafNodeInfo;

struct {
    uint8 hash_type = 0;
    optional<LeafNodeInfo> info;
} LeafNodeHashInput;

The public_key and credential fields represent the leaf public key and the credential for the member holding that leaf, respectively. The info field is equal to the null optional value when the leaf is blank (i.e., no member occupies that leaf).

Likewise, the hash of a parent node (including the root) is the hash of a ParentNodeHashInput struct:

struct {
    uint8 hash_type = 1;
    optional<HPKEPublicKey> public_key;
    opaque left_hash<0..255>;
    opaque right_hash<0..255>;
} ParentNodeHashInput

The left_hash and right_hash fields hold the hashes of the node’s left and right children, respectively. The public_key field holds the hash of the public key stored at this node, represented as an optional<HPKEPublicKey> object, which is null if and only if the node is blank.

6.4. Group State

Each member of the group maintains a representation of the state of the group:

struct {
    opaque group_id<0..255>;
    uint32 epoch;
    opaque tree_hash<0..255>;
    opaque transcript_hash<0..255>;
} GroupState;

The fields in this state have the following semantics:

When a new member is added to the group, an existing member of the group provides the new member with a Welcome message. The Welcome message provides the information the new member needs to initialize its GroupState.

Different group operations will have different effects on the group state. These effects are described in their respective subsections of Section 9. The following rules apply to all operations:

transcript_hash_[n] = Hash(transcript_hash_[n-1] || operation)

When a new one-member group is created (which requires no GroupOperation), the transcript_hash field is set to an all-zero vector of length Hash.length, where the Hash algorithm is defined by the ciphersuite.

6.5. Direct Paths

As described in Section 5.4, each MLS message needs to transmit node values along the direct path of a leaf. The path contains a public key for the leaf node, and a public key and encrypted secret value for intermediate nodes in the path. In both cases, the path is ordered from the leaf to the root; each node MUST be the parent of its predecessor.

struct {
    HPKEPublicKey ephemeral_key;
    opaque ciphertext<0..2^16-1>;
} HPKECiphertext;

struct {
    HPKEPublicKey public_key;
    HPKECiphertext encrypted_path_secrets<0..2^16-1>;
} DirectPathNode;

struct {
    DirectPathNode nodes<0..2^16-1>;
} DirectPath;

The length of the node\_secrets vector MUST be zero for the first node in the path. For the remaining elements in the vector, the number of ciphertexts in the node\_secrets vector MUST be equal to the length of the resolution of the corresponding copath node. Each ciphertext in the list is the encryption to the corresponding node in the resolution.

The HPKECiphertext values are computed according to the Encrypt function defined in [I-D.barnes-cfrg-hpke].

Decryption is performed in the corresponding way, using the private key of the resolution node and the ephemeral public key transmitted in the message.

6.6. Key Schedule

Group keys are derived using the HKDF-Extract and HKDF-Expand functions as defined in [RFC5869], as well as the functions defined below:

HKDF-Expand-Label(Secret, Label, Context, Length) =
    HKDF-Expand(Secret, HkdfLabel, Length)

Where HkdfLabel is specified as:

struct {
    uint16 length = Length;
    opaque label<7..255> = "mls10 " + Label;
    opaque context<0..2^32-1> = Context;
} HkdfLabel;

Derive-Secret(Secret, Label, Context) =
    HKDF-Expand-Label(Secret, Label, Hash(Context), Hash.length)

The Hash function used by HKDF is the ciphersuite hash algorithm. Hash.length is its output length in bytes. In the below diagram:

When processing a handshake message, a client combines the following information to derive new epoch secrets:

Given these inputs, the derivation of secrets for an epoch proceeds as shown in the following diagram:

               init_secret_[n-1] (or 0)
update_secret -> HKDF-Extract = epoch_secret
                     +--> Derive-Secret(., "sender data", GroupState_[n])
                     |    = sender_data_secret
                     +--> Derive-Secret(., "handshake", GroupState_[n])
                     |    = handshake_secret
                     +--> Derive-Secret(., "app", GroupState_[n])
                     |    = application_secret
                     +--> Derive-Secret(., "confirm", GroupState_[n])
                     |    = confirmation_key
               Derive-Secret(., "init", GroupState_[n])

6.7. Encryption Keys

As described in Section 8, MLS encrypts three different types of information:

The sender information used to look up the key for the content encryption is encrypted under AEAD using a random nonce and the sender_data_key which is derived from the sender_data_secret as follows:

sender_data_key =
    HKDF-Expand-Label(sender_data_secret, "sd key", "", key_length)

Each handshake message is encrypted using a key and a nonce derived from the handshake_secret for a specific sender to prevent two senders to perform in the following way:

handshake_nonce_[sender] =
    HKDF-Expand-Label(handshake_secret, "hs nonce", [sender], nonce_length)

handshake_key_[sender] =
    HKDF-Expand-Label(handshake_secret, "hs key", [sender], key_length)

Here the value [sender] represents the index of the member that will use this key to send, encoded as a uint32.

For application messages, a chain of keys is derived for each sender in a similar fashion. This allows forward secrecy at the level of application messages within and out of an epoch. A step in this chain (the second subscript) is called a “generation”.

           HKDF-Expand-Label(., "app sender", [sender], Hash.length)
                     +--> HKDF-Expand-Label(.,"nonce", "", nonce_length)
                     |    = write_nonce_[sender]_[N-1]
                     +--> HKDF-Expand-Label(.,"key", "", key_length)
                     |    = write_key_[sender]_[N-1]
           HKDF-Expand-Label(., "app sender", [sender], Hash.length)

As before the value [sender] represents the index of the member that will use this key to send, encoded as a uint32.

[[ OPEN ISSUE: The HKDF context field is left empty for now. A proper security study is needed to make sure that we do not need more information in the context to achieve the security goals.]]

[[ OPEN ISSUE: At the moment there is no contributivity of Application secrets chained from the initial one to the next generation of Epoch secret. While this seems safe because cryptographic operations using the application secrets can’t affect the group init_secret, it remains to be proven correct. ]]

The following rules apply to the usage of the secrets, keys, and nonces derived above:

7. Initialization Keys

In order to facilitate asynchronous addition of clients to a group, it is possible to pre-publish initialization keys that provide some public information about a user. UserInitKey messages provide information about a client that any existing member can use to add this client to the group asynchronously.

A UserInitKey object specifies what ciphersuites a client supports, as well as providing public keys that the client can use for key derivation and signing. The client’s identity key is intended to be stable throughout the lifetime of the group; there is no mechanism to change it. Init keys are intended to be used a very limited number of times, potentially once. (see Section 12.4). UserInitKeys also contain an identifier chosen by the client, which the client MUST assure uniquely identifies a given UserInitKey object among the set of UserInitKeys created by this client.

The init_keys array MUST have the same length as the cipher_suites array, and each entry in the init_keys array MUST be a public key for the asymmetric encryption scheme defined in the cipher_suites array and used in the HPKE construction for TreeKEM.

The whole structure is signed using the client’s identity key. A UserInitKey object with an invalid signature field MUST be considered malformed. The input to the signature computation comprises all of the fields except for the signature field.

uint8 ProtocolVersion;

struct {
    opaque user_init_key_id<0..255>;
    ProtocolVersion supported_versions<0..255>;
    CipherSuite cipher_suites<0..255>;
    HPKEPublicKey init_keys<1..2^16-1>;
    Credential credential;
    opaque signature<0..2^16-1>;
} UserInitKey;

8. Message Framing

Handshake and application messages use a common framing structure. This framing provides encryption to assure confidentiality within the group, as well as signing to authenticate the sender within the group.

The two main structures involved are MLSPlaintext and MLSCiphertext. MLSCiphertext represents a signed and encrypted message, with protections for both the content of the message and related metadata. MLSPlaintext represents a message that is only signed, and not encrypted. Applications SHOULD use MLSCiphertext to encode both application and handshake messages, but MAY transmit handshake messages encoded as MLSPlaintext objects in cases where it is necessary for the delivery service to examine such messages.

enum {
} ContentType;

struct {
    opaque group_id<0..255>;
    uint32 epoch;
    uint32 sender;
    ContentType content_type;

    select (MLSPlaintext.content_type) {
        case handshake:
            GroupOperation operation;

        case application:
            opaque application_data<0..2^32-1>;

    opaque signature<0..2^16-1>;
} MLSPlaintext;

struct {
    opaque group_id<0..255>;
    uint32 epoch;
    ContentType content_type;
    opaque sender_data_nonce<0..255>;
    opaque encrypted_sender_data<0..255>;
    opaque ciphertext<0..2^32-1>;
} MLSCiphertext;

The remainder of this section describe how to compute the signature of an MLSPlaintext object and how to convert it to an MLSCiphertext object. The overall process is as follows:

The group identifier, epoch and content_type fields are copied from the MLSPlaintext object directly. The content encryption process populates the ciphertext field of the MLSCiphertext object. The metadata encryption step populates the encrypted_sender_data field.

Decryption follows the same step in reverse: Decrypt the metadata, then the message and verify the content signature.

8.1. Metadata Encryption

The “sender data” used to look up the key for the content encryption is encrypted under AEAD using the MLSCiphertext sender_data_nonce and the sender_data_key from the keyschedule. It is encoded as an object of the following form:

struct {
    uint32 sender;
    uint32 generation;
} MLSSenderData;

The Additional Authenticated Data (AAD) for the SenderData ciphertext computation is its prefix in the MLSCiphertext, namely:

struct {
    opaque group_id<0..255>;
    uint32 epoch;
    ContentType content_type;
    opaque sender_data_nonce<0..255>;
} MLSCiphertextSenderDataAAD;

When parsing a SenderData struct as part of message decryption, the recipient MUST verify that the sender field represents an occupied leaf in the ratchet tree. In particular, the sender index value MUST be less than the number of leaves in the tree.

8.2. Content Signing and Encryption

The signature field in an MLSPlaintext object is computed using the signing private key corresponding to the credential at the leaf in the tree indicated by the sender field. The signature covers the metadata and message content, with the signature field truncated.

The ciphertext field of the MLSCiphertext object is produced by supplying the inputs described below to the AEAD function specified by the ciphersuite in use. The plaintext input contains content and signature of the MLSPlaintext, plus optional padding. These values are encoded in the following form:

struct {
    opaque content[length\_of\_content];
    uint8 signature[MLSInnerPlaintext.sig_len];
    uint16 sig_len;
    uint8  marker = 1;
    uint8  zero\_padding[length\_of\_padding];
} MLSCiphertextContent;

The key and nonce used for the encryption of the message depend on the content type of the message. The sender chooses the handshake key for a handshake message or an ununsed generation from its (per-sender) application key chain for the current epoch, according to the type of message being encrypted.

The Additional Authenticated Data (AAD) input to the encryption contains an object of the following form, with the values used to identify the key and nonce:

struct {
    opaque group_id<0..255>;
    uint32 epoch;
    ContentType content_type;
    opaque sender_data_nonce<0..255>;
    opaque encrypted_sender_data<0..255>;
} MLSCiphertextContentAAD;

The ciphertext field of the MLSCiphertext object is produced by supplying these inputs to the AEAD function specified by the ciphersuite in use.

9. Handshake Messages

Over the lifetime of a group, its state will change for:

In MLS, these changes are accomplished by broadcasting “handshake” messages to the group. Note that unlike TLS and DTLS, there is not a consolidated handshake phase to the protocol. Rather, handshake messages are exchanged throughout the lifetime of a group, whenever a change is made to the group state. This means an unbounded number of interleaved application and handshake messages.

An MLS handshake message encapsulates a specific GroupOperation message that accomplishes a change to the group state. It is carried in an MLSPlaintext message that provides a signature by the sender of the message. Applications may choose to send handshake messages in encrypted form, as MLSCiphertext messages.

enum {
} GroupOperationType;

struct {
    GroupOperationType msg_type;
    select (GroupOperation.msg_type) {
        case init:      Init;
        case add:       Add;
        case update:    Update;
        case remove:    Remove;
    opaque confirmation<0..255>;
} GroupOperation;

The high-level flow for processing a handshake message is as follows:

  1. If the handshake message is encrypted (i.e., encoded as an MLSCiphertext object), decrypt it following the procedures described in Section 8.
  2. Verify that the epoch field of enclosing MLSPlaintext message is equal the epoch field of the current GroupState object.
  3. Verify that the signature on the MLSPlaintext message verifies using the public key from the credential stored at the leaf in the tree indicated by the sender field.
  4. Use the operation message to produce an updated, provisional GroupState object incorporating the proposed changes.
  5. Use the confirmation_key for the new epoch to compute the confirmation MAC for this message, as described below, and verify that it is the same as the confirmation field in the GroupOperation object.
  6. If the the above checks are successful, consider the updated GroupState object as the current state of the group.

The confirmation value confirms that the members of the group have arrived at the same state of the group:

GroupOperation.confirmation =
    HMAC(confirmation_key, GroupState.transcript\_hash)

HMAC [RFC2104] uses the Hash algorithm for the ciphersuite in use. Sign uses the signature algorithm indicated by the signer’s credential.

[[ OPEN ISSUE: It is not possible for the recipient of a handshake message to verify that ratchet tree information in the message is accurate, because each node can only compute the secret and private key for nodes in its direct path. This creates the possibility that a malicious participant could cause a denial of service by sending a handshake message with invalid values for public keys in the ratchet tree. ]]

9.1. Init

[[ OPEN ISSUE: Direct initialization is currently undefined. A client can create a group by initializing its own state to reflect a group including only itself, then adding the initial members. This has computation and communication complexity O(N log N) instead of the O(N) complexity of direct initialization. ]]

9.2. Add

In order to add a new member to the group, an existing member of the group must take two actions:

  1. Send a Welcome message to the new member
  2. Send an Add message to the group (including the new member)

The Welcome message contains the information that the new member needs to initialize a GroupState object that can be updated to the current state using the Add message. This information is encrypted for the new member using HPKE. The recipient key pair for the HPKE encryption is the one included in the indicated UserInitKey, corresponding to the indicated ciphersuite.

struct {
    HPKEPublicKey public_key;
    optional<Credential> credential;
} RatchetNode;

struct {
    ProtocolVersion version;
    opaque group_id<0..255>;
    uint32 epoch;
    optional<RatchetNode> tree<1..2^32-1>;
    opaque transcript_hash<0..255>;
    opaque init_secret<0..255>;
} WelcomeInfo;

struct {
    opaque user_init_key_id<0..255>;
    CipherSuite cipher_suite;
    HPKECiphertext encrypted_welcome_info;
} Welcome;

In the description of the tree as a list of nodes, the credential field for a node MUST be populated if and only if that node is a leaf in the tree.

Note that the init_secret in the Welcome message is the init_secret at the output of the key schedule diagram in Section 6.6. That is, if the epoch value in the Welcome message is n, then the init_secret value is init_secret_[n]. The new member can combine this init secret with the update secret transmitted in the corresponding Add message to get the epoch secret for the epoch in which it is added. No secrets from prior epochs are revealed to the new member.

Since the new member is expected to process the Add message for itself, the Welcome message should reflect the state of the group before the new user is added. The sender of the Welcome message can simply copy all fields from their GroupState object.

[[ OPEN ISSUE: The Welcome message needs to be synchronized in the same way as the Add. That is, the Welcome should be sent only if the Add succeeds, and is not in conflict with another, simultaneous Add. ]]

An Add message provides existing group members with the information they need to update their GroupState with information about the new member:

struct {
    uint32 index;
    UserInitKey init_key;
    opaque welcome_info_hash<0..255>;
} Add;

The index field indicates where in the tree the new member should be added. The new member can be added at an existing, blank leaf node, or at the right edge of the tree. In any case, the index value MUST satisfy 0 <= index <= n, where n is the size of the group. The case index = n indicates an add at the right edge of the tree). If index < n and the leaf node at position index is not blank, then the recipient MUST reject the Add as malformed.

The welcome_info_hash field contains a hash of the WelcomeInfo object sent in a Welcome message to the new member.

A group member generates this message by requesting a UserInitKey from the directory for the user to be added, and encoding it into an Add message.

The client joining the group processes Welcome and Add messages together as follows:

An existing member receiving a Add message first verifies the signature on the message, then updates its state as follows:

The update secret resulting from this change is an all-zero octet string of length Hash.length.

After processing an Add message, the new member SHOULD send an Update immediately to update its key. This will help to limit the tree structure degrading into subtrees, and thus maintain the protocol’s efficiency.

9.3. Update

An Update message is sent by a group member to update its leaf secret and key pair. This operation provides post-compromise security with regard to the member’s prior leaf private key.

struct {
    DirectPath path;
} Update;

The sender of an Update message creates it in the following way:

A member receiving a Update message first verifies the signature on the message, then updates its state as follows:

The update secret resulting from this change is the path secret for the root node of the ratchet tree.

9.4. Remove

A Remove message is sent by a group member to remove one or more other members from the group. A member MUST NOT use a Remove message to remove themselves from the group. If a member of a group receives a Remove message where the removed index is equal to the signer index, the recipient MUST reject the message as malformed.

struct {
    uint32 removed;
    DirectPath path;
} Remove;

The sender of a Remove message generates it as as follows:

A member receiving a Remove message first verifies the signature on the message. The member then updates its state as follows:

Note that, in step 4, there must be at least one non-null element in the tree, since any valid GroupState must have the current member in the tree and self-removal is prohibited. The same reasoning justifies the existence of a non-blank leaf in the ratchet tree in step 5.

The update secret resulting from this change is the path secret computed for the root node of the ratchet tree in the first step.

10. Sequencing of State Changes

[[ OPEN ISSUE: This section has an initial set of considerations regarding sequencing. It would be good to have some more detailed discussion, and hopefully have a mechanism to deal with this issue. ]]

Each handshake message is premised on a given starting state, indicated in its prior_epoch field. If the changes implied by a handshake messages are made starting from a different state, the results will be incorrect.

This need for sequencing is not a problem as long as each time a group member sends a handshake message, it is based on the most current state of the group. In practice, however, there is a risk that two members will generate handshake messages simultaneously, based on the same state.

When this happens, there is a need for the members of the group to deconflict the simultaneous handshake messages. There are two general approaches:

As long as handshake messages cannot be merged, there is a risk of starvation. In a sufficiently busy group, a given member may never be able to send a handshake message, because he always loses to other members. The degree to which this is a practical problem will depend on the dynamics of the application.

It might be possible, because of the non-contributivity of intermediate nodes, that update messages could be applied one after the other without the Delivery Service having to reject any handshake message, which would make MLS more resilient regarding the concurrency of handshake messages. The Messaging system can decide to choose the order for applying the state changes. Note that there are certain cases (if no total ordering is applied by the Delivery Service) where the ordering is important for security, ie. all updates must be executed before removes.

Regardless of how messages are kept in sequence, implementations MUST only update their cryptographic state when valid handshake messages are received. Generation of handshake messages MUST be stateless, since the endpoint cannot know at that time whether the change implied by the handshake message will succeed or not.

10.1. Server-Enforced Ordering

With this approach, the delivery service ensures that incoming messages are added to an ordered queue and outgoing messages are dispatched in the same order. The server is trusted to resolve conflicts during race-conditions (when two members send a message at the same time), as the server doesn’t have any additional knowledge thanks to the confidentiality of the messages.

Messages should have a counter field sent in clear-text that can be checked by the server and used for tie-breaking. The counter starts at 0 and is incremented for every new incoming message. If two group members send a message with the same counter, the first message to arrive will be accepted by the server and the second one will be rejected. The rejected message needs to be sent again with the correct counter number.

To prevent counter manipulation by the server, the counter’s integrity can be ensured by including the counter in a signed message envelope.

This applies to all messages, not only state changing messages.

10.2. Client-Enforced Ordering

Order enforcement can be implemented on the client as well, one way to achieve it is to use a two step update protocol: the first client sends a proposal to update and the proposal is accepted when it gets 50%+ approval from the rest of the group, then it sends the approved update. Clients which didn’t get their proposal accepted, will wait for the winner to send their update before retrying new proposals.

While this seems safer as it doesn’t rely on the server, it is more complex and harder to implement. It also could cause starvation for some clients if they keep failing to get their proposal accepted.

10.3. Merging Updates

It is possible in principle to partly address the problem of concurrent changes by having the recipients of the changes merge them, rather than having the senders retry. Because the value of intermediate node is determined by its last updated child, updates can be merged by recipients as long as the recipients agree on an order – the only question is which node was last updated.

Recall that the processing of an update proceeds in two steps:

  1. Compute updated secret values by hashing up the tree
  2. Update the tree with the new secret and public values

To merge an ordered list of updates, a recipient simply performs these updates in the specified order.

For example, suppose we have a tree in the following configuration:

     /       \
  KDF(B)    KDF(D)
  /  \      /  \
 A    B    C    D

Now suppose B and C simultaneously decide to update to X and Y, respectively. They will send out updates of the following form:

  Update from B      Update from C
  =============      =============
      KDF(KDF(X))             KDF(KDF(Y))
     /                         \
  KDF(X)                        KDF(Y)
     \                         /
      X                       Y

Assuming that the ordering agreed by the group says that B’s update should be processed before C’s, the other members in the group will overwrite the root value for B with the root value from C, and all arrive at the following state:

     /       \
  KDF(X)    KDF(Y)
  /  \      /  \
 A    X    Y    D

11. Application Messages

The primary purpose of the handshake protocol is to provide an authenticated group key exchange to clients. In order to protect Application messages sent among those members of a group, the Application secret provided by the handshake key schedule is used to derive encryption keys for the Message Protection Layer.

Application messages MUST be protected with the Authenticated-Encryption with Associated-Data (AEAD) encryption scheme associated with the MLS ciphersuite. Note that “Authenticated” in this context does not mean messages are known to be sent by a specific client but only from a legitimate member of the group. To authenticate a message from a particular member, signatures are required. Handshake messages MUST use asymmetric signatures to strongly authenticate the sender of a message.

Each member maintains their own chain of Application secrets, where the first one is derived based on a secret chained from the Epoch secret. As shown in Section 6.6, the initial Application secret is bound to the identity of each client to avoid collisions and allow support for decryption of reordered messages.

Subsequent Application secrets MUST be rotated for each message sent in order to provide stronger cryptographic security guarantees. The Application Key Schedule use this rotation to generate fresh AEAD encryption keys and nonces used to encrypt and decrypt future Application messages. In all cases, a participant MUST NOT encrypt more than expected by the security bounds of the AEAD scheme used.

Note that each change to the Group through a handshake message will cause a change of the group Secret. Hence this change MUST be applied before encrypting any new Application message. This is required for confidentiality reasons in order for members to avoid receiving messages from the group after leaving, being added to, or excluded from the group.

11.1. Message Encryption and Decryption

The group members MUST use the AEAD algorithm associated with the negotiated MLS ciphersuite to AEAD encrypt and decrypt their Application messages according to the Message Framing section.

The group identifier and epoch allow a device to know which group secrets should be used and from which Epoch secret to start computing other secrets and keys. The sender identifier is used to derive the member’s Application secret chain from the initial group Application secret. The application generation field is used to determine which Application secret should be used from the chain to compute the correct AEAD keys before performing decryption.

Application messages SHOULD be padded to provide some resistance against traffic analysis techniques over encrypted traffic. [CLINIC] [HCJ16] While MLS might deliver the same payload less frequently across a lot of ciphertexts than traditional web servers, it might still provide the attacker enough information to mount an attack. If Alice asks Bob: “When are we going to the movie ?” the answer “Wednesday” might be leaked to an adversary by the ciphertext length. An attacker expecting Alice to answer Bob with a day of the week might find out the plaintext by correlation between the question and the length.

Similarly to TLS 1.3, if padding is used, the MLS messages MUST be padded with zero-valued bytes before AEAD encryption. Upon AEAD decryption, the length field of the plaintext is used to compute the number of bytes to be removed from the plaintext to get the correct data. As the padding mechanism is used to improve protection against traffic analysis, removal of the padding SHOULD be implemented in a “constant-time” manner at the MLS layer and above layers to prevent timing side-channels that would provide attackers with information on the size of the plaintext. The padding length length_of_padding can be chosen at the time of the message encryption by the sender. Recipients can calculate the padding size from knowing the total size of the ApplicationPlaintext and the length of the content.

[[ TODO: A preliminary formal security analysis has yet to be performed on this authentication scheme.]]

[[ OPEN ISSUE: Currently, the group identifier, epoch and generation are contained as meta-data of the Signature. A different solution could be to include the GroupState instead, if more information is required to achieve the security goals regarding cross-group attacks. ]]

[[ OPEN ISSUE: Should the padding be required for handshake messages ? Can an adversary get more than the position of a participant in the tree without padding ? Should the base ciphertext block length be negotiated or is is reasonable to allow to leak a range for the length of the plaintext by allowing to send a variable number of ciphertext blocks ? ]]

11.2. Delayed and Reordered Application messages

Since each Application message contains the group identifier, the epoch and a message counter, a client can receive messages out of order. If they are able to retrieve or recompute the correct AEAD decryption key from currently stored cryptographic material clients can decrypt these messages.

For usability, MLS clients might be required to keep the AEAD key and nonce for a certain amount of time to retain the ability to decrypt delayed or out of order messages, possibly still in transit while a decryption is being done.

[[TODO: Describe here or in the Architecture spec the details. Depending on which Secret or key is kept alive, the security guarantees will vary.]]

12. Security Considerations

The security goals of MLS are described in [I-D.ietf-mls-architecture]. We describe here how the protocol achieves its goals at a high level, though a complete security analysis is outside of the scope of this document.

12.1. Confidentiality of the Group Secrets

Group secrets are derived from (i) previous group secrets, and (ii) the root key of a ratcheting tree. Only group members know their leaf private key in the group, therefore, the root key of the group’s ratcheting tree is secret and thus so are all values derived from it.

Initial leaf keys are known only by their owner and the group creator, because they are derived from an authenticated key exchange protocol. Subsequent leaf keys are known only by their owner. [[TODO: or by someone who replaced them.]]

Note that the long-term identity keys used by the protocol MUST be distributed by an “honest” authentication service for clients to authenticate their legitimate peers.

12.2. Authentication

There are two forms of authentication we consider. The first form considers authentication with respect to the group. That is, the group members can verify that a message originated from one of the members of the group. This is implicitly guaranteed by the secrecy of the shared key derived from the ratcheting trees: if all members of the group are honest, then the shared group key is only known to the group members. By using AEAD or appropriate MAC with this shared key, we can guarantee that a member in the group (who knows the shared secret key) has sent a message.

The second form considers authentication with respect to the sender, meaning the group members can verify that a message originated from a particular member of the group. This property is provided by digital signatures on the messages under identity keys.

[[ OPEN ISSUE: Signatures under the identity keys, while simple, have the side-effect of preclude deniability. We may wish to allow other options, such as (ii) a key chained off of the identity key, or (iii) some other key obtained through a different manner, such as a pairwise channel that provides deniability for the message contents.]]

12.3. Forward and post-compromise security

Message encryption keys are derived via a hash ratchet, which provides a form of forward secrecy: learning a message key does not reveal previous message or root keys. Post-compromise security is provided by Update operations, in which a new root key is generated from the latest ratcheting tree. If the adversary cannot derive the updated root key after an Update operation, it cannot compute any derived secrets.

12.4. Init Key Reuse

Initialization keys are intended to be used only once and then deleted. Reuse of init keys is not believed to be inherently insecure [dhreuse], although it can complicate protocol analyses.

13. IANA Considerations

TODO: Registries for protocol parameters, e.g., ciphersuites

14. Contributors

15. References

15.1. Normative References

[I-D.barnes-cfrg-hpke] Barnes, R. and K. Bhargavan, "Hybrid Public Key Encryption", Internet-Draft draft-barnes-cfrg-hpke-01, March 2019.
[IEEE1363] "IEEE Standard Specifications for Password-Based Public-Key Cryptographic Techniques", IEEE standard, DOI 10.1109/ieeestd.2009.4773330, n.d..
[RFC2104] Krawczyk, H., Bellare, M. and R. Canetti, "HMAC: Keyed-Hashing for Message Authentication", RFC 2104, DOI 10.17487/RFC2104, February 1997.
[RFC2119] Bradner, S., "Key words for use in RFCs to Indicate Requirement Levels", BCP 14, RFC 2119, DOI 10.17487/RFC2119, March 1997.
[RFC5116] McGrew, D., "An Interface and Algorithms for Authenticated Encryption", RFC 5116, DOI 10.17487/RFC5116, January 2008.
[RFC5869] Krawczyk, H. and P. Eronen, "HMAC-based Extract-and-Expand Key Derivation Function (HKDF)", RFC 5869, DOI 10.17487/RFC5869, May 2010.
[RFC7748] Langley, A., Hamburg, M. and S. Turner, "Elliptic Curves for Security", RFC 7748, DOI 10.17487/RFC7748, January 2016.
[RFC8174] Leiba, B., "Ambiguity of Uppercase vs Lowercase in RFC 2119 Key Words", BCP 14, RFC 8174, DOI 10.17487/RFC8174, May 2017.
[RFC8446] Rescorla, E., "The Transport Layer Security (TLS) Protocol Version 1.3", RFC 8446, DOI 10.17487/RFC8446, August 2018.
[X962] ANSI, "Public Key Cryptography For The Financial Services Industry: The Elliptic Curve Digital Signature Algorithm (ECDSA)", ANSI X9.62, 1998.

15.2. Informative References

[art] Cohn-Gordon, K., Cremers, C., Garratt, L., Millican, J. and K. Milner, "On Ends-to-Ends Encryption: Asynchronous Group Messaging with Strong Security Guarantees", January 2018.
[CLINIC] Miller, B., Huang, L., Joseph, A. and J. Tygar, "I Know Why You Went to the Clinic: Risks and Realization of HTTPS Traffic Analysis", Privacy Enhancing Technologies pp. 143-163, DOI 10.1007/978-3-319-08506-7_8, 2014.
[dhreuse] Menezes, A. and B. Ustaoglu, "On reusing ephemeral keys in Diffie-Hellman key agreement protocols", International Journal of Applied Cryptography Vol. 2, pp. 154, DOI 10.1504/ijact.2010.038308, 2010.
[doubleratchet] Cohn-Gordon, K., Cremers, C., Dowling, B., Garratt, L. and D. Stebila, "A Formal Security Analysis of the Signal Messaging Protocol", 2017 IEEE European Symposium on Security and Privacy (EuroS&P), DOI 10.1109/eurosp.2017.27, April 2017.
[HCJ16] Husák, M., Čermák, M., Jirsík, T. and P. Čeleda, "HTTPS traffic analysis and client identification using passive SSL/TLS fingerprinting", EURASIP Journal on Information Security Vol. 2016, DOI 10.1186/s13635-016-0030-7, February 2016.
[I-D.ietf-trans-rfc6962-bis] Laurie, B., Langley, A., Kasper, E., Messeri, E. and R. Stradling, "Certificate Transparency Version 2.0", Internet-Draft draft-ietf-trans-rfc6962-bis-31, February 2019.
[keyagreement] Barker, E., Chen, L., Roginsky, A. and M. Smid, "Recommendation for Pair-Wise Key Establishment Schemes Using Discrete Logarithm Cryptography", National Institute of Standards and Technology report, DOI 10.6028/nist.sp.800-56ar2, May 2013.
[signal] Perrin(ed), T. and M. Marlinspike, "The Double Ratchet Algorithm", n.d..

Appendix A. Tree Math

One benefit of using left-balanced trees is that they admit a simple flat array representation. In this representation, leaf nodes are even-numbered nodes, with the n-th leaf at 2*n. Intermediate nodes are held in odd-numbered nodes. For example, a 11-element tree has the following structure:

         X                       X                       X
   X           X           X           X           X
X     X     X     X     X     X     X     X     X     X     X
0  1  2  3  4  5  6  7  8  9 10 11 12 13 14 15 16 17 18 19 20

This allows us to compute relationships between tree nodes simply by manipulating indices, rather than having to maintain complicated structures in memory, even for partial trees. The basic rule is that the high-order bits of parent and child nodes have the following relation (where x is an arbitrary bit string):

parent=01x => left=00x, right=10x

The following python code demonstrates the tree computations necessary for MLS. Test vectors can be derived from the diagram above.

# The largest power of 2 less than n.  Equivalent to:
#   int(math.floor(math.log(x, 2)))
def log2(x):
    if x == 0:
        return 0

    k = 0
    while (x >> k) > 0:
        k += 1
    return k-1

# The level of a node in the tree.  Leaves are level 0, their
# parents are level 1, etc.  If a node's children are at different
# level, then its level is the max level of its children plus one.
def level(x):
    if x & 0x01 == 0:
        return 0

    k = 0
    while ((x >> k) & 0x01) == 1:
        k += 1
    return k

# The number of nodes needed to represent a tree with n leaves
def node_width(n):
    return 2*(n - 1) + 1

# The index of the root node of a tree with n leaves
def root(n):
    w = node_width(n)
    return (1 << log2(w)) - 1

# The left child of an intermediate node.  Note that because the
# tree is left-balanced, there is no dependency on the size of the
# tree.  The child of a leaf node is itself.
def left(x):
    k = level(x)
    if k == 0:
        return x

    return x ^ (0x01 << (k - 1))

# The right child of an intermediate node.  Depends on the size of
# the tree because the straightforward calculation can take you
# beyond the edge of the tree.  The child of a leaf node is itself.
def right(x, n):
    k = level(x)
    if k == 0:
        return x

    r = x ^ (0x03 << (k - 1))
    while r >= node_width(n):
        r = left(r)
    return r

# The immediate parent of a node.  May be beyond the right edge of
# the tree.
def parent_step(x):
    k = level(x)
    b = (x >> (k + 1)) & 0x01
    return (x | (1 << k)) ^ (b << (k + 1))

# The parent of a node.  As with the right child calculation, have
# to walk back until the parent is within the range of the tree.
def parent(x, n):
    if x == root(n):
        return x

    p = parent_step(x)
    while p >= node_width(n):
        p = parent_step(p)
    return p

# The other child of the node's parent.  Root's sibling is itself.
def sibling(x, n):
    p = parent(x, n)
    if x < p:
        return right(p, n)
    elif x > p:
        return left(p)

    return p

# The direct path of a node, ordered from the root
# down, not including the root or the terminal node
def direct_path(x, n):
    d = []
    p = parent(x, n)
    r = root(n)
    while p != r:
        p = parent(p, n)
    return d

# The copath of the node is the siblings of the nodes on its direct
# path (including the node itself)
def copath(x, n):
    d = dirpath(x, n)
    if x != sibling(x, n):

    return [sibling(y, n) for y in d]

# Frontier is is the list of full subtrees, from left to right.  A
# balance binary tree with n leaves has a full subtree for every
# power of two where n has a bit set, with the largest subtrees
# furthest to the left.  For example, a tree with 11 leaves has full
# subtrees of size 8, 2, and 1.
def frontier(n):
    st = [1 << k for k in range(log2(n) + 1) if n & (1 << k) != 0]
    st = reversed(st)

    base = 0
    f = []
    for size in st:
        f.append(root(size) + base)
        base += 2*size
    return f

# Leaves are in even-numbered nodes
def leaves(n):
    return [2*i for i in range(n)]

# The resolution of a node is the collection of non-blank
# descendants of this node.  Here the tree is represented by a list
# of nodes, where blank nodes are represented by None
def resolve(tree, x, n):
    if tree[x] != None:
        return [x]

    if level(x) == 0:
        return []

    L = resolve(tree, left(x), n)
    R = resolve(tree, right(x, n), n)
    return L + R

Authors' Addresses

Richard Barnes Cisco EMail:
Jon Millican Facebook EMail:
Emad Omara Google EMail:
Katriel Cohn-Gordon University of Oxford EMail:
Raphael Robert Wire EMail: