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Versions: (draft-barnes-mls-protocol) 00 01 02 03 04 05

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


              The Messaging Layer Security (MLS) Protocol
                       draft-ietf-mls-protocol-05

Abstract

   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.

   Internet-Drafts are working documents of the Internet Engineering
   Task Force (IETF).  Note that other groups may also distribute
   working documents as Internet-Drafts.  The list of current Internet-
   Drafts is at https://datatracker.ietf.org/drafts/current/.

   Internet-Drafts are draft documents valid for a maximum of six months
   and may be updated, replaced, or obsoleted by other documents at any
   time.  It is inappropriate to use Internet-Drafts as reference
   material or to cite them other than as "work in progress."

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






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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
   (https://trustee.ietf.org/license-info) 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  . . . . . . . . . . . . . . . . . . . . . . . .   3
     1.1.  Change Log  . . . . . . . . . . . . . . . . . . . . . . .   4
   2.  Terminology . . . . . . . . . . . . . . . . . . . . . . . . .   5
   3.  Basic Assumptions . . . . . . . . . . . . . . . . . . . . . .   6
   4.  Protocol Overview . . . . . . . . . . . . . . . . . . . . . .   7
   5.  Ratchet Trees . . . . . . . . . . . . . . . . . . . . . . . .  10
     5.1.  Tree Computation Terminology  . . . . . . . . . . . . . .  10
     5.2.  Ratchet Tree Nodes  . . . . . . . . . . . . . . . . . . .  13
     5.3.  Views of a Ratchet Tree . . . . . . . . . . . . . . . . .  14
     5.4.  Ratchet Tree Updates  . . . . . . . . . . . . . . . . . .  15
     5.5.  Synchronizing Views of the Tree . . . . . . . . . . . . .  16
   6.  Cryptographic Objects . . . . . . . . . . . . . . . . . . . .  17
     6.1.  Curve25519, SHA-256, and AES-128-GCM  . . . . . . . . . .  18
       6.1.1.  P-256, SHA-256, and AES-128-GCM . . . . . . . . . . .  18
     6.2.  Credentials . . . . . . . . . . . . . . . . . . . . . . .  19
     6.3.  Tree Hashes . . . . . . . . . . . . . . . . . . . . . . .  20
     6.4.  Group State . . . . . . . . . . . . . . . . . . . . . . .  21
     6.5.  Direct Paths  . . . . . . . . . . . . . . . . . . . . . .  23
     6.6.  Key Schedule  . . . . . . . . . . . . . . . . . . . . . .  23
     6.7.  Encryption Keys . . . . . . . . . . . . . . . . . . . . .  25
   7.  Initialization Keys . . . . . . . . . . . . . . . . . . . . .  27
   8.  Message Framing . . . . . . . . . . . . . . . . . . . . . . .  28
     8.1.  Metadata Encryption . . . . . . . . . . . . . . . . . . .  30
     8.2.  Content Signing and Encryption  . . . . . . . . . . . . .  31
   9.  Handshake Messages  . . . . . . . . . . . . . . . . . . . . .  31
     9.1.  Init  . . . . . . . . . . . . . . . . . . . . . . . . . .  33
     9.2.  Add . . . . . . . . . . . . . . . . . . . . . . . . . . .  33
     9.3.  Update  . . . . . . . . . . . . . . . . . . . . . . . . .  36
     9.4.  Remove  . . . . . . . . . . . . . . . . . . . . . . . . .  37
   10. Sequencing of State Changes . . . . . . . . . . . . . . . . .  37
     10.1.  Server-Enforced Ordering . . . . . . . . . . . . . . . .  38



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     10.2.  Client-Enforced Ordering . . . . . . . . . . . . . . . .  39
     10.3.  Merging Updates  . . . . . . . . . . . . . . . . . . . .  39
   11. Application Messages  . . . . . . . . . . . . . . . . . . . .  40
     11.1.  Message Encryption and Decryption  . . . . . . . . . . .  41
     11.2.  Delayed and Reordered Application messages . . . . . . .  42
   12. Security Considerations . . . . . . . . . . . . . . . . . . .  42
     12.1.  Confidentiality of the Group Secrets . . . . . . . . . .  43
     12.2.  Authentication . . . . . . . . . . . . . . . . . . . . .  43
     12.3.  Forward and post-compromise security . . . . . . . . . .  43
     12.4.  Init Key Reuse . . . . . . . . . . . . . . . . . . . . .  44
   13. IANA Considerations . . . . . . . . . . . . . . . . . . . . .  44
   14. Contributors  . . . . . . . . . . . . . . . . . . . . . . . .  44
   15. References  . . . . . . . . . . . . . . . . . . . . . . . . .  45
     15.1.  Normative References . . . . . . . . . . . . . . . . . .  45
     15.2.  Informative References . . . . . . . . . . . . . . . . .  46
   Appendix A.  Tree Math  . . . . . . . . . . . . . . . . . . . . .  47
   Authors' Addresses  . . . . . . . . . . . . . . . . . . . . . . .  50

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 https://github.com/mlswg/mls-protocol.
   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



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   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

   RFC EDITOR PLEASE DELETE THIS SECTION.

   draft-05

   o  Common framing for handshake and application messages (*)

   o  Handshake message encryption (*)

   o  Convert from literal state to a commitment via the "tree hash" (*)

   o  Add credentials to the tree and remove the "roster" concept (*)

   o  Remove the secret field from tree node values

   draft-04

   o  Updating the language to be similar to the Architecture document

   o  ECIES is now renamed in favor of HPKE (*)

   o  Using a KDF instead of a Hash in TreeKEM (*)

   draft-03

   o  Added ciphersuites and signature schemes (*)

   o  Re-ordered fields in UserInitKey to make parsing easier (*)

   o  Fixed inconsistencies between Welcome and GroupState (*)

   o  Added encryption of the Welcome message (*)

   draft-02



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   o  Removed ART (*)

   o  Allowed partial trees to avoid double-joins (*)

   o  Added explicit key confirmation (*)

   draft-01

   o  Initial description of the Message Protection mechanism. (*)

   o  Initial specification proposal for the Application Key Schedule
      using the per-participant chaining of the Application Secret
      design. (*)

   o  Initial specification proposal for an encryption mechanism to
      protect Application Messages using an AEAD scheme. (*)

   o  Initial specification proposal for an authentication mechanism of
      Application Messages using signatures. (*)

   o  Initial specification proposal for a padding mechanism to
      improving protection of Application Messages against traffic
      analysis. (*)

   o  Inversion of the Group Init Add and Application Secret derivations
      in the Handshake Key Schedule to be ease chaining in case we
      switch design. (*)

   o  Removal of the UserAdd construct and split of GroupAdd into Add
      and Welcome messages (*)

   o  Initial proposal for authenticating handshake messages by signing
      over group state and including group state in the key schedule (*)

   o  Added an appendix with example code for tree math

   o  Changed the ECIES mechanism used by TreeKEM so that it uses nonces
      generated from the shared secret

   draft-00

   o  Initial adoption of draft-barnes-mls-protocol-01 as a WG item.

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



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   14 [RFC2119] [RFC8174] when, and only when, they appear in all
   capitals, as shown here.

   Client:  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.

   Group:  A collection of clients with shared cryptographic state.

   Member:  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:

   o  A long-term identity key provider which allows clients to
      authenticate protocol messages in a group.  These keys MUST be
      kept for the lifetime of the group as there is no mechanism in the
      protocol for changing a client's identity key.

   o  A broadcast channel, for each group, which will relay a message to
      all members of a group.  For the most part, we assume that this
      channel delivers messages in the same order to all participants.
      (See Section 10 for further considerations.)





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   o  A directory to which clients can publish initialization keys and
      download initialization keys for other participants.

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:

   o  Adding a member, initiated by a current member;

   o  Updating the leaf secret of a member;

   o  Removing a member.

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













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                                                             Group
   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.




























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                                                                  Group
   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.




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                                                             Group
   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.

                                                             Group
   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



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   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:

   o  The direct path of C is (C, CD, ABCD)

   o  The copath of C is (D, AB, EFG)

   o  The frontier of the tree is (ABCD, EF, G)













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               ABCDEFG
              /      \
             /        \
            /          \
        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:

   o  0 = A

   o  1 = AB

   o  2 = B

   o  3 = ABCD

   o  4 = C

   o  5 = CD

   o  6 = D

   o  7 = ABCDEFG

   o  8 = E

   o  9 = EF

   o  10 = F

   o  11 = EFG

   o  12 = G

   (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].)



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   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:

   o  0 = A

   o  1 = B

   o  2 = C

   o  3 = D

   o  4 = E

   o  5 = F

   o  6 = G

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:

   o  An HPKE ciphersuite, which specifies a Key Encapsulation Method
      (KEM), an AEAD encryption scheme, and a hash function

   o  A Derive-Key-Pair function that produces an asymmetric key pair
      for the specified KEM from a symmetric secret, using the specified
      hash function.

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

   o  A private key (only within direct path, see below)

   o  A public key

   o  A credential (only for leaf nodes)

   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




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   of the node.  The nodes in a resolution are ordered according to
   their indices.

   o  The resolution of a non-blank node is a one element list
      containing the node itself

   o  The resolution of a blank leaf node is the empty list

   o  The resolution of a blank intermediate node is the result of
      concatinating the resolution of its left child with the resolution
      of its right child, in that order

   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:

   o  The resolution of node 5 is the list [CD]

   o  The resolution of node 2 is the empty list []

   o  The resolution of node 3 is the list [A, CD]

   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




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   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:

         G
        / \
       /   \
      /     \
     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:









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       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[2]
            /     \
        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:

   o  The public key for the node

   o  Zero or more encrypted copies of the path secret corresponding to
      the node

   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-



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       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])                  |
             |            |                                  |
             | pk(ns[0])  |                                  |
             +------------+----------------------------------+

   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:

   o  A hash function

   o  A Diffie-Hellman finite-field group or elliptic curve

   o  An AEAD encryption algorithm [RFC5116]

   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.





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   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 {
       ecdsa_secp256r1_sha256(0x0403),
       ed25519(0x0807),
       (0xFFFF)
   } SignatureScheme;

   enum {
       P256_SHA256_AES128GCM(0x0000),
       X25519_SHA256_AES128GCM(0x0001),
       (0xFFFF)
   } CipherSuite;

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

   This ciphersuite uses the following primitives:

   o  Hash function: SHA-256

   o  Diffie-Hellman group: Curve25519 [RFC7748]

   o  AEAD: AES-128-GCM

   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:

   o  Hash function: SHA-256



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   o  Diffie-Hellman group: secp256r1 (NIST P-256)

   o  AEAD: AES-128-GCM

   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 5.6.2.3 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:

   o  The public key of a signature key pair

   o  The identity of the holder of the private key

   o  The signature scheme that the holder will use to sign MLS messages




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

   enum {
       basic(0),
       x509(1),
       (255)
   } CredentialType;

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

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

           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:















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   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:








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   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:

   o  The "group_id" field is an application-defined identifier for the
      group.

   o  The "epoch" field represents the current version of the group key.

   o  The "tree_hash" field contains a commitment to the contents of the
      group's rachet tree and the credentials for the members of the
      group, as described in Section 6.3.

   o  The "transcript_hash" field contains the list of "GroupOperation"
      messages that led to this state.

   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:

   o  The "group_id" field is constant

   o  The "epoch" field increments by one for each GroupOperation that
      is processed

   o  The "tree_hash" is updated to represent the current tree and
      credentials

   o  The "transcript_hash" is updated by a GroupOperation message
      "operation" in the following way:

   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.




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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:









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   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:

   o  HKDF-Extract takes its salt argument from the top and its IKM
      argument from the left

   o  Derive-Secret takes its Secret argument from the incoming arrow

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

   o  The init secret from the previous epoch

   o  The update secret for the current epoch

   o  The GroupState object for current epoch

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


















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               init_secret_[n-1] (or 0)
                     |
                     V
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
                     |
                     V
               Derive-Secret(., "init", GroupState_[n])
                     |
                     V
               init_secret_[n]

6.7.  Encryption Keys

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

   o  Metadata (sender information)

   o  Handshake messages

   o  Application messages

   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:






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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".

           application_secret
                     |
                     V
           HKDF-Expand-Label(., "app sender", [sender], Hash.length)
                     |
                     V
           application_secret_[sender]_[0]
                     |
                    ...
                     |
                     V
           application_secret_[sender]_[N-1]
                     |
                     +--> HKDF-Expand-Label(.,"nonce", "", nonce_length)
                     |    = write_nonce_[sender]_[N-1]
                     |
                     +--> HKDF-Expand-Label(.,"key", "", key_length)
                     |    = write_key_[sender]_[N-1]
                     V
           HKDF-Expand-Label(., "app sender", [sender], Hash.length)
                     |
                     V
           application_secret_[sender]_[N]

   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



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   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:

   o  Senders MUST only use a given secret once and monotonically
      increment the generation of their secret.  This is important to
      provide Forward Secrecy at the level of Application messages.  An
      attacker getting hold of a member specific Application Secret at
      generation [N+1] will not be able to derive the member's
      Application Secret [N] nor the associated AEAD key and nonce.

   o  Receivers MUST delete an Application Secret once it has been used
      to derive the corresponding AEAD key and nonce as well as the next
      Application Secret.  Receivers MAY keep the AEAD key and nonce
      around for some reasonable period.

   o  Receivers MUST delete AEAD keys and nonces once they have been
      used to successfully decrypt a message.

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.



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   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.

























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   enum {
       invalid(0),
       handshake(1),
       application(2),
       (255)
   } 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:

   o  Gather the required metadata:

      *  Group ID

      *  Epoch

      *  Content Type

      *  Nonce

      *  Sender index



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      *  Key generation

   o  Sign the protected content and metadata

   o  Encrypt the sender information using the random nonce and the key
      derived from the sender_data_secret

   o  Encrypt the content using a content encryption key identified by
      the metadata

   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.







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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:

   o  Group initialization



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   o  A member adding a new client

   o  A member updating its leaf key

   o  A member deleting another member

   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 {
       init(0),
       add(1),
       update(2),
       remove(3),
       (255)
   } 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.





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   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




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   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.





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   [[ 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:

   o  Prepare a new GroupState object based on the Welcome message

   o  Process the Add message as an existing member would

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

   o  If the "index" value is equal to the size of the group, increment
      the size of the group, and extend the tree accordingly

   o  Verify the signature on the included UserInitKey; if the signature
      verification fails, abort






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   o  Generate a WelcomeInfo object describing the state prior to the
      add, and verify that its hash is the same as the value of the
      "welcome_info_hash" field

   o  Update the ratchet tree by setting to blank all nodes in the
      direct path of the new node

   o  Set the leaf node in the tree at position "index" to a new node
      containing the public key from the UserInitKey in the Add
      corresponding to the ciphersuite in use, as well as the credential
      under which the UserInitKey was signed

   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:

   o  Generate a fresh leaf key pair

   o  Compute its direct path in the current ratchet tree

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

   o  Update the cached ratchet tree by replacing nodes in the direct
      path from the updated leaf using the information contained in the
      Update message

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






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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:

   o  Generate a fresh leaf key pair

   o  Compute its direct path in the current ratchet tree, starting from
      the removed leaf

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

   o  Update the ratchet tree by replacing nodes in the direct path from
      the removed leaf using the information in the Remove message

   o  Update the ratchet tree by setting to blank all nodes in the
      direct path of the removed leaf, and also setting the root node to
      blank

   o  Truncate the tree such that the rightmost non-blank leaf is the
      last node of the tree

   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.
   ]]



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   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:

   o  Have the delivery service enforce a total order

   o  Have a signal in the message that clients can use to break ties

   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




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   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.




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

        KDF(KDF(D))
        /       \
     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(KDF(Y))
        /       \
     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.





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   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



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   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




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   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



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   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

   o  Benjamin Beurdouche
      INRIA
      benjamin.beurdouche@ens.fr

   o  Karthikeyan Bhargavan
      INRIA
      karthikeyan.bhargavan@inria.fr

   o  Cas Cremers
      University of Oxford
      cas.cremers@cs.ox.ac.uk

   o  Alan Duric
      Wire
      alan@wire.com

   o  Srinivas Inguva
      Twitter
      singuva@twitter.com

   o  Albert Kwon
      MIT
      kwonal@mit.edu

   o  Eric Rescorla
      Mozilla
      ekr@rtfm.com

   o  Thyla van der Merwe
      Royal Holloway, University of London
      thyla.van.der@merwe.tech



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15.  References

15.1.  Normative References

   [I-D.barnes-cfrg-hpke]
              Barnes, R. and K. Bhargavan, "Hybrid Public Key
              Encryption", draft-barnes-cfrg-hpke-01 (work in progress),
              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,
              <https://www.rfc-editor.org/info/rfc2104>.

   [RFC2119]  Bradner, S., "Key words for use in RFCs to Indicate
              Requirement Levels", BCP 14, RFC 2119,
              DOI 10.17487/RFC2119, March 1997,
              <https://www.rfc-editor.org/info/rfc2119>.

   [RFC5116]  McGrew, D., "An Interface and Algorithms for Authenticated
              Encryption", RFC 5116, DOI 10.17487/RFC5116, January 2008,
              <https://www.rfc-editor.org/info/rfc5116>.

   [RFC5869]  Krawczyk, H. and P. Eronen, "HMAC-based Extract-and-Expand
              Key Derivation Function (HKDF)", RFC 5869,
              DOI 10.17487/RFC5869, May 2010,
              <https://www.rfc-editor.org/info/rfc5869>.

   [RFC7748]  Langley, A., Hamburg, M., and S. Turner, "Elliptic Curves
              for Security", RFC 7748, DOI 10.17487/RFC7748, January
              2016, <https://www.rfc-editor.org/info/rfc7748>.

   [RFC8174]  Leiba, B., "Ambiguity of Uppercase vs Lowercase in RFC
              2119 Key Words", BCP 14, RFC 8174, DOI 10.17487/RFC8174,
              May 2017, <https://www.rfc-editor.org/info/rfc8174>.

   [RFC8446]  Rescorla, E., "The Transport Layer Security (TLS) Protocol
              Version 1.3", RFC 8446, DOI 10.17487/RFC8446, August 2018,
              <https://www.rfc-editor.org/info/rfc8446>.

   [X962]     ANSI, "Public Key Cryptography For The Financial Services
              Industry: The Elliptic Curve Digital Signature Algorithm
              (ECDSA)", ANSI X9.62, 1998.



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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, <https://eprint.iacr.org/2017/666.pdf>.

   [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]    Husak, M., &#268;ermak, M., Jirsik, T., and P.
              &#268;eleda, "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", draft-
              ietf-trans-rfc6962-bis-31 (work in progress), 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.,
              <https://www.signal.org/docs/specifications/
              doubleratchet/>.




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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     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



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       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)



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       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:
           d.append(p)
           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):
           d.append(x)

       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



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       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: rlb@ipv.sx


   Jon Millican
   Facebook

   Email: jmillican@fb.com


   Emad Omara
   Google

   Email: emadomara@google.com


   Katriel Cohn-Gordon
   University of Oxford

   Email: me@katriel.co.uk







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   Raphael Robert
   Wire

   Email: raphael@wire.com















































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