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Network Working Group                                 P. M. Hallam-Baker
Internet-Draft                                           2 November 2020
Intended status: Informational
Expires: 6 May 2021

       Mathematical Mesh 3.0 Part VIII: Cryptographic Algorithms


   The Mathematical Mesh 'The Mesh' is an infrastructure that
   facilitates the exchange of configuration and credential data between
   multiple user devices and provides end-to-end security.  This
   document describes the cryptographic algorithm suites used in the
   Mesh and the implementation of Multi-Party Encryption and Multi-Party
   Key Generation used in the Mesh.

   [Note to Readers]

   Discussion of this draft takes place on the MATHMESH mailing list
   (mathmesh@ietf.org), which is archived at

   This document is also available online at

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
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   Internet-Drafts are draft documents valid for a maximum of six months
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   material or to cite them other than as "work in progress."

   This Internet-Draft will expire on 6 May 2021.

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Copyright Notice

   Copyright (c) 2020 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.

Table of Contents

   1.  Introduction  . . . . . . . . . . . . . . . . . . . . . . . .   3
   2.  Definitions . . . . . . . . . . . . . . . . . . . . . . . . .   3
     2.1.  Requirements Language . . . . . . . . . . . . . . . . . .   3
     2.2.  Defined Terms . . . . . . . . . . . . . . . . . . . . . .   3
     2.3.  Related Specifications  . . . . . . . . . . . . . . . . .   4
     2.4.  Implementation Status . . . . . . . . . . . . . . . . . .   4
   3.  Recommended and Required Algorithms . . . . . . . . . . . . .   4
     3.1.  Mesh Device . . . . . . . . . . . . . . . . . . . . . . .   4
     3.2.  Constrained Device  . . . . . . . . . . . . . . . . . . .   5
   4.  Multi-Party Cryptography  . . . . . . . . . . . . . . . . . .   6
     4.1.  Application to Diffie Hellman (not normative) . . . . . .   6
     4.2.  Multi-Party Key Generation  . . . . . . . . . . . . . . .   6
     4.3.  Multi-Party Decryption  . . . . . . . . . . . . . . . . .   7
     4.4.  Mutually Authenticated Key Exchange.  . . . . . . . . . .   7
     4.5.  Implementation  . . . . . . . . . . . . . . . . . . . . .   7
       4.5.1.  Implementation for Ed25519 and Ed448  . . . . . . . .   8
       4.5.2.  Implementation for X25519 and X448  . . . . . . . . .   8
   5.  Multi-Party Key Generation  . . . . . . . . . . . . . . . . .   9
   6.  Multi-Party Decryption  . . . . . . . . . . . . . . . . . . .   9
     6.1.  Mechanism . . . . . . . . . . . . . . . . . . . . . . . .  11
     6.2.  Implementation  . . . . . . . . . . . . . . . . . . . . .  12
       6.2.1.  Group Creation  . . . . . . . . . . . . . . . . . . .  12
       6.2.2.  Provisioning a Member . . . . . . . . . . . . . . . .  13
       6.2.3.  Message Encryption and Decryption . . . . . . . . . .  13
   7.  Mutually Authenticated Key Agreement  . . . . . . . . . . . .  14
   8.  Security Considerations . . . . . . . . . . . . . . . . . . .  15
   9.  IANA Considerations . . . . . . . . . . . . . . . . . . . . .  15
   10. Acknowledgements  . . . . . . . . . . . . . . . . . . . . . .  15
   11. Examples  . . . . . . . . . . . . . . . . . . . . . . . . . .  15
     11.1.  Key Combination  . . . . . . . . . . . . . . . . . . . .  15
       11.1.1.  Ed25519  . . . . . . . . . . . . . . . . . . . . . .  16
       11.1.2.  Ed448  . . . . . . . . . . . . . . . . . . . . . . .  16
       11.1.3.  X25519 . . . . . . . . . . . . . . . . . . . . . . .  16
       11.1.4.  X448 . . . . . . . . . . . . . . . . . . . . . . . .  16
     11.2.  Group Encryption . . . . . . . . . . . . . . . . . . . .  16

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       11.2.1.  X25519 . . . . . . . . . . . . . . . . . . . . . . .  16
       11.2.2.  X448 . . . . . . . . . . . . . . . . . . . . . . . .  16
   12. Normative References  . . . . . . . . . . . . . . . . . . . .  16
   13. Informative References  . . . . . . . . . . . . . . . . . . .  17

1.  Introduction

   This document describes the cryptographic algorithm suites used in
   the Mesh and the implementation of Multi-Party Encryption and Multi-
   Party Key Generation used in the Mesh.

   To allow use of Mesh capabilities on the least capable computing
   devices currently in use, separate schedules of recommended and
   required algorithms are specified for Standard Devices and
   Constrained Devices.

   The Constrained device class may be considered to include most 8-bit
   CPUs equipped with sufficient memory to support the necessary
   operations.  For example an Ardunino Mega 2560 which can perform ECDH
   key agreement and signature operations in times ranging from 3 to 8
   seconds.  While such a device is clearly not suited to perform such
   operations routinely, a one-time connection process that takes
   several minutes to complete need not be of major concern.

   The Standard device class may be considered to include the vast
   majority of general purpose and personal computing devices
   manufactured since 2010.  Even a Raspberry Pi Zero which currently
   retails at $5 is capable of performing the cryptographic functions
   required to implement the Mesh with negligible impact on the user.

2.  Definitions

   This section presents the related specifications and standard, the
   terms that are used as terms of art within the documents and the
   terms used as requirements language.

2.1.  Requirements Language

   The key words "MUST", "MUST NOT", "REQUIRED", "SHALL", "SHALL NOT",
   document are to be interpreted as described in [RFC2119].

2.2.  Defined Terms

   The terms of art used in this document are described in the _Mesh
   Architecture Guide_ [draft-hallambaker-mesh-architecture].

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2.3.  Related Specifications

   The architecture of the Mathematical Mesh is described in the _Mesh
   Architecture Guide_ [draft-hallambaker-mesh-architecture].  The Mesh
   documentation set and related specifications are described in this

2.4.  Implementation Status

   The implementation status of the reference code base is described in
   the companion document [draft-hallambaker-mesh-developer].

3.  Recommended and Required Algorithms

   To allow implementation of Mesh capabilities on the widest possible
   range of devices, separate algorithm requirements and recommendations
   are specified for four classes of device:

   Administration Device  A general-purpose computing device that is
      used for Mesh administration functions.

   Mesh Device  A general-purpose computing device that is not used for
      Mesh administration functions with sufficient memory and
      processing power to perform public key cryptography operations
      without paying particular attention to the impact on performance.

   Constrained Device  An embedded computing device with limited memory
      and computing power that offers sufficient processing capabilities
      to perform occasional public key operations (e.g. during device
      initialization) but is not suited to repeated operations.

   Bridge Device  A trusted device that enables Mesh Devices to
      interoperate with Constrained devices.

   Since Administration Devices and Mesh Devices MUST support
   communication with Mesh Devices and Constrained devices, they MUST
   meet all the REQUIRED algorithms for both types of device.

3.1.  Mesh Device

   Support for the following algorithms is REQUIRED:

   *  SHA-2-512 [SHA-2]

   *  HMAC-SHA-2-512 [RFC2104]

   *  HMAC-based Extract-and-Expand Key Derivation Function [RFC5869]

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   *  AES-CBC-256 Encryption [FIPS197]

   *  Advanced Encryption Standard (AES) Key Wrap Algorithm [RFC3394]

   *  Montgomery Curve Diffie-Hellman Key Agreement X25519 and X448

   *  Edwards-Curve Digital Signature Algorithm Ed25519 and Ed448

   Support for the following algorithms is RECOMMENDED:

   *  AES-GCM-256 Encryption [AES-GCM]

   *  SHA-3-512 [SHA-3]

   *  KMAC SHA-3-512 [SHA-3-Derived]

   While the use of GCM is generally preferred over CBC mode in IETF
   security protocols, this mode is not currently supported by the
   reference implementation platform.

3.2.  Constrained Device

   Support for the following algorithms is REQUIRED:

   *  SHA-2-512 [SHA-2]

   *  HMAC-SHA-2-512 [RFC2104]

   *  HMAC-based Extract-and-Expand Key Derivation Function [RFC5869]

   *  Poly1035 Authenticated Encryption [RFC8439]

   *  ChaCha20 Encryption [RFC8439]

   *  Advanced Encryption Standard (AES) Key Wrap Algorithm [RFC3394]

   *  Edwards-Curve Digital Signature Algorithm Ed25519 [RFC8032]

   *  Edwards-Curve Diffie-Hellman Key Agreement Ed25519 [RFC8032]

   Use of the Edwards Curves for Signature and Key Agreement allows both
   functions to be supported by a single library with no reduction in

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4.  Multi-Party Cryptography

   The multi-party key generation and multi-party decryption mechanisms
   used in the Mesh protocols are made possible by the fact that Diffie
   Hellman key agreement and elliptic curve variants thereof support
   properties we call the Key Combination Law and the Result Combination

   Let {_X_, _x_}, {_Y_, _y_}, {_E_, _e_} be {public, private} key

   The Key Combination law states that we can define an operator ? such
   that there is a keypair {_Z_, _z_} such that:

   _Z_ = _X_ ? _Y_ and _z_ = (_x_ + _y_) mod _o_ (where _o_ is the order
   of the group)

   The Result Combination Law states that we can define an operator ?
   such that:

   (_x_ ? _E_) ? (_y_ ? _E_) = (_z_ ? _E_) = (_e_ ? _Z_).

4.1.  Application to Diffie Hellman (not normative)

   For the Diffie Hellman system in a modular field p, o = p-1 and _a_?
   _b_ = _a_? _b_ = _a_._b_mod _p_.


   By definition, X = e^(x) mod p, Y = e^(y) mod p, and Z = e^(z)mod p.


   Z = e^(z) mod p = e^(x+y) mod p = (e^(x)e^(y)) mod p = e^(x)mod
   p.e^(y) mod p = X.Y

   A similar proof may be constructed for the operator ?.

4.2.  Multi-Party Key Generation

   The Key Combination Law provides the basis for the Key Co-Generation
   technique used to ensure that the cryptographic keys used in devices
   connected to a Mesh profile are sufficiently random and have not been
   compromised by malware or other 'backdoor' compromise to the machine
   during or after manufacture.

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   For the Diffie Hellman system, the Key Combination law provides all
   the mechanism needed to implement a Key Co-Generation mechanism.  If
   the Device key is {_X_, _x_}, the administration device can generate
   a Co-Generation Key Pair {_Y_, _y_} and generate a Device Connection
   Assertion for the final public key E calculated from knowledge of X
   and Y alone.  Passing the value _y_ to the device (using a secure
   channel) allows it to calculate the corresponding private key _e_
   required to make use of the Device Connection Assertion.

   This approach ensures that a party with knowledge of either _x_ or
   _y_ but not both obtains no knowledge of _e_.

   Section REF _Ref5309729 \w \h 5 describes the implementation of these
   schemes in the Mesh

4.3.  Multi-Party Decryption

   The Key Combination Law and Result Combination Law provide the basis
   for the Multi-Party Decryption technique used to support Mesh
   Encryption Groups.

   Section REF _Ref5309538 \w \h 6 describes the application of this
   technique in the Mesh

4.4.  Mutually Authenticated Key Exchange.

   The Result Combination Law is used to provide a Key Exchange
   mechanism that provides mutual authentication of the parties while
   preserving forward secrecy.

4.5.  Implementation

   For elliptic curve cryptosystems, the operators ? and ? are point

   Implementing a robust Key Co-Generation for the Elliptic Curve
   Cryptography schemes described in [RFC7748] and [RFC8032] requires
   some additional considerations to be addressed.

   *  The secret scalar used in the EdDSA algorithm is calculated from
      the private key using a digest function.  It is therefore
      necessary to specify the Key Co-Generation mechanism by reference
      to operations on the secret scalar values rather than operations
      on the private keys.

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   *  The Montgomery Ladder traditionally used to perform X25519 and
      X448 point multiplication does not require implementation of a
      function to add two arbitrary points.  While the steps required to
      create such a function are fully constrained by the specification,
      the means of satisfying the constraints is not.

4.5.1.  Implementation for Ed25519 and Ed448

   The data structures used to implement co-generation of public keys
   are defined in the main Mesh Reference Guide.  This document
   describes only the additional implementation details.

   Note that the 'private key' described in [RFC8032] is in fact a seed
   used to generate a 'secret scalar' value that is the value that has
   the function of being the private key in the ECDH algorithm.

   To provision a new public key to a device, the provisioning device:

   0.  Obtains the device profile of the device(s) to be provisioned to
       determine the type of key to perform co-generation for.  Let the
       device {public, private} key be {D, d}.

   1.  Generates a private key _m_ with the specified number of bytes
       (32 or 57].

   2.  Calculates the corresponding public key _M_.

   3.  Calculates the Application public key A = D+M where + is point

   4.  Constructs the application device entry containing the private
       key value m and encrypts under the device encryption key d.

   On receipt, the device may at its option use its knowledge of the
   secret scalar corresponding to d and m to calculate the application
   secret scalar a or alternatively maintain the two secrets separately
   and make use of the result combination law to perform private key

4.5.2.  Implementation for X25519 and X448

   While the point addition function can be defined for any elliptic
   curve system, it is not necessary to implement point addition to
   support ECDH key agreement.

   In particular, point multiplication using the Montgomery ladder
   technique over Montgomery curves only operate on the x co-ordinate
   and only require point doubling operations.

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   For expediency, the current implementation of the Mesh reference code
   uses the Edwards curves for both signature and encryption pending
   announcement of platform support for both algorithms.

5.  Multi-Party Key Generation

   Multi-Party Key Generation is a capability that is used in the Mesh
   to enable provisioning of application specific private key pairs to
   connected devices without revealing any information concerning the
   application private key of the device.

   For example, Alice provisions the confirmation service to her watch.
   The provisioning device could generate a signature key for the device
   and encrypt it under the encryption key of the device.  But this
   means that we cannot attribute signatures to the watch with absolute
   certainty as the provisioning device has had knowledge of the watch
   signature key.  Nor do we wish to use the device signature key for
   the confirmation service.

   Multi-Party Key Generation allows an administration device to
   provision a connected device with an application specific private key
   that is specific to that application and no other such that the
   application can determine the public key of the device but has no
   knowledge of the private key.

   Provisioning an application private key to a device requires the
   administration device to:

   *  Generate a new application public key for the device.

   *  Construct and publish whatever application specific credentials
      the device requires to use the application.

   *  Providing the information required to make use of the private key
      to the device.

   Note that while the administration device needs to know the device
   application public key, it does not require knowledge of the device
   application private key.

6.  Multi-Party Decryption

   A key limitation of most deployed messaging systems is that true end-
   to-end confidentiality is only achieved for a limited set of
   communication patterns.  Specifically, bilateral communications
   (Alice sends a message to Bob) or broadcast communications to a known
   set of recipients (Alice sends a message to Bob, Carol and Doug).
   These capabilities do not support communication patterns where the

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   set of recipients changes over time or is confidential.  Yet such
   requirements commonly occur in situations such as sending a message
   to a mailing list whose membership isn't known to the sender, or
   creating a spreadsheet whose readership is to be limited to
   authorized members of the 'accounting' team.

   The mathematical approach that makes key co-generation possible may
   be applied to support a public key encryption mode in which
   encryption is performed as usual but decryption requires the use of
   multiple keys.  This approach is variously described in the
   literature as distributed key generation and proxy re-encryption

   The approach specified in this document borrows aspects of both these
   techniques.  This combined approach is called 'recryption'.  Using
   recryption allows a sender to send a message to a group of users
   whose membership is not known to the sender at the time the message
   is sent and can change at any time.

   Proxy re-encryption provides a technical capability that meets the
   needs of such communication patterns.  Conventional symmetric key
   cryptography uses a single key to encrypt and decrypt data.  Public
   key cryptography uses two keys, the key used to encrypt data is
   separate from the key used to decrypt.  Proxy re-encryption
   introduces a third key (the recryption key) that allows a party to
   permit an encrypted data packet to be decrypted using a different key
   without permitting the data to be decrypted.

   The introduction of a recryption key permits end-to-end
   confidentiality to be preserved when a communication pattern requires
   that some part of the communication be supported by a service.

   The introduction of a third type of key, the recryption key permits
   two new roles to be established, that of an administrator and
   recryption service.  There are thus four parties:

   Administrator  Holder of Decryption Key, Creator of Recryption Keys

   Sender  Holder of Encryption Key

   Recryption Service  Holder of Recryption keys

   Receiver  Holder of personal decryption key

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   The information stored at the recryption service is necessary but not
   sufficient to decrypt the message.  Thus, no disclosure of the
   message plaintext occurs even in the event that an attacker gains
   full knowledge of all the information stored by the recryption

6.1.  Mechanism

   The mechanism used to support recryption is the same as the mechanism
   used to support key co-generation except that this time, instead of
   combining two keys to create one, the private component of a
   decryption key (i.e. the private key) is split into two parts, a
   recryption key and a decryption key.

   Recall that the key combination law for Diffie Hellman crypto-systems
   states that we can add two private keys to get a third.  It follows
   that we can split the private key portion of a keypair {_G_, _g_}
   into two parts by choosing a random number that is less than the
   order of the Diffie-Hellman group to be our first key _x_. Our second
   key is _y_ = _g_ - _r_ mod _o_, where _o_ is the order of the group.

   Having generated _x_, _y_, we can use these to perform private key
   agreement operations on a public key _E_ and then use the result
   combination law to obtain the same result that we would have obtained
   using _g_.

   One means of applying this mechanism to recryption would be to
   generate a different random value x for each member of the group and
   store it at the recryption service and communicate the value y to the
   member via a secure channel.  Applying this approach, we can clearly
   see that the recryption service gains no information about the value
   of the private key since the only information it holds is a random
   number which could have been generated without any knowledge of the
   group private key.

   [RFC8032] requires that implementations derive the scalar secret by
   taking a cryptographic digest of the private key.  This means that
   either the client or the service must use a non-compliant
   implementation.  Given this choice, it is preferable to require that
   the non-standard implementation be required at the service rather
   than the client.  This limits the scope of the non-conformant key
   derivation approach to the specialist recryption service and ensures
   that the client enforce the requirement to generate the private key
   component by means of a digest.

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6.2.  Implementation

   Implementation of recryption in the Mesh has four parts:

   *  Creation and management of the recryption group.

   *  Provisioning of members to a recryption group.

   *  Message encryption.

   *  Message decryption.

   These operations are all performed using the same catalog and
   messaging infrastructure provided by the Mesh for other purposes.

   Each recryption group has its own independent Mesh account.  This has
   many advantages:

   *  Administration of the recryption group may be spread across
      multiple Mesh users or transferred from one user to another
      without requiring specification of a separate management protocol
      to support these operations.

   *  The recryption account address can be used by Mesh applications
      such as group messaging, conferencing, etc. as a contact address.

   *  The contact request service can be used to notify members that
      they have been granted membership in the group.

6.2.1.  Group Creation

   Creation of a Recryption group requires the steps of:

   *  Generating the recryption group key pair

   *  Creating the recryption group account

   *  Generating administrator record for each administrator.

   *  Publishing the administrator records to the recryption catalog.

   Note that in principle, we could make use of the key combination law
   to enable separation of duties controls on administrators so that
   provisioning of members required multiple administrators to
   participate in the process.  This is left to future versions.

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6.2.2.  Provisioning a Member

   To provision a user as a member of the recryption group, the
   administrator requires their current recryption profile.  The
   administrator MAY obtain this by means of a contact service request.
   As with any contact service request, this request is subject to
   access control and MAY require authorization by the intended user
   before the provisioning can proceed.

   Having obtained the user's recryption profile, the administration
   tool generates a decryption private key for the user and encrypts it
   under the member's key to create the encrypted decryption key entry.

   The administration tool then computes the secret scalar from the
   private key and uses this together with the secret scalar of the
   recryption group to compute the recryption key for the member.  This
   value and the encrypted decryption key entry are combined to form the
   recryption group membership record which is published to the catalog.

6.2.3.  Message Encryption and Decryption

   Encryption of a messages makes use of DARE Message in exactly the
   same manner as any other encryption.  The sole difference being that
   the recipient entry for the recryption operation MUST specify the
   recryption group address an not just the key fingerprint.  This
   allows the recipient to determine which recryption service to contact
   to perform the recryption operation.

   To decrypt a message, the recipient makes an authenticated recryption
   request to the specified recryption service specifying:

   *  The recipient entry to be used for decryption

   *  The fingerprint of the decryption key(s) the device would like to
      make use of.

   *  Whether or not the encrypted decryption key entry should be

   The recryption service searches the catalog for the corresponding
   recryption group to find a matching entry.  If found and if the
   recipient and proposed decryption key are dully authorized for the
   purpose, the service performs the key agreement operation using the
   recryption key specified in the entry and returns the result to the

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   The recipient then decrypts the recryption data entry using its
   device decryption key and uses the group decryption key to calculate
   the other half of the result.  The two halves of the result are then
   added to obtain the key agreement value that is then used to decrypt
   the message.

7.  Mutually Authenticated Key Agreement

   Diffie Hellman key agreement using the authenticated public keys of
   the principals provides mutual authentication of those principals.

   For example, if Alice's key pair is {_a_, _A_} and Bob's key pair is
   {_b_, _B_}, the Diffie Hellman key agreement value _DH_ (_a_, _B_) =
   _DH_ (_b_, _A_) can only be generated from the public information if
   _a_ or _b_ is known.

   The chief disadvantage of this approach is that it only allows Alice
   and Bob to establish a single shared secret that will never vary and
   does not provide forward secrecy.  To avoid this, cryptographic
   protocols usually perform the key agreement against an ephemeral key
   and either accept that the client key is not authenticated or perform
   multiple key agreements and combine the results.

   Using the Result Combination Law allows a key agreement mechanism to
   combine the benefits of mutual authentication with the use of
   ephemeral keys without the need for multiple private key operations
   or additional round trips.

   In its simplest form, the key exchange has two parties which we refer
   to as the client and the server.  The client being the party that
   initiates the protocol exchange and the server being the party that
   responds.  Let the public key pair of the client be {_a_, _A_} and
   that of the server {_b_, _B_}.

   Two versions of the key agreement mechanism are specified:

   Client ephemeral  The client contributes an ephemeral key pair
      {_n_(A)_, _N_(A)_}. The effective public key of the client is _A_
      ? _N_(A)_.

      The server uses its public key _B_.

      The key agreement value is _DH_(_a_ + _n_(A)_, B) = _DH_ (_b_, _A_
      ? _N_(A)_)

   Dual ephemeral  The client contributes an ephemeral key pair
      {_n_(A)_, _N_(A)_}. The effective public key of the client is _A_
      ? _N_(A)_.

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      The server contributes an ephemeral key pair {_n_(B)_, _N_(B)_}.
      The effective public key of the client is _B_ ? _N_(B)_.

      The key agreement value is _DH_(_a_ + _n_(A)_, _B_ ? _N_(B)_) =
      _DH_ (_b_ + _n_(B)_, _A_ ? _N_(A)_)

   The function of the ephemeral key is effectively that of a nonce but
   it is shared with the counter-party as a public key value.

   The dual ephemeral approach has the advantage that it limits the
   scope for side channel attacks as both sides have contributed unknown
   information to the key agreement value.  The disadvantage of this
   approach is that the key agreement value can only be calculated after
   the server has provided its ephemeral.

   Implementations MAY take advantage of the result combination law to
   enable private key operations involving the authenticated key (or a
   contribution to it) to be performed in trustworthy hardware.

   An advantage of this key exchange mechanism over the traditional TLS
   key exchange approach is that no signature operation is involved,
   thus ensuring that either party can repudiate the exchange and thus
   the claim that they were in communication.

   The master secret is calculated from the key agreement value in the
   usual fashion.  For ECDH algorithms, this comprises the steps of
   converting the key agreement value to an octet string which forms the
   input to a Key Derivation Function.

8.  Security Considerations

   The security considerations for use and implementation of Mesh
   services and applications are described in the Mesh Security
   Considerations guide [draft-hallambaker-mesh-security].

9.  IANA Considerations

   This document requires no IANA actions.

10.  Acknowledgements

   A list of people who have contributed to the design of the Mesh is
   presented in [draft-hallambaker-mesh-architecture].

11.  Examples

11.1.  Key Combination

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11.1.1.  Ed25519

11.1.2.  Ed448

11.1.3.  X25519

11.1.4.  X448

11.2.  Group Encryption

11.2.1.  X25519

11.2.2.  X448

12.  Normative References

   [AES-GCM]  Dworkin, M. J., "Recommendation for Block Cipher Modes of
              Operation: Galois/Counter Mode (GCM) and GMAC", November

              Hallam-Baker, P., "Mathematical Mesh 3.0 Part I:
              Architecture Guide", Work in Progress, Internet-Draft,
              draft-hallambaker-mesh-architecture-14, 27 July 2020,

              Hallam-Baker, P., "Mathematical Mesh 3.0 Part VII:
              Security Considerations", Work in Progress, Internet-
              Draft, draft-hallambaker-mesh-security-05, 27 July 2020,

   [FIPS197]  NIST, "Advanced Encryption Standard (AES)", November 2001.

   [RFC2104]  Krawczyk, H., Bellare, M., and R. Canetti, "HMAC: Keyed-
              Hashing for Message Authentication", RFC 2104,
              DOI 10.17487/RFC2104, February 1997,

   [RFC2119]  Bradner, S., "Key words for use in RFCs to Indicate
              Requirement Levels", BCP 14, RFC 2119,
              DOI 10.17487/RFC2119, March 1997,

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   [RFC3394]  Schaad, J. and R. Housley, "Advanced Encryption Standard
              (AES) Key Wrap Algorithm", RFC 3394, DOI 10.17487/RFC3394,
              September 2002, <https://www.rfc-editor.org/rfc/rfc3394>.

   [RFC5869]  Krawczyk, H. and P. Eronen, "HMAC-based Extract-and-Expand
              Key Derivation Function (HKDF)", RFC 5869,
              DOI 10.17487/RFC5869, May 2010,

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

   [RFC8032]  Josefsson, S. and I. Liusvaara, "Edwards-Curve Digital
              Signature Algorithm (EdDSA)", RFC 8032,
              DOI 10.17487/RFC8032, January 2017,

   [RFC8439]  Nir, Y. and A. Langley, "ChaCha20 and Poly1305 for IETF
              Protocols", RFC 8439, DOI 10.17487/RFC8439, June 2018,

   [SHA-2]    NIST, "Secure Hash Standard", August 2015.

   [SHA-3]    Dworkin, M. J., "SHA-3 Standard: Permutation-Based Hash
              and Extendable-Output Functions", August 2015.

              Kelsey, J. M., Chang, S. H., and R. A. Perlner, "SHA-3
              Derived Functions: cSHAKE, KMAC, TupleHash and
              ParallelHash SHARE", December 2016.

13.  Informative References

   [Blaze98]  "[Reference Not Found!]".

              Hallam-Baker, P., "Mathematical Mesh: Reference
              Implementation", Work in Progress, Internet-Draft, draft-
              hallambaker-mesh-developer-10, 27 July 2020,

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