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RFC 9370: Multiple Key Exchanges in the Internet Key Exchange Protocol Version 2 (IKEv2)
- CJ. Tjhai,
- M. Tomlinson,
- G. Bartlett,
- S. Fluhrer,
- D. Van Geest,
- O. Garcia-Morchon,
- V. Smyslov
Abstract
This document describes how to extend the Internet Key Exchange Protocol Version 2 (IKEv2) to allow multiple key exchanges to take place while computing a shared secret during a Security Association (SA) setup.¶
This document utilizes the IKE_
This document updates RFC 7296 by renaming a Transform Type 4 from "Diffie-Hellman Group (D-H)" to "Key Exchange Method (KE)" and renaming a field in the Key Exchange Payload from "Diffie-Hellman Group Num" to "Key Exchange Method". It also renames an IANA registry for this Transform Type from "Transform Type 4 - Diffie- Hellman Group Transform IDs" to "Transform Type 4 - Key Exchange Method Transform IDs". These changes generalize key exchange algorithms that can be used in IKEv2.¶
Status of This Memo
This is an Internet Standards Track document.¶
This document is a product of the Internet Engineering Task Force (IETF). It represents the consensus of the IETF community. It has received public review and has been approved for publication by the Internet Engineering Steering Group (IESG). Further information on Internet Standards is available in Section 2 of RFC 7841.¶
Information about the current status of this document, any
errata, and how to provide feedback on it may be obtained at
https://
Copyright Notice
Copyright (c) 2023 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://
1. Introduction
1.1. Problem Description
The Internet Key Exchange Protocol version 2 (IKEv2), as specified in [RFC7296], uses
the Diffie-Hellman (DH) or the Elliptic Curve Diffie-Hellman (ECDH)
algorithm, which shall be referred to as "(EC)DH" collectively, to
establish a shared secret between an initiator and a responder. The
security of the (EC)DH algorithms relies on the difficulty to solve a
discrete logarithm problem in multiplicative (and, respectively,
elliptic curve) groups when the order of the group parameter is large
enough. While solving such a problem remains infeasible with current
computing power, it is believed that general-purpose quantum
computers will be able to solve this problem, implying that the
security of IKEv2 is compromised. There are, however, a number of
cryptosystems that are conjectured to be resistant to quantum
It is essential to have the ability to perform one or more post-quantum key exchanges in conjunction with an (EC)DH key exchange so that the resulting
shared key is resistant to quantum
1.2. Proposed Extension
This document describes a method to perform multiple successive key
exchanges in IKEv2. This method allows integration of PQC in IKEv2,
while maintaining backward compatibility, to derive a set of IKE keys
that is resistant to quantum
IKE peers perform multiple successive key exchanges to establish an IKE SA. Each exchange produces some shared secret, and these secrets are combined in a way such that:¶
- (a)
- the final shared secret is computed from all of the component key exchange secrets;¶
- (b)
- unless both peers support and agree to use the additional key exchanges introduced in this specification, the final shared secret equivalent to the shared secret specified in [RFC7296] is obtained; and¶
- (c)
- if any part of the component key exchange method is a post-quantum algorithm, the final shared secret is post-quantum secure.¶
Some post-quantum key exchange payloads may have sizes larger than
the standard maximum transmission unit (MTU) size. Therefore, there
could be issues with fragmentation at the IP layer. In order to allow
the use of those larger payload sizes, this mechanism relies on the
IKE_
While this extension is primarily aimed at IKE SAs due to the
potential fragmentation issue discussed above, it also applies to
CREATE_
Note that readers should consider the approach defined in this document as providing a long-term solution in upgrading the IKEv2 protocol to support post-quantum algorithms. A short-term solution to make IKEv2 key exchange quantum secure is to use post-quantum pre-shared keys as specified in [RFC8784].¶
Note also that the proposed approach of performing multiple successive key exchanges in such a way, when the resulting session keys depend on all of them, is not limited to only addressing the threat of quantum computers. It can also be used when all of the performed key exchanges are classical (EC)DH primitives, where, for various reasons (e.g., policy requirements), it is essential to perform multiple key exchanges.¶
This specification does not attempt to address key exchanges with KE payloads longer than 64 KB; the current IKE payload format does not allow such a possibility. At the time of writing, it appears likely that there are a number of key exchanges available that would not have such a requirement. [BEYOND-64K] discusses approaches that could be taken to exchange huge payloads if such a requirement were needed.¶
1.3. Document Organization
The remainder of this document is organized as follows. Section 2 describes how multiple key exchanges are performed between two IKE peers and how keying materials are derived for both SAs and Child SAs. Section 3 discusses IANA considerations for the namespaces introduced in this document. Section 4 discusses security considerations. In the Appendices, some examples of multiple key exchanges are illustrated in Appendix A. Appendix B summarizes design criteria and alternative approaches that have been considered. These approaches are later discarded, as described in Appendix C.¶
The key words "MUST", "MUST NOT", "REQUIRED", "SHALL", "SHALL NOT", "SHOULD", "SHOULD NOT", "RECOMMENDED", "NOT RECOMMENDED", "MAY", and "OPTIONAL" in this document are to be interpreted as described in BCP 14 [RFC2119] [RFC8174] when, and only when, they appear in all capitals, as shown here.¶
2. Multiple Key Exchanges
2.1. Design Overview
Most post-quantum key agreement algorithms are relatively new. Thus, they are not fully trusted. There are also many proposed algorithms that have different trade-offs and that rely on different hard problems. The concern is that some of these hard problems may turn out to be easier to solve than anticipated; thus, the key agreement algorithm may not be as secure as expected. A hybrid solution, when multiple key exchanges are performed and the calculated shared key depends on all of them, allows us to deal with this uncertainty by combining a classical key exchange with a post-quantum one, as well as leaving open the possibility of combining it with multiple post-quantum key exchanges.¶
In order to be able to use IKE fragmentation [RFC7383] for those key exchanges that may
have long public keys, this specification utilizes the
IKE_
Note that this document assumes that each key exchange method
requires one round trip and consumes exactly one IKE_
In order to minimize communication overhead, only the key shares that are agreed upon are actually exchanged. To negotiate additional key exchanges, seven new Transform Types are defined. These transforms and Transform Type 4 share the same Transform IDs.¶
It is assumed that new Transform Type 4 identifiers will be assigned later for various post-quantum key exchanges [IKEV2TYPE4ID]. This specification does not make a distinction between classical (EC)DH and post-quantum key exchanges, nor between post-quantum algorithms that are true key exchanges and post-quantum algorithms that act as key transport mechanisms: all are treated equivalently by the protocol. This document renames a field in the Key Exchange Payload from "Diffie-Hellman Group Num" to "Key Exchange Method". This document also renames Transform Type 4 from "Diffie-Hellman Group (D-H)" to "Key Exchange Method (KE)". The corresponding renaming to the IANA registry is described in Section 3.¶
The fact that newly defined transforms share the same registry for
possible Transform IDs with Transform Type 4 allows additional key
exchanges to be of any type: either post-quantum or classical
(EC)DH. This approach allows any combination of the defined key
exchange methods to take place. This also allows IKE peers to perform
a single post-quantum key exchange in the IKE_
The SA payload in the IKE_
Then, provided that additional key exchanges are negotiated, the
initiator and the responder perform one or more IKE_
Initiator Responder
---------------------------------------------------------------------
<-- IKE_SA_INIT (additional key exchanges negotiation) -->
<-- {IKE_INTERMEDIATE (additional key exchange)} -->
...
<-- {IKE_INTERMEDIATE (additional key exchange)} -->
<-- {IKE_AUTH} -->
2.2. Protocol Details
In the simplest case, the initiator starts a single key exchange
(and has no interest in supporting multiple), and it is not concerned
with possible fragmentation of the IKE_
In this case, the initiator performs the IKE_
If the initiator wants to negotiate multiple key exchanges, then the initiator uses the protocol behavior listed below.¶
2.2.1. IKE_SA_INIT Round: Negotiation
Multiple key exchanges are negotiated using the standard IKEv2
mechanism via SA payload. For this purpose, seven new transform
types are defined: Additional Key Exchange 1 (ADDKE1) with IANA-assigned value
6, Additional Key Exchange 2 (ADDKE2) (7), Additional Key Exchange 3 (ADDKE3) (8),
Additional Key Exchange 4 (ADDKE4) (9), Additional Key Exchange 5 (ADDKE5) (10),
Additional Key Exchange 6 (ADDKE6) (11), and Additional Key Exchange 7 (ADDKE7) (12).
They are collectively called "Additional Key Exchange (ADDKE)
Transform Types" in this document and have slightly different semantics
than the existing IKEv2 Transform Types. They are interpreted as an
indication of additional key exchange methods that peers agree to
perform in a series of IKE_
The key exchange method negotiated via Transform Type 4 always
takes place in the IKE_
With these semantics, note that ADDKE Transform Types are not associated with any particular type of key exchange and do not have any Transform IDs that are specific per Transform Type IANA registry. Instead, they all share a single registry for Transform IDs, namely "Transform Type 4 - Key Exchange Method Transform IDs". All key exchange algorithms (both classical or post-quantum) should be added to this registry. This approach gives peers flexibility in defining the ways they want to combine different key exchange methods.¶
When forming a proposal, the initiator adds transforms for the
IKE_
The responder performs the negotiation using the standard IKEv2
procedure described in Section 3.3 of [RFC7296]. However, for the ADDKE Transform Types,
the responder's choice MUST NOT contain duplicated
algorithms (those with an identical Transform ID and attributes),
except for the Transform ID of NONE. An algorithm is represented as
a transform. In some cases, the transform could include a set of
associated attributes that define details of the algorithm. In this
case, two transforms can be the same, but the attributes must be
different. Additionally, the order of the attributes does not
affect the equality of the algorithm, so the following two
transforms define the same algorithm:
"ID=alg1, ATTR1=attr1, ATTR2=attr2" and
"ID=alg1, ATTR2=attr2, ATTR1=attr1". If the
responder is unable to select algorithms that are not duplicated for each
proposed key exchange (either because the proposal contains too few
choices or due to the local policy restrictions on using the
proposed algorithms), then the responder MUST reject
the message with an error notification of type NO_
If the responder selects NONE for some ADDKE
Transform Types (provided they are proposed by the initiator), then any
corresponding additional key exchanges MUST NOT take place.
Therefore, if the
initiator includes NONE in all of the ADDKE
Transform Types and the responder selects this value for all of them,
then no IKE_
The initiator MAY propose ADDKE Transform Types that are not consecutive, for example, proposing ADDKE2 and ADDKE5 Transform Types only. The responder MUST treat all of the omitted ADDKE transforms as if they were proposed with Transform ID NONE.¶
Below is an example of the SA payload in the initiator's IKE_
SA Payload
|
+--- Proposal #1 ( Proto ID = IKE(1), SPI Size = 8,
| 9 transforms, SPI = 0x35a1d6f22564f89d )
|
+-- Transform ENCR ( ID = ENCR_AES_GCM_16 )
| +-- Attribute ( Key Length = 256 )
|
+-- Transform KE ( ID = 4096-bit MODP Group )
|
+-- Transform PRF ( ID = PRF_HMAC_SHA2_256 )
|
+-- Transform ADDKE2 ( ID = PQ_KEM_1 )
|
+-- Transform ADDKE2 ( ID = PQ_KEM_2 )
|
+-- Transform ADDKE3 ( ID = PQ_KEM_1 )
|
+-- Transform ADDKE3 ( ID = PQ_KEM_2 )
|
+-- Transform ADDKE5 ( ID = PQ_KEM_3 )
|
+-- Transform ADDKE5 ( ID = NONE )
In this example, the initiator proposes performing the
initial key exchange using a 4096-bit MODP Group followed by two
mandatory additional key exchanges (i.e., ADDKE2 and ADDKE3 Transform Types)
using PQ_
The responder might return the following SA payload, indicating
that it agrees to perform two additional key exchanges, PQ_
SA Payload
|
+--- Proposal #1 ( Proto ID = IKE(1), SPI Size = 8,
| 6 transforms, SPI = 0x8df52b331a196e7b )
|
+-- Transform ENCR ( ID = ENCR_AES_GCM_16 )
| +-- Attribute ( Key Length = 256 )
|
+-- Transform KE ( ID = 4096-bit MODP Group )
|
+-- Transform PRF ( ID = PRF_HMAC_SHA2_256 )
|
+-- Transform ADDKE2 ( ID = PQ_KEM_2 )
|
+-- Transform ADDKE3 ( ID = PQ_KEM_1 )
|
+-- Transform ADDKE5 ( ID = NONE )
If the initiator includes any ADDKE Transform
Types into the SA payload in the IKE_
Initiator Responder
---------------------------------------------------------------------
HDR, SAi1(.. ADDKE*...), KEi, Ni,
N(INTERMEDIATE_EXCHANGE_SUPPORTED) --->
HDR, SAr1(.. ADDKE*...), KEr, Nr,
[CERTREQ],
<--- N(INTERMEDIATE_EXCHANGE_SUPPORTED)
It is possible for an attacker to manage to send a response to
the initiator's IKE_
2.2.2. IKE_INTERMEDIATE Round: Additional Key Exchanges
For each additional key exchange agreed to in the IKE_
Initiator Responder
---------------------------------------------------------------------
HDR, SK {KEi(n)} -->
<-- HDR, SK {KEr(n)}
The initiator sends key exchange data in the KEi(n) payload.
This message is protected with the current SK_
On receiving this, the responder sends back key exchange payload
KEr(n); "KEr(n)" denotes the n-th IKE_
The former "Diffie-Hellman Group Num" (now called "Key Exchange Method") field in the KEi(n) and KEr(n) payloads MUST match the n-th negotiated additional key exchange.¶
Once this exchange is done, both sides compute an updated keying material:¶
SKEYSEED(n) = prf(SK_d(n-1), SK(n) | Ni | Nr)
From this exchange, SK(n) is the resulting shared secret. Ni and Nr are nonces from the IKE_
{SK_d(n) | SK_ai(n) | SK_ar(n) | SK_ei(n) | SK_er(n) | SK_pi(n) |
SK_pr(n)} = prf+ (SKEYSEED(n), Ni | Nr | SPIi | SPIr)
Both the initiator and the responder use these updated key
values in the next exchange
2.2.3. IKE_AUTH Exchange
After all IKE_
2.2.4. CREATE_CHILD_SA Exchange
The CREATE_
Using multiple key exchanges with a CREATE_
The key exchange negotiated via Transform Type 4 always takes
place in the CREATE_
After an IKE SA is created, the window size may be greater than one; thus, multiple concurrent exchanges may be in progress. Therefore, it is
essential to link the IKE_
The responder MUST include this notification in a
CREATE_
Below is an example of CREATE_
Initiator Responder
---------------------------------------------------------------------
HDR(CREATE_CHILD_SA), SK {SA, Ni, KEi} -->
<-- HDR(CREATE_CHILD_SA), SK {SA, Nr, KEr,
N(ADDITIONAL_KEY_EXCHANGE)(link1)}
HDR(IKE_FOLLOWUP_KE), SK {KEi(1),
N(ADDITIONAL_KEY_EXCHANGE)(link1)} -->
<-- HDR(IKE_FOLLOWUP_KE), SK {KEr(1),
N(ADDITIONAL_KEY_EXCHANGE)(link2)}
HDR(IKE_FOLLOWUP_KE), SK {KEi(2),
N(ADDITIONAL_KEY_EXCHANGE)(link2)} -->
<-- HDR(IKE_FOLLOWUP_KE), SK {KEr(2),
N(ADDITIONAL_KEY_EXCHANGE)(link3)}
HDR(IKE_FOLLOWUP_KE), SK {KEi(3),
N(ADDITIONAL_KEY_EXCHANGE)(link3)} -->
<-- HDR(IKE_FOLLOWUP_KE), SK {KEr(3)}
The former "Diffie-Hellman Group Num" (now called "Key Exchange Method") field in the KEi(n) and KEr(n) payloads MUST match the n-th negotiated additional key exchange.¶
Due to some unexpected events (e.g., a
reboot), it is possible that the initiator may lose its state, forget that it is in
the process of performing additional key exchanges, and never
start the remaining IKE_
It may also take too long for the initiator to prepare
and to send the next IKE_
It is possible that
the peers start rekeying the IKE SA or the Child SA at the same time, which is called
"simultaneous rekeying". Sections 2.8.1 and 2.8.2 of [RFC7296]
describe how IKEv2 handles this situation. In a nutshell, IKEv2
follows the rule that, in the case of simultaneous rekeying, if two
identical new IKE SAs (or two pairs of Child SAs) are created, then
one of them should be deleted. Which one to delete is
determined by comparing the values of four nonces that are used in
the colliding CREATE_
With multiple key exchanges, the SAs are not yet created when
the CREATE_
In most cases, rekey collisions are resolved in the
CREATE_
If multiple key exchanges are negotiated in the CREATE_
In the case of an IKE SA rekey:¶
SKEYSEED = prf(SK_d, SK(0) | Ni | Nr | SK(1) | ... SK(n))
In the case of a Child SA creation or rekey:¶
KEYMAT = prf+ (SK_d, SK(0) | Ni | Nr | SK(1) | ... SK(n))
In both cases, SK_
2.2.5. Interaction with IKEv2 Extensions
It is believed that this specification requires no modification to the IKEv2 extensions defined so far. In particular, the IKE SA resumption mechanism defined in [RFC5723] can be used to resume IKE SAs created using this specification.¶
2.2.5.1. Interaction with Childless IKE SA
It is possible to establish IKE SAs with post-quantum algorithms by
only using IKE_
It is also possible for the initiator to send proposals without
any ADDKE Transform Types in the IKE_
The support for childless IKE SA is not negotiated, but it is
the responder that indicates the support for this mode. As such,
the responder cannot enforce that the initiator use this mode.
Therefore, it is entirely possible that the initiator does not
support this extension and sends IKE_
Note that if the initial IKE SA is used to transfer sensitive information, then this information will not be protected using the additional key exchanges, which may use post-quantum algorithms. In this arrangement, the peers will have to use post-quantum algorithm in Transform Type 4 in order to mitigate the risk of quantum attack.¶
3. IANA Considerations
This document adds a new exchange type into the "IKEv2 Exchange Types" registry:¶
44 IKE_FOLLOWUP_KE
This document renames Transform Type 4 defined in the "Transform Type Values" registry from "Diffie-Hellman Group (D-H)" to "Key Exchange Method (KE)".¶
This document renames the IKEv2 registry originally titled "Transform Type 4 - Diffie-Hellman Group Transform IDs" to "Transform Type 4 - Key Exchange Method Transform IDs".¶
This document adds the following Transform Types to the "Transform Type Values" registry:¶
| Type | Description | Used In |
|---|---|---|
| 6 | Additional Key Exchange 1 (ADDKE1) | (optional in IKE, AH, ESP) |
| 7 | Additional Key Exchange 2 (ADDKE2) | (optional in IKE, AH, ESP) |
| 8 | Additional Key Exchange 3 (ADDKE3) | (optional in IKE, AH, ESP) |
| 9 | Additional Key Exchange 4 (ADDKE4) | (optional in IKE, AH, ESP) |
| 10 | Additional Key Exchange 5 (ADDKE5) | (optional in IKE, AH, ESP) |
| 11 | Additional Key Exchange 6 (ADDKE6) | (optional in IKE, AH, ESP) |
| 12 | Additional Key Exchange 7 (ADDKE7) | (optional in IKE, AH, ESP) |
This document defines a new Notify Message Type in the "IKEv2 Notify Message Types - Status Types" registry:¶
16441 ADDITIONAL_KEY_EXCHANGE
This document also defines a new Notify Message Type in the "IKEv2 Notify Message Types - Error Types" registry:¶
47 STATE_NOT_FOUND
IANA has added the following instructions for designated experts for the "Transform Type 4 - Key Exchange Method Transform IDs" subregistry:¶
- While adding new Key Exchange (KE) methods, the following considerations must be applied. A KE method must take exactly one round-trip (one IKEv2 exchange), and at the end of this exchange, both peers must be able to derive the shared secret. In addition, any public value that peers exchanged during a KE method must fit into a single IKEv2 payload. If these restrictions are not met for a KE method, then there must be documentation on how this KE method is used in IKEv2.¶
IANA has also completed the following changes. It is assumed that [RFC9370] refers to this specification.¶
-
Added a reference to [RFC9370] in what was the "Transform Type 4 - Diffie-Hellman Group Transform IDs" registry.¶
-
Replaced the Note on what was the "Transform Type 4 - Diffie-Hellman Group Transform IDs" registry with the following notes:¶
This registry was originally named "Transform Type 4 - Diffie-Hellman Group Transform IDs" and was referenced using that name in a number of RFCs published prior to [RFC9370], which gave it the current title.¶
This registry is used by the "Key Exchange Method (KE)" transform type and by all "Additional Key Exchange (ADDKE)" transform types.¶
To find out requirement levels for Key Exchange Methods for IKEv2, see [RFC8247].¶
-
Appended [RFC9370] to the Reference column of Transform Type 4 in the "Transform Type Values" registry.¶
-
Added these notes to the "Transform Type Values" registry:¶
"Key Exchange Method (KE)" transform type was originally named "Diffie-Hellman Group (D-H)" and was referenced by that name in a number of RFCs published prior to [RFC9370], which gave it the current title.¶
All "Additional Key Exchange (ADDKE)" entries use the same "Transform Type 4 - Key Exchange Method Transform IDs" registry as the "Key Exchange Method (KE)" entry.¶
4. Security Considerations
The extension in this document is intended to mitigate two possible threats in IKEv2: the compromise of (EC)DH key exchange using Shor's algorithm while remaining backward compatible and the potential compromise of existing or future PQC key exchange algorithms. To address the former threat, this extension allows the establishment of a shared secret by using multiple key exchanges: typically, one classical (EC)DH and the other one post-quantum algorithm. In order to address the latter threat, multiple key exchanges using a post-quantum algorithm can be performed to form the shared key.¶
Unlike key exchange methods (Transform Type 4), the Encryption Algorithm (Transform Type 1), the Pseudorandom Function (Transform Type 2), and the Integrity Algorithm (Transform Type 3) are not susceptible to Shor's algorithm. However, they are susceptible to Grover's attack [GROVER], which allows a quantum computer to perform a brute force key search, using quadratically fewer steps than the classical counterpart. Simply increasing the key length can mitigate this attack. It was previously believed that one needed to double the key length of these algorithms. However, there are a number of factors that suggest that it is quite unlikely to achieve the quadratic speedup using Grover's algorithm. According to NIST [NISTPQCFAQ], current applications can continue using an AES algorithm with the minimum key length of 128 bits. Nevertheless, if the data needs to remain secure for many years to come, one may want to consider using a longer key size for the algorithms in Transform Types 1-3.¶
SKEYSEED is calculated from shared SK(x), using an algorithm defined in Transform Type 2. While a quantum attacker may learn the value of SK(x), if this value is obtained by means of a classical key exchange, other SK(x) values generated by means of a post-quantum algorithm ensure that the final SKEYSEED is not compromised. This assumes that the algorithm defined in the Transform Type 2 is quantum resistant.¶
The ordering of the additional key exchanges should not matter in general, as only the final shared secret is of interest. Nonetheless, because the strength of the running shared secret increases with every additional key exchange, an implementer may want to first perform the most secure method (in some metrics) followed by less secure methods.¶
The main focus of this document is to prevent a passive attacker from
performing a "harvest
- The first reason is that the authentication step remains classical. In particular, the authenticity of the SAs established under IKEv2 is protected by using a pre-shared key or digital signature algorithms. While the pre-shared key option, provided the key is long enough, is post-quantum secure, the other algorithms are not. Moreover, in implementations where scalability is a requirement, the pre-shared key method may not be suitable. Post-quantum authenticity may be provided by using a post-quantum digital signature.¶
- Secondly, it should be noted that the purpose of post-quantum algorithms is
to provide resistance to attacks mounted in the future. The current
threat is that encrypted sessions are subject to eavesdropping and are
archived with decryption by quantum computers at some
point in the future. Until quantum computers become available, there
is no point in attacking the authenticity of a connection because
there are no possibilities for exploitation. These only occur at
the time of the connection, for example, by mounting an on-path
attack. Consequently, there is less urgency for
post-quantum authenticity compared to post-quantum confidentiality
.¶
Performing multiple key exchanges while establishing an IKE SA increases the responder's susceptibility to DoS attacks because of an increased amount of resources needed before the initiator is authenticated. This is especially true for post-quantum key exchange methods, where many of them are more memory and/or CPU intensive than the classical counterparts.¶
Responders may consider recommendations from [RFC8019] to deal with increased DoS-attack susceptibility. It
is also possible that the responder only agrees to create an initial IKE SA
without performing additional key exchanges if the initiator includes
such an option in its proposals. Then, peers immediately rekey the
initial IKE SA with the CREATE_
5. References
5.1. Normative References
- [RFC2119]
-
Bradner, S., "Key words for use in RFCs to Indicate Requirement Levels", BCP 14, RFC 2119, DOI 10
.17487 , , <https:///RFC2119 www >..rfc -editor .org /info /rfc2119 - [RFC7296]
-
Kaufman, C., Hoffman, P., Nir, Y., Eronen, P., and T. Kivinen, "Internet Key Exchange Protocol Version 2 (IKEv2)", STD 79, RFC 7296, DOI 10
.17487 , , <https:///RFC7296 www >..rfc -editor .org /info /rfc7296 - [RFC8174]
-
Leiba, B., "Ambiguity of Uppercase vs Lowercase in RFC 2119 Key Words", BCP 14, RFC 8174, DOI 10
.17487 , , <https:///RFC8174 www >..rfc -editor .org /info /rfc8174 - [RFC9242]
-
Smyslov, V., "Intermediate Exchange in the Internet Key Exchange Protocol Version 2 (IKEv2)", RFC 9242, DOI 10
.17487 , , <https:///RFC9242 www >..rfc -editor .org /info /rfc9242
5.2. Informative References
- [BEYOND-64K]
-
Tjhai, CJ., Heider, T., and V. Smyslov, "Beyond 64KB Limit of IKEv2 Payloads", Work in Progress, Internet-Draft, draft
-tjhai , , <https://-ikev2 -beyond -64k -limit -03 datatracker >..ietf .org /doc /html /draft -tjhai -ikev2 -beyond -64k -limit -03 - [G-IKEV2]
-
Smyslov, V. and B. Weis, "Group Key Management using IKEv2", Work in Progress, Internet-Draft, draft
-ietf , , <https://-ipsecme -g -ikev2 -09 datatracker >..ietf .org /doc /html /draft -ietf -ipsecme -g -ikev2 -09 - [GROVER]
-
Grover, L., "A fast quantum mechanical algorithm for database search", Proc. of the Twenty-Eighth Annual ACM Symposium on the Theory of Computing (STOC), pp. 212-219, DOI 10
.48550 , , <https:///ar Xiv .quant -ph /9605043 doi >..org /10 .48550 /ar Xiv .quant -ph /9605043 - [IKEV2TYPE4ID]
-
IANA, "Internet Key Exchange Version 2 (IKEv2) Parameters: Transform Type 4 - Diffie-Hellman Group Transform IDs", <https://
www >..iana .org /assignments /ikev2 -parameters / - [NISTPQCFAQ]
-
NIST, "Post-Quantum Cryptography Standard", , <https://
csrc >..nist .gov /Projects /post -quantum -cryptography /faqs - [RFC5723]
-
Sheffer, Y. and H. Tschofenig, "Internet Key Exchange Protocol Version 2 (IKEv2) Session Resumption", RFC 5723, DOI 10
.17487 , , <https:///RFC5723 www >..rfc -editor .org /info /rfc5723 - [RFC6023]
-
Nir, Y., Tschofenig, H., Deng, H., and R. Singh, "A Childless Initiation of the Internet Key Exchange Version 2 (IKEv2) Security Association (SA)", RFC 6023, DOI 10
.17487 , , <https:///RFC6023 www >..rfc -editor .org /info /rfc6023 - [RFC7383]
-
Smyslov, V., "Internet Key Exchange Protocol Version 2 (IKEv2) Message Fragmentation", RFC 7383, DOI 10
.17487 , , <https:///RFC7383 www >..rfc -editor .org /info /rfc7383 - [RFC8019]
-
Nir, Y. and V. Smyslov, "Protecting Internet Key Exchange Protocol Version 2 (IKEv2) Implementations from Distributed Denial
-of , RFC 8019, DOI 10-Service Attacks" .17487 , , <https:///RFC8019 www >..rfc -editor .org /info /rfc8019 - [RFC8247]
-
Nir, Y., Kivinen, T., Wouters, P., and D. Migault, "Algorithm Implementation Requirements and Usage Guidance for the Internet Key Exchange Protocol Version 2 (IKEv2)", RFC 8247, DOI 10
.17487 , , <https:///RFC8247 www >..rfc -editor .org /info /rfc8247 - [RFC8784]
-
Fluhrer, S., Kampanakis, P., McGrew, D., and V. Smyslov, "Mixing Preshared Keys in the Internet Key Exchange Protocol Version 2 (IKEv2) for Post-quantum Security", RFC 8784, DOI 10
.17487 , , <https:///RFC8784 www >..rfc -editor .org /info /rfc8784
Appendix A. Sample Multiple Key Exchanges
This appendix shows some examples of multiple key exchanges. These examples are not normative, and they describe some message flow scenarios that may occur in establishing an IKE or Child SA. Note that some payloads that are not relevant to multiple key exchanges may be omitted for brevity.¶
A.1. IKE_INTERMEDIATE Exchanges Carrying Additional Key Exchange Payloads
The exchanges below show that the initiator proposes the use of additional key exchanges to establish an IKE SA. The initiator proposes three sets of additional key exchanges, all of which are optional. Therefore, the responder can choose NONE for some or all of the additional exchanges if the proposed key exchange methods are not supported or for whatever reasons the responder decides not to perform the additional key exchange.¶
Initiator Responder
---------------------------------------------------------------------
HDR(IKE_SA_INIT), SAi1(.. ADDKE*...), --->
KEi(Curve25519), Ni, N(IKEV2_FRAG_SUPPORTED),
N(INTERMEDIATE_EXCHANGE_SUPPORTED)
Proposal #1
Transform ECR (ID = ENCR_AES_GCM_16,
256-bit key)
Transform PRF (ID = PRF_HMAC_SHA2_512)
Transform KE (ID = Curve25519)
Transform ADDKE1 (ID = PQ_KEM_1)
Transform ADDKE1 (ID = PQ_KEM_2)
Transform ADDKE1 (ID = NONE)
Transform ADDKE2 (ID = PQ_KEM_3)
Transform ADDKE2 (ID = PQ_KEM_4)
Transform ADDKE2 (ID = NONE)
Transform ADDKE3 (ID = PQ_KEM_5)
Transform ADDKE3 (ID = PQ_KEM_6)
Transform ADDKE3 (ID = NONE)
<--- HDR(IKE_SA_INIT), SAr1(.. ADDKE*...),
KEr(Curve25519), Nr, N(IKEV2_FRAG_SUPPORTED),
N(INTERMEDIATE_EXCHANGE_SUPPORTED)
Proposal #1
Transform ECR (ID = ENCR_AES_GCM_16,
256-bit key)
Transform PRF (ID = PRF_HMAC_SHA2_512)
Transform KE (ID = Curve25519)
Transform ADDKE1 (ID = PQ_KEM_2)
Transform ADDKE2 (ID = NONE)
Transform ADDKE3 (ID = PQ_KEM_5)
HDR(IKE_INTERMEDIATE), SK {KEi(1)(PQ_KEM_2)} -->
<--- HDR(IKE_INTERMEDIATE), SK {KEr(1)(PQ_KEM_2)}
HDR(IKE_INTERMEDIATE), SK {KEi(2)(PQ_KEM_5)} -->
<--- HDR(IKE_INTERMEDIATE), SK {KEr(2)(PQ_KEM_5)}
HDR(IKE_AUTH), SK{ IDi, AUTH, SAi2, TSi, TSr } --->
<--- HDR(IKE_AUTH), SK{ IDr, AUTH, SAr2,
TSi, TSr }
In this particular example, the responder chooses to perform two
additional key exchanges. It selects PQ_
SKEYSEED(1) = prf(SK_d, SK(1) | Ni | Nr)
The updated SKEYSEED value is then used to derive the following keying materials.¶
{SK_d(1) | SK_ai(1) | SK_ar(1) | SK_ei(1) | SK_er(1) | SK_pi(1) |
SK_pr(1)} = prf+ (SKEYSEED(1), Ni | Nr | SPIi | SPIr)
As per [RFC9242],
both peers compute Int
In the next IKE_
SKEYSEED(2) = prf(SK_d(1), SK(2) | Ni | Nr)
{SK_d(2) | SK_ai(2) | SK_ar(2) | SK_ei(2) | SK_er(2) | SK_pi(2) |
SK_pr(2)} = prf+ (SKEYSEED(2), Ni | Nr | SPIi | SPIr)
In this update, SK(2) is the shared secret from the third
additional key exchange, i.e., PQ_
After the completion of the second IKE_
A.2. No Additional Key Exchange Used
The initiator proposes two sets of optional additional key
exchanges, but the responder does not support any of them. The
responder chooses NONE for each set. Consequently, the
IKE_
Initiator Responder
---------------------------------------------------------------------
HDR(IKE_SA_INIT), SAi1(.. ADDKE*...), --->
KEi(Curve25519), Ni, N(IKEV2_FRAG_SUPPORTED),
N(INTERMEDIATE_EXCHANGE_SUPPORTED)
Proposal #1
Transform ECR (ID = ENCR_AES_GCM_16,
256-bit key)
Transform PRF (ID = PRF_HMAC_SHA2_512)
Transform KE (ID = Curve25519)
Transform ADDKE1 (ID = PQ_KEM_1)
Transform ADDKE1 (ID = PQ_KEM_2)
Transform ADDKE1 (ID = NONE)
Transform ADDKE2 (ID = PQ_KEM_3)
Transform ADDKE2 (ID = PQ_KEM_4)
Transform ADDKE2 (ID = NONE)
<--- HDR(IKE_SA_INIT), SAr1(.. ADDKE*...),
KEr(Curve25519), Nr, N(IKEV2_FRAG_SUPPORTED),
N(INTERMEDIATE_EXCHANGE_SUPPORTED)
Proposal #1
Transform ECR (ID = ENCR_AES_GCM_16,
256-bit key)
Transform PRF (ID = PRF_HMAC_SHA2_512)
Transform KE (ID = Curve25519)
Transform ADDKE1 (ID = NONE)
Transform ADDKE2 (ID = NONE)
HDR(IKE_AUTH), SK{ IDi, AUTH, SAi2, TSi, TSr } --->
<--- HDR(IKE_AUTH), SK{ IDr, AUTH, SAr2,
TSi, TSr }
A.3. Additional Key Exchange in the CREATE_CHILD_SA Exchange Only
The exchanges below show that the initiator does not propose the
use of additional key exchanges to establish an IKE SA, but they are
required in order to establish a Child SA. In order to establish a
fully quantum
Initiator Responder
---------------------------------------------------------------------
HDR(IKE_SA_INIT), SAi1, --->
KEi(Curve25519), Ni, N(IKEV2_FRAG_SUPPORTED)
<--- HDR(IKE_SA_INIT), SAr1,
KEr(Curve25519), Nr, N(IKEV2_FRAG_SUPPORTED),
N(CHILDLESS_IKEV2_SUPPORTED)
HDR(IKE_AUTH), SK{ IDi, AUTH } --->
<--- HDR(IKE_AUTH), SK{ IDr, AUTH }
HDR(CREATE_CHILD_SA),
SK{ SAi(.. ADDKE*...), Ni, KEi(Curve25519) } --->
Proposal #1
Transform ECR (ID = ENCR_AES_GCM_16,
256-bit key)
Transform PRF (ID = PRF_HMAC_SHA2_512)
Transform KE (ID = Curve25519)
Transform ADDKE1 (ID = PQ_KEM_1)
Transform ADDKE1 (ID = PQ_KEM_2)
Transform ADDKE2 (ID = PQ_KEM_5)
Transform ADDKE2 (ID = PQ_KEM_6)
Transform ADDKE2 (ID = NONE)
<--- HDR(CREATE_CHILD_SA), SK{ SAr(.. ADDKE*...),
Nr, KEr(Curve25519),
N(ADDITIONAL_KEY_EXCHANGE)(link1) }
Proposal #1
Transform ECR (ID = ENCR_AES_GCM_16,
256-bit key)
Transform PRF (ID = PRF_HMAC_SHA2_512)
Transform KE (ID = Curve25519)
Transform ADDKE1 (ID = PQ_KEM_2)
Transform ADDKE2 (ID = PQ_KEM_5)
HDR(IKE_FOLLOWUP_KE), SK{ KEi(1)(PQ_KEM_2), --->
N(ADDITIONAL_KEY_EXCHANGE)(link1) }
<--- HDR(IKE_FOLLOWUP_KE), SK{ KEr(1)(PQ_KEM_2),
N(ADDITIONAL_KEY_EXCHANGE)(link2) }
HDR(IKE_FOLLOWUP_KE), SK{ KEi(2)(PQ_KEM_5), --->
N(ADDITIONAL_KEY_EXCHANGE)(link2) }
<--- HDR(IKE_FOLLOWUP_KE), SK{ KEr(2)(PQ_KEM_5) }
A.4. No Matching Proposal for Additional Key Exchanges
The initiator proposes the combination of PQ_
Initiator Responder
---------------------------------------------------------------------
HDR(IKE_SA_INIT), SAi1(.. ADDKE*...), --->
KEi(Curve25519), Ni, N(IKEV2_FRAG_SUPPORTED),
N(INTERMEDIATE_EXCHANGE_SUPPORTED)
Proposal #1
Transform ECR (ID = ENCR_AES_GCM_16,
256-bit key)
Transform PRF (ID = PRF_HMAC_SHA2_512)
Transform KE (ID = Curve25519)
Transform ADDKE1 (ID = PQ_KEM_1)
Transform ADDKE1 (ID = PQ_KEM_2)
Transform ADDKE2 (ID = PQ_KEM_3)
Transform ADDKE2 (ID = PQ_KEM_4)
Transform ADDKE2 (ID = NONE)
<--- HDR(IKE_SA_INIT), N(NO_PROPOSAL_CHOSEN)
Appendix B. Design Criteria
The design of the extension is driven by the following criteria:¶
- 1)
-
Need for PQC in IPsec¶
Quantum computers, which might become feasible in the near future, pose a threat to our classical public key cryptography. PQC, a family of public key cryptography that is believed to be resistant to these computers, needs to be integrated into the IPsec protocol suite to restore confidentiality and authenticity.¶
- 2)
-
Hybrid¶
There is currently no post-quantum key exchange that is trusted at the level that (EC)DH is trusted for defending against conventional (non-quantum) adversaries. A hybrid post-quantum algorithm to be introduced, along with the well
-established primitives, addresses this concern, since the overall security is at least as strong as each individual primitive.¶ - 3)
-
Focus on post-quantum confidentiality¶
A passive attacker can store all monitored encrypted IPsec communication today and decrypt it once a quantum computer is available in the future. This attack can have serious consequences that will not be visible for years to come. On the other hand, an attacker can only perform active attacks, such as impersonation of the communicating peers, once a quantum computer is available sometime in the future. Thus, this specification focuses on confidentiality due to the urgency of this problem and presents a defense against the serious attack described above, but it does not address authentication because it is less urgent at this stage.¶
- 4)
-
Limit the amount of exchanged data¶
The protocol design should be such that the amount of exchanged data, such as public keys, is kept as small as possible, even if the initiator and the responder need to agree on a hybrid group or if multiple public keys need to be exchanged.¶
- 5)
-
Not post-quantum specific¶
Any cryptographic algorithm could be potentially broken in the future by currently unknown or impractical attacks. Quantum computers are merely the most concrete example of this. The design does not categorize algorithms as "post-quantum" or "non
-post -quantum", nor does it create assumptions about the properties of the algorithms; meaning that if algorithms with different properties become necessary in the future, this extension can be used unchanged to facilitate migration to those algorithms.¶ - 6)
-
Limited amount of changes¶
A key goal is to limit the number of changes required when enabling a post-quantum handshake. This ensures easier and quicker adoption in existing implementations
.¶ - 7)
-
Localized changes¶
Another key requirement is that changes to the protocol are limited in scope, in particular, limiting changes in the exchanged messages and in the state machine, so that they can be easily implemented.¶
- 8)
-
Deterministic operation¶
This requirement means that the hybrid post-quantum exchange and, thus, the computed keys will be based on algorithms that both client and server wish to support.¶
- 9)
-
Fragmentation support¶
Some PQC algorithms could be relatively bulky and might require fragmentation. Thus, a design goal is the adaptation and adoption of an existing fragmentation method or the design of a new method that allows for the fragmentation of the key shares.¶
- 10)
-
Backward compatibility and interoperabilit
y¶ This is a fundamental requirement to ensure that hybrid post-quantum IKEv2 and standard IKEv2 implementations as per [RFC7296] are interoperable.¶
- 11)
-
Compliance with USA Federal Information Processing Standards (FIPS)¶
IPsec is widely used in Federal Information Systems, and FIPS certification is an important requirement. However, at the time of writing, none of the algorithms that is believed to be post-quantum is yet FIPS compliant. Nonetheless, it is possible to combine this post-quantum algorithm with a FIPS-compliant key establishment method so that the overall design remains FIPS compliant [NISTPQCFAQ].¶
- 12)
-
Ability to use this method with multiple classical (EC)DH key exchanges¶
In some situations, peers have no single, mutually trusted, key exchange algorithm (e.g., due to local policy restrictions). The ability to combine two (or more) key exchange methods in such a way that the resulting shared key depends on all of them allows peers to communicate in this situation.¶
Appendix C. Alternative Design
This section gives an overview on a number of alternative approaches that have been considered but later discarded. These approaches are as follows.¶
-
Sending the classical and post-quantum key exchanges as a single transform¶
A method to combine the various key exchanges into a single large KE payload was considered. This effort is documented in a previous version of this document
(draft -tjhai -ipsecme -hybrid -qske -ikev2 -01 ). This method allows us to cleanly apply hybrid key exchanges during the Child SA. However, it does add considerable complexity and requires an independent fragmentation solution.¶ -
Sending post-quantum proposals and policies in the KE payload only¶
With the objective of not introducing unnecessary notify payloads, a method to communicate the hybrid post-quantum proposal in the KE payload during the first pass of the protocol exchange was considered. Unfortunately, this design is susceptible to the following downgrade attack. Consider the scenario where there is an on-path attacker sitting between an initiator and a responder. Through the SAi payload, the initiator proposes using a hybrid post-quantum group and, as a fallback, a Diffie-Hellman group; and through the KEi payload, the initiator proposes a list of hybrid post-quantum proposals and policies. The on-path attacker intercepts this traffic and replies with N
(INVALID_ KE_ PAYLOAD ), suggesting a downgrade to the fallback Diffie-Hellman group instead. The initiator then resends the same SAi payload and the KEi payload containing the public value of the fallback Diffie-Hellman group. Note that the attacker may forward the second IKE_ SA_ INIT message only to the responder. Therefore, at this point in time, the responder will not have the information that the initiator prefers the hybrid group. Of course, it is possible for the responder to have a policy to reject an IKE_ SA_ INIT message that (a) offers a hybrid group but does not offer the corresponding public value in the KEi payload and (b) the responder has not specifically acknowledged that it does not support the requested hybrid group. However, the checking of this policy introduces unnecessary protocol complexity. Therefore, in order to fully prevent any downgrade attacks, using a KE payload alone is not sufficient, and the initiator MUST always indicate its preferred post-quantum proposals and policies in a notify payload in the subsequent IKE_ SA_ INIT messages following an N (INVALID_ KE_ PAYLOAD ) response.¶ -
New payload types to negotiate hybrid proposals and to carry post-quantum public values¶
Semantically, it makes sense to use a new payload type, which mimics the SA payload, to carry a hybrid proposal. Likewise, another new payload type that mimics the KE payload could be used to transport hybrid public value. Although, in theory, a new payload type could be made backward compatible by not setting its critical flag as per Section 2.5 of [RFC7296], it is believed that it may not be that simple in practice. Since the original release of IKEv2 in RFC 4306, no new payload type has ever been proposed; therefore, this creates a potential risk of having a backward
-compatibility issue from nonconformant IKEv2 implementations . Since there appears to be no other compelling advantages apart from a semantic one, the existing Transform Type and notify payloads are used instead.¶ -
Hybrid public value payload¶
One way to transport the negotiated hybrid public payload, which contains one classical Diffie-Hellman public value and one or more post-quantum public values, is to bundle these into a single KE payload. Alternatively, these could also be transported in a single new hybrid public value payload. However, following the same reasoning as above may not be a good idea from a backward
-compatibility perspective. Using a single KE payload would require encoding or formatting to be defined so that both peers are able to compose and extract the individual public values. However, it is believed that it is cleaner to send the hybrid public values in multiple KE payloads: one for each group or algorithm. Furthermore, at this point in the protocol exchange, both peers should have indicated support for handling multiple KE payloads.¶ -
Fragmentation¶
The handling of large IKE_
SA_ INIT messages has been one of the most challenging tasks. A number of approaches have been considered, and the two prominent ones that have been discarded are outlined as follows.¶ The first approach is to treat the entire IKE_
SA_ INIT message as a stream of bytes, which is then split into a number of fragments, each of which is wrapped onto a payload that will fit into the size of the network MTU. The payload that wraps each fragment has a new payload type, and it is envisaged that this new payload type will not cause a backward -compatibility issue because, at this stage of the protocol, both peers should have indicated support of fragmentation in the first pass of the IKE_ SA_ INIT exchange. The negotiation of fragmentation is performed using a notify payload, which also defines supporting parameters, such as the size of fragment in octets and the fragment identifier. The new payload that wraps each fragment of the messages in this exchange is assigned the same fragment identifier. Furthermore, it also has other parameters, such as a fragment index and total number of fragments. This approach has been discarded due to its blanket approach to fragmentation. In cases where only a few payloads need to be fragmented, this approach appears to be overly complicated.¶ Another idea that has been discarded is fragmenting an individual payload without introducing a new payload type. The idea is to use the 9-th bit (the bit after the critical flag in the RESERVED field) in the generic payload header as a flag to mark that this payload is fragmented. As an example, if a KE payload is to be fragmented, it may look as follows.¶
1 2 3 0 1 2 3 4 5 6 7 8 9 0 1 2 3 4 5 6 7 8 9 0 1 2 3 4 5 6 7 8 9 0 1 +-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+ | Next Payload |C|F| RESERVED | Payload Length | +-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+ | Diffie-Hellman Group Number | Fragment Identifier | +-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+ | Fragment Index | Total Fragments | +-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+ | Total KE Payload Data Length | +-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+ | | ~ Fragmented KE Payload ~ | | +-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+
Figure 1: Example of How to Fragment a KE Payload When the flag F is set, the current KE payload is a fragment of a larger KE payload. The Payload Length field denotes the size of this payload fragment in octets: including the size of the generic payload header. The 2-octet RESERVED field following Diffie-Hellman Group Number was to be used as a fragment identifier to help the assembly and disassembly of fragments. The Fragment Index and Total Fragments fields are self
-explanatory . The Total KE Payload Data Length indicates the size of the assembled KE payload data in octets. Finally, the actual fragment is carried in Fragment KE Payload field.¶ This approach has been discarded because it is believed that the working group may not want to use the RESERVED field to change the format of a packet, and that implementers may not like the added complexity from checking the fragmentation flag in each received payload. More importantly, fragmenting the messages in this way may leave the system to be more prone to denial
-of -service (DoS) attacks. This issue can be solved using IKE_ INTERMEDIATE [RFC9242] to transport the large post-quantum key exchange payloads and using the generic IKEv2 fragmentation protocol [RFC7383].¶ -
Group sub-identifier¶
As discussed before, each group identifier is used to distinguish a post-quantum algorithm. Further classification could be made on a particular post-quantum algorithm by assigning an additional value alongside the group identifier. This sub-identifier value may be used to assign different security
-parameter sets to a given post-quantum algorithm. However, this level of detail does not fit the principles of the document where it should deal with generic hybrid key exchange protocol and not a specific ciphersuite. Furthermore, there are enough Diffie-Hellman group identifiers should this be required in the future.¶
Acknowledgements
The authors would like to thank Frederic Detienne and Olivier Pelerin for their comments and suggestions, including the idea to negotiate the post-quantum algorithms using the existing KE payload. The authors are also grateful to Tobias Heider and Tobias Guggemos for valuable comments. Thanks to Paul Wouters for reviewing the document.¶