<?xml version="1.0"?>
<!DOCTYPE rfc SYSTEM "rfc2629.dtd" [
<!ENTITY RFC2119 PUBLIC '' 'http://xml2rfc.ietf.org/public/rfc/bibxml/reference.RFC.2119.xml'>
<!ENTITY RFC4648 PUBLIC '' 'http://xml2rfc.ietf.org/public/rfc/bibxml/reference.RFC.4648.xml'>
<!ENTITY RFC7517 PUBLIC '' 'http://xml2rfc.ietf.org/public/rfc/bibxml/reference.RFC.7517.xml'>
<!ENTITY RFC7518 PUBLIC '' 'http://xml2rfc.ietf.org/public/rfc/bibxml/reference.RFC.7518.xml'>
<!ENTITY RFC7638 PUBLIC '' 'http://xml2rfc.ietf.org/public/rfc/bibxml/reference.RFC.7638.xml'>
<!ENTITY RFC7748 PUBLIC '' 'http://xml2rfc.ietf.org/public/rfc/bibxml/reference.RFC.7748.xml'>
<!ENTITY EDDSA PUBLIC '' 'http://xml2rfc.ietf.org/public/rfc/bibxml3/reference.I-D.irtf-cfrg-eddsa.xml'>
]>

<?rfc compact="no"?>
<?rfc toc="yes"?>
<?rfc symrefs="yes"?>

<rfc category="std" ipr="trust200902"
     docName="draft-ietf-jose-cfrg-curves-01">

	<front>

		<title abbrev="CFRG ECDH and signatures in JOSE">
			CFRG ECDH and signatures in JOSE
		</title>

		<author initials="I." surname="Liusvaara"
			fullname="Ilari Liusvaara">
			<organization>Independent</organization>
			<address>
				<email>ilariliusvaara@welho.com</email>
			</address>
		</author>
    
		<date year="2016"/>

		<abstract>
			<t>This document defines how to use Diffie-Hellman algorithms "X25519" and "X448" as well
			as signature algorithms "Ed25519" and "Ed448" from IRTF CFRG elliptic
			curves work in JOSE.</t>
		</abstract>
    
	</front>

	<middle>

		<section anchor="intro" title="Introduction">
			<t>Internet Research Task Force (IRTF) Crypto Forum Research Group (CFRG) selected new
			Diffie-Hellman algorithms ("X25519" and "X448"; <xref target="RFC7748"/>) and
			signature algorithms ("Ed25519" and "Ed448");
			<xref target="I-D.irtf-cfrg-eddsa"/>) for asymmetric key cryptography. This document
			defines how those algorithms are to be used in JOSE in interoperable manner.</t>

			<t>This document defines the conventions to be used in context of <xref target="RFC7517" />
			and <xref target="RFC7518" /></t>

			<t>While the CFRG also defined two pairs of isogenous elliptic curves that underlie these
			algorithms, these curves are not directly exposed, as the algorithms laid on top are
			sufficient for the purposes of JOSE and are much easier to use (e.g. trying to apply ECDSA
			to those curves leads to nasty corner-cases and produces odd results).</t>

			<t>All inputs to and outputs from the the ECDH and signature functions are defined to
			be octet strings, with the exception of output of verification function, which is a
			boolean.</t>

			<section title="Requirements Terminology">
				<t>The key words "MUST", "MUST NOT", "REQUIRED", "SHALL", "SHALL NOT", "SHOULD",
				"SHOULD NOT", "RECOMMENDED", "MAY", and "OPTIONAL" in this document are to be
				interpreted as described in <xref target="RFC2119" />.</t>
			</section>
		</section>

		<section anchor="okp-keytype" title="Key type &apos;OKP&apos;">
			<t>A new key type (kty) value &quot;OKP&quot; (Octet Key Pair) is defined for public key
			algorithms that use octet strings as private and public keys. It has the following parameters:
			<list style="symbols">
				<t>The parameter "kty" MUST be "OKP".</t>
				<t>The parameter "crv" MUST be present, and contain the subtype of the key (from "JSON
				Web Elliptic curve" registry).</t>
				<t>The parameter "x" MUST be present, and contain the public key encoded using
				base64url <xref target="RFC4648" /> encoding.</t>
				<t>The parameter "d" MUST be present for private keys, and contain the private key
				encoded using base64url encoding. This parameter MUST NOT be present for public
				keys.</t>
			</list>
			Note: Do not assume that there is an underlying elliptic curve, despite the existence of the
			"crv" and "x" parameters (for instance, this key type could be extended to represent DH
			algorithms based on hyperelliptic surfaces).
			</t>
			<t>When calculating thumbprints <xref target="RFC7638" />, the three public key fields are
			included in the hash. That is, in lexicographic order: &quot;crv&quot;, &quot;kty&quot; and
			&quot;x&quot;.</t>
			<t>[TBD: Switch to "alg" parameter for subtyping? But normally "alg" is not included in JWK
			thumbprints and there are multiple "ECDH-ES" algorithms already in JWA.]</t>
		</section>
		<section anchor="algorithms" title="Algorithms">
			<section title="Signatures">
				<section title="Algorithms">
					<t>The following signature algorithms are defined here (to be applied as
					values of "alg" parameter). All these have keys with subtype ("crv") of the
					same name:</t>

<figure><artwork>
  "alg"/"crv"            The signature algorithm applied
  Ed25519                Ed25519
  Ed448                  Ed448
</artwork></figure>

					<t>The key type for these keys is &quot;OKP&quot; and key subtype for
					these algorithms MUST be the same as the algorithm name.</t>
					<t>The keys of these subtypes MUST NOT be used for ECDH-ES.</t>
					<t>[TBD: Merge the alg values into a single one that can perform signing
					with any signature-capable OKP subtype? That would remove a source of
					possible errors, since then the message and key could not mismatch in
					algorithm.]</t>
				</section>
				<section title="Signing">
					<t>Signing for these is preformed by applying the signing algorithm
					defined in <xref target="I-D.irtf-cfrg-eddsa"/> to the private key (as private
					key), public key (as public key) and the JWS Signing Input (as message).
					The resulting signature is the JWS Signature value. All inputs and outputs
					are octet strings.</t>
				</section>
				<section title="Verification">
					<t>Verification is performed by applying the verification algorithm
					defined in <xref target="I-D.irtf-cfrg-eddsa"/> to the public key (as public
					key), the JWS Signing Input (as message) and the JWS Signature value (as 
					signature). All inputs are octet strings. If the algorithm accepts, the
					signature is valid, otherwise signature is invalid.</t>
				</section>
			</section>
			<section title="ECDH-ES">
				<t>The following key subtypes defined here for purpose of "Key Agreement with
				Elliptic Curve Diffie-Hellman Ephemeral Static" (ECDH-ES).</t>

<figure><artwork>
   "crv"             ECDH Function applied
   X25519            X25519
   X448              X448
</artwork></figure>

				<t>The key type used with these keys is &quot;OKP&quot;. These subtypes MUST NOT
				be used for signing.</t>

				<t><xref target="RFC7518" /> section 4.6 defines the ECDH-ES algorithms
				"ECDH-ES+A128KW", "ECDH-ES+A192KW", "ECDH-ES+A256KW" and "ECDH-ES".</t>

				<section title="Performing the ECDH operation">

					<t>The "x" parameter of "epk" field is set as follows:</t>

					<t>Apply the appropriate ECDH function to the ephemeral private key (as scalar
					input) and the standard basepoint (as u-coordinate input). The output is the
					value for "x" parameter of "epk" field. All inputs and outputs are octet
					strings.</t>

					<t>The Z value (raw key agreement output) for key agreement (to be used in
					subsequent KDF as per <xref target="RFC7518" /> section 4.6.2) is determined
					as follows:</t>

					<t>Apply the appropriate ECDH function to the ephemeral private key (as scalar
					input) and receiver public key (as u-coordinate input). The output is the Z
					value. All inputs and outputs are octet strings.</t>
				</section>
			</section>
		</section>

		<section anchor="security" title="Security considerations">
			<t>Security considerations from <xref target="RFC7748"/> and
			<xref target="I-D.irtf-cfrg-eddsa"/> apply here.</t>

			<t>Do not separate key material from information about what key subtype it is for.
			When using keys, check that the algorithm is compatible with the key subtype for the
			key. To do otherwise opens system up to attacks via mixing up algorithms. It is particularly
			dangerous to mix up signature and MAC algorithms.</t>

			<t>Although for Ed25519 and Ed448 the signature binds the key used for signing, do not
			assume this, as there are many signature algorithms that fail to make such binding. If
			key-binding is desired, include the key used for signing either inside the JWS protected
			header or the data to sign.</t>

			<t>If key generation or batch signature verification is performed, a well-seed
			cryptographic random number generator is REQUIRED. Signing and non-batch signature
			verification are deterministic operations and do not need random numbers of any kind.</t>

			<t>The JWA ECDH-ES KDF construction does not mix keys into the final shared secret. While
			in key exchange such could be a bad mistake, here either receiver public key has to be
			chosen maliciously or the sender has to be malicious in order to cause problems. And in
			either case, all security evaporates anyway.</t>

			<t>The nominal security strengths of X25519 and X448 are ~126 and ~223 bits. Therefore,
			using 256-bit symmetric encryption (especially key wrapping and encryption) with X448 is
			RECOMMENDED.</t>
		</section>

		<section anchor="acknowledgements" title="Acknowledgements">
			<t>Mike Jones for comments on initial pre-draft.</t>
		</section>

		<section anchor="IANA" title="IANA considerations">
			<?rfc subcompact="yes" ?>
			<t>The following is added to JSON Web Key Types Registry:</t>
			<t><list style="symbols">
				<t>"kty" Parameter Value: "OKP"</t>
				<t>Key Type Description: Octet string key pairs</t>
				<t>JOSE Implementation Requirements: Optional</t>
				<t>Change Controller: IESG</t>
				<t>Specification Document(s): Section 2 of [RFC-THIS]</t>
			</list><vspace blankLines="1" /></t>

			<t>The following is added to JSON Web Key Parameters Registry:</t>
			<t><list style="symbols">
				<t>Parameter Name: "crv"</t>
				<t>Parameter Description: The subtype of keypair</t>
				<t>Parameter Information Class: Public</t>
				<t>Used with "kty" Value(s): "OKP"</t>
				<t>Change Controller: IESG</t>
				<t>Specification Document(s): Section 2 of [RFC-THIS]</t>
			</list></t>

			<t><list style="symbols">
				<t>Parameter Name: "d"</t>
				<t>Parameter Description: The private key</t>
				<t>Parameter Information Class: Private</t>
				<t>Used with "kty" Value(s): "OKP"</t>
				<t>Change Controller: IESG</t>
				<t>Specification Document(s): Section 2 of [RFC-THIS]</t>
			</list></t>

			<t><list style="symbols">
				<t>Parameter Name: "x"</t>
				<t>Parameter Description: The public key</t>
				<t>Parameter Information Class: Public</t>
				<t>Used with "kty" Value(s): "OKP"</t>
				<t>Change Controller: IESG</t>
				<t>Specification Document(s): Section 2 of [RFC-THIS]</t>
			</list><vspace blankLines="1" /></t>

			<t>The following is added to JSON Web Signature and Encryption Algorithms Registry:</t>
			<t><list style="symbols">
				<t>Algorithm Name: "Ed25519"</t>
				<t>Algorithm Description: Ed25519 signature algorithm</t>
				<t>Algorithm Usage Location(s): "alg"</t>
				<t>JOSE Implementation Requirements: Optional</t>
				<t>Change Controller: IESG</t>
				<t>Specification Document(s): Section 3.1 of [RFC-THIS]</t>
				<t>Algorithm Analysis Documents(s): <xref target="I-D.irtf-cfrg-eddsa"/></t>
			</list></t>
			
			<t><list style="symbols">
				<t>Algorithm Name: "Ed448"</t>
				<t>Algorithm Description: Ed448 signature algorithm</t>
				<t>Algorithm Usage Location(s): "alg"</t>
				<t>JOSE Implementation Requirements: Optional</t>
				<t>Change Controller: IESG</t>
				<t>Specification Document(s): Section 3.1 of [RFC-THIS]</t>
				<t>Algorithm Analysis Documents(s): <xref target="I-D.irtf-cfrg-eddsa"/></t>
			</list></t>

			<t>The following is added to JSON Web Key Elliptic Curve Registry:</t>
			<t><list style="symbols">
				<t>Curve Name: "Ed25519"</t>
				<t>Curve Description: Ed25519 signature algorithm keypairs</t>
				<t>JOSE Implementation Requirements: Optional</t>
				<t>Change Controller: IESG</t>
				<t>Specification Document(s): Section 3.1 of [RFC-THIS]</t>
			</list></t>
			
			<t><list style="symbols">
				<t>Curve Name: "Ed448"</t>
				<t>Curve Description: Ed448 signature algorithm keypairs</t>
				<t>JOSE Implementation Requirements: Optional</t>
				<t>Change Controller: IESG</t>
				<t>Specification Document(s): Section 3.1 of [RFC-THIS]</t>
			</list></t>

			<t><list style="symbols">
				<t>Curve name: "X25519"</t>
				<t>Curve Description: X25519 function keypairs</t>
				<t>JOSE Implementation Requirements: Optional</t>
				<t>Change Controller: IESG</t>
				<t>Specification Document(s): Section 3.2 of [RFC-THIS]</t>
				<t>Analysis Documents(s): <xref target="RFC7748"/></t>
			</list></t>

			<t><list style="symbols">
				<t>Curve Name: "X448"</t>
				<t>Curve Description: X448 function keypairs</t>
				<t>JOSE Implementation Requirements: Optional</t>
				<t>Change Controller: IESG</t>
				<t>Specification Document(s): Section 3.2 of [RFC-THIS]</t>
				<t>Analysis Documents(s): <xref target="RFC7748"/></t>
			</list></t>
			<?rfc subcompact="no" ?>
		</section>
	</middle>
	<back>
		<references title="Normative References">

			&RFC2119;
			&RFC4648;
			&RFC7748;
			&EDDSA;

		</references>

		<references title="Informative References">
			&RFC7517;
			&RFC7518;
			&RFC7638;
		</references>

		<section anchor="Examples" title="Examples">
			<t>To the extent possible, the examples use material lifted from test vectors of
			<xref target="RFC7748"/> and <xref target="I-D.irtf-cfrg-eddsa"/></t>
			<section title="Ed25519 private key">
<figure><artwork>
{"kty":"OKP","crv":"Ed25519",
"d":"nWGxne_9WmC6hEr0kuwsxERJxWl7MmkZcDusAxyuf2A"
"x":"11qYAYKxCrfVS_7TyWQHOg7hcvPapiMlrwIaaPcHURo"}
</artwork></figure>
				<t>The hexadecimal dump of private key is:</t>
<figure><artwork>
9d 61 b1 9d ef fd 5a 60 ba 84 4a f4 92 ec 2c c4
44 49 c5 69 7b 32 69 19 70 3b ac 03 1c ae 7f 60
</artwork></figure>
				<t>And of the public key:</t>
<figure><artwork>
d7 5a 98 01 82 b1 0a b7 d5 4b fe d3 c9 64 07 3a
0e e1 72 f3 da a6 23 25 af 02 1a 68 f7 07 51 1a
</artwork></figure>
			</section>
			<section title="Ed25519 public key">
				<t>This is the public parts of the previous private key (just omits "d"):</t>
<figure><artwork>
{"kty":"OKP","crv":"Ed25519",
"x":"11qYAYKxCrfVS_7TyWQHOg7hcvPapiMlrwIaaPcHURo"}
</artwork></figure>
			</section>
			<section title="JWK thumbprint canonicalization">
				<t>The JWK thumbprint canonicalization of the two above examples is (linebreak
				inserted for formatting reasons)</t>
<figure><artwork>
{"crv":"Ed25519","kty":"OKP","x":"11qYAYKxCrfVS_7TyWQHOg7hcvPapiMlrwI
aaPcHURo"}
</artwork></figure>
				<t>Which has the SHA-256 hash of:
				90facafea9b1556698540f70c0117a22ea37bd5cf3ed3c47093c1707282b4b89</t>
			</section>
			<section title="Ed25519 Signing">
				<t>The JWS protected header is:</t>
<figure><artwork>
{"alg":"Ed25519"}
</artwork></figure>
				<t>This has base64url encoding of:</t>
<figure><artwork>
eyJhbGciOiJFZDI1NTE5In0
</artwork></figure>
				<t>The payload is (text):</t>
<figure><artwork>
Example of Ed25519 signing
</artwork></figure>
				<t>This has base64url encoding of:</t>
<figure><artwork>
RXhhbXBsZSBvZiBFZDI1NTE5IHNpZ25pbmc
</artwork></figure>
				<t>The JWS signing input is (concatenation of base64url encoding of the (protected)
				header, a dot and base64url encoding of the payload) is:</t>
<figure><artwork>
eyJhbGciOiJFZDI1NTE5In0.RXhhbXBsZSBvZiBFZDI1NTE5IHNpZ25pbmc
</artwork></figure>
				<t>Applying Ed25519 signing algorithm to the private key, public key and the JWS
				signing input yields signature (hex):</t>
<figure><artwork>
53 18 48 60 b1 c6 83 7f 4d 54 22 e9 40 05 43 fd
47 1f 3a 69 c6 48 2c cb 15 9a 17 62 42 e2 21 b1
5c 72 63 9b fe a3 9b b2 08 f3 2c ab 1f 27 0f b8
36 57 1c 52 0b d8 ac 41 eb 45 b3 55 d0 77 19 01
</artwork></figure>
				<t>Converting this to base64url yields:</t>
<figure><artwork>
UxhIYLHGg39NVCLpQAVD_UcfOmnGSCzLFZoXYkLiIbFccmOb_qObsgjzLKsfJw-4NlccU
gvYrEHrRbNV0HcZAQ
</artwork></figure>
				<t>So the compact serialization of JWS is (concatenation of signing input, a dot and
				base64url encoding of the signature:</t>
<figure><artwork>
eyJhbGciOiJFZDI1NTE5In0.RXhhbXBsZSBvZiBFZDI1NTE5IHNpZ25pbmc.UxhIYLHGg
39NVCLpQAVD_UcfOmnGSCzLFZoXYkLiIbFccmOb_qObsgjzLKsfJw-4NlccUgvYrEHrRb
NV0HcZAQ
</artwork></figure>
			</section>
			<section title="Ed25519 Validation">
				<t>The JWS from above example is:</t>
<figure><artwork>
eyJhbGciOiJFZDI1NTE5In0.RXhhbXBsZSBvZiBFZDI1NTE5IHNpZ25pbmc.UxhIYLHGg
39NVCLpQAVD_UcfOmnGSCzLFZoXYkLiIbFccmOb_qObsgjzLKsfJw-4NlccUgvYrEHrRb
NV0HcZAQ
</artwork></figure>
				<t>This has 2 dots in it, so it might be valid JWS. Base64url decoding the protected
				header yields:</t>
<figure><artwork>
{"alg":"Ed25519"}
</artwork></figure>
				<t>So this is Ed25519 signature. Now the key has: "kty":"OKP" and "crv":"Ed25519", so
				the key is valid for the algorithm (if it had other values, the validation would have
				failed).</t>
				<t>The signing input is the part before second dot:</t>
<figure><artwork>
eyJhbGciOiJFZDI1NTE5In0.RXhhbXBsZSBvZiBFZDI1NTE5IHNpZ25pbmc
</artwork></figure>
				<t>Applying Ed25519 verification algorithm to the public key, JWS signing input and
				the signature yields true. So the signature is valid. The message is base64 decoding
				of the part between the dots:</t>
<figure><artwork>
Example of Ed25519 signing
</artwork></figure>
			</section>
			<section title="ECDH-ES with X25519">
				<t>The public key to encrypt to is:</t>
<figure><artwork>
{"kty":"OKP","crv":"X25519","kid":"Bob"
"x":"3p7bfXt9wbTTW2HC7OQ1Nz-DQ8hbeGdNrfx-FG-IK08"}
</artwork></figure>
				<t>The public key from target key is (hex):</t>
<figure><artwork>
de 9e db 7d 7b 7d c1 b4 d3 5b 61 c2 ec e4 35 37
3f 83 43 c8 5b 78 67 4d ad fc 7e 14 6f 88 2b 4f
</artwork></figure>
				<t>The ephemeral secret happens to be (hex):</t>
<figure><artwork>
77 07 6d 0a 73 18 a5 7d 3c 16 c1 72 51 b2 66 45
df 4c 2f 87 eb c0 99 2a b1 77 fb a5 1d b9 2c 2a
</artwork></figure>
				<t>So the ephemeral public key is X25519(ephkey,G) (hex):</t>
<figure><artwork>
85 20 f0 09 89 30 a7 54 74 8b 7d dc b4 3e f7 5a
0d bf 3a 0d 26 38 1a f4 eb a4 a9 8e aa 9b 4e 6a
</artwork></figure>
				<t>This is packed into ephemeral public key value:</t>
<figure><artwork>
{"kty":"OKP","crv":"X25519",
"x":"hSDwCYkwp1R0i33ctD73Wg2_Og0mOBr066SpjqqbTmo"}
</artwork></figure>
				<t>So the protected header could for example be:</t>
<figure><artwork>
{"alg":"ECDH-ES+A128KW","epk":{"kty":"OKP","crv":"X25519",
"x":"hSDwCYkwp1R0i33ctD73Wg2_Og0mOBr066SpjqqbTmo"},
"enc":"A128GCM","kid":"Bob"}
</artwork></figure>
				<t>And sender computes as the DH Z value as X25519(ephkey,recv_pub) (hex):</t>
<figure><artwork>
4a 5d 9d 5b a4 ce 2d e1 72 8e 3b f4 80 35 0f 25
e0 7e 21 c9 47 d1 9e 33 76 f0 9b 3c 1e 16 17 42
</artwork></figure>
				<t>The receiver computes as the DH Z value as X25519(seckey,ephkey_pub) (hex):</t>
<figure><artwork>
4a 5d 9d 5b a4 ce 2d e1 72 8e 3b f4 80 35 0f 25
e0 7e 21 c9 47 d1 9e 33 76 f0 9b 3c 1e 16 17 42
</artwork></figure>
				<t>Which is the same as sender's value (the both sides run this through KDF before
				using as direct encryption key or AES128-KW key).</t>
			</section>
			<section title="ECDH-ES with X448">
				<t>The public key to encrypt to is (linebreak inserted for formatting reasons):</t>
<figure><artwork>
{"kty":"OKP","crv":"X448","kid":"Dave"
"x":"PreoKbDNIPW8_AtZm2_sz22kYnEHvbDU80W0MCfYuXL8PjT7QjKhPKcG3LV67D2
uB73BxnvzNgk"}
</artwork></figure>
				<t>The public key from target key is (hex):</t>
<figure><artwork>
3e b7 a8 29 b0 cd 20 f5 bc fc 0b 59 9b 6f ec cf
6d a4 62 71 07 bd b0 d4 f3 45 b4 30 27 d8 b9 72
fc 3e 34 fb 42 32 a1 3c a7 06 dc b5 7a ec 3d ae
07 bd c1 c6 7b f3 36 09
</artwork></figure>
				<t>The ephemeral secret happens to be (hex):</t>
<figure><artwork>
9a 8f 49 25 d1 51 9f 57 75 cf 46 b0 4b 58 00 d4
ee 9e e8 ba e8 bc 55 65 d4 98 c2 8d d9 c9 ba f5
74 a9 41 97 44 89 73 91 00 63 82 a6 f1 27 ab 1d
9a c2 d8 c0 a5 98 72 6b
</artwork></figure>
				<t>So the ephemeral public key is X448(ephkey,G) (hex):</t>
<figure><artwork>
9b 08 f7 cc 31 b7 e3 e6 7d 22 d5 ae a1 21 07 4a
27 3b d2 b8 3d e0 9c 63 fa a7 3d 2c 22 c5 d9 bb
c8 36 64 72 41 d9 53 d4 0c 5b 12 da 88 12 0d 53
17 7f 80 e5 32 c4 1f a0
</artwork></figure>
				<t>This is packed into ephemeral public key value (linebreak inserted for formatting
				purposes):</t>
<figure><artwork>
{"kty":"OKP","crv":"X448",
"x":"mwj3zDG34-Z9ItWuoSEHSic70rg94Jxj-qc9LCLF2bvINmRyQdlT1AxbEtqIEg1
TF3-A5TLEH6A"}
</artwork></figure>
				<t>So the protected header could for example be (linebreak inserted for formatting
				purposes):</t>
<figure><artwork>
{"alg":"ECDH-ES+A256KW","epk":{"kty":"OKP","crv":"X448",
"x":"mwj3zDG34-Z9ItWuoSEHSic70rg94Jxj-qc9LCLF2bvINmRyQdlT1AxbEtqIEg1
TF3-A5TLEH6A"},"enc":"A256GCM","kid":"Dave"}
</artwork></figure>
				<t>And sender computes as the DH Z value as X448(ephkey,recv_pub) (hex):</t>
<figure><artwork>
07 ff f4 18 1a c6 cc 95 ec 1c 16 a9 4a 0f 74 d1
2d a2 32 ce 40 a7 75 52 28 1d 28 2b b6 0c 0b 56
fd 24 64 c3 35 54 39 36 52 1c 24 40 30 85 d5 9a
44 9a 50 37 51 4a 87 9d
</artwork></figure>
				<t>The receiver computes as the DH Z value as X448(seckey,ephkey_pub) (hex):</t>
<figure><artwork>
07 ff f4 18 1a c6 cc 95 ec 1c 16 a9 4a 0f 74 d1
2d a2 32 ce 40 a7 75 52 28 1d 28 2b b6 0c 0b 56
fd 24 64 c3 35 54 39 36 52 1c 24 40 30 85 d5 9a
44 9a 50 37 51 4a 87 9d
</artwork></figure>
				<t>Which is the same as sender's value (the both sides run this through KDF before
				using as direct encryption key or AES256-KW key).</t>
			</section>
		</section>
	</back>
</rfc>
