draft-irtf-cfrg-spake2.xml

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<rfc category="info" ipr="trust200902" docName="draft-irtf-cfrg-spake2-24" submissionType="IRTF">
  <front>
    <title>SPAKE2, a PAKE</title>
    <author fullname="Watson Ladd" initials="W." surname="Ladd">
      <organization>Cloudflare</organization>
      <address>
        <email>watsonbladd@gmail.com</email>
      </address>
    </author>
    <author role="editor" initials="B." surname="Kaduk" fullname="Benjamin Kaduk">
      <organization abbrev="Akamai">Akamai Technologies</organization>
      <address>
        <email>kaduk@mit.edu</email>
      </address>
    </author>
    <date month="September" year="2021"/>
    <workgroup>
      Internet Research Task Force
    </workgroup>
    <abstract>
      <t>This document describes SPAKE2 which is a protocol for two
      parties that share a password to derive a strong shared key
      without disclosing the password. This method is compatible with
      any group, is computationally efficient, and SPAKE2 has a
      security proof. This document predated the CFRG PAKE competition
      and it was not selected, however, given existing use of variants
      in Kerberos and other applications it was felt publication was
      beneficial. Applications that need a symmetric PAKE and where hashing onto an elliptic curve
      at execution time is not possible can use SPAKE2. This document is a product of the Crypto Forum
      Research Group (CFRG) in the IRTF.</t>
    </abstract>
  </front>
  <middle>
    <section anchor="intro" title="Introduction">
      <t>This document describes SPAKE2, a means for two parties that
      share a password to derive a strong shared key without
      disclosing the password.  This password-based key exchange
      protocol is compatible with any group (requiring only a 
      scheme to map a random input of fixed length per group to a random
      group element), is computationally efficient, and has a security
      proof.  Predetermined parameters for a selection of commonly
      used groups are also provided for use by other protocols.</t>
      <t>SPAKE2 was not selected as the result of the CFRG PAKE selection 
      competition. However, given existing use of variants in Kerberos and 
      other applications it was felt publication was beneficial. This RFC
      represents the individual opinion(s) of one or more members of
      the Crypto Forum Research Group of the Internet Research Task
      Force (IRTF).</t>
      <t> Many of these applications predated methods to hash to
      elliptic curves being available or predated the publication of
      the PAKEs that were chosen as an outcome of the PAKE selection
      competition. In cases where a symmetric PAKE is needed, and
      hashing onto an elliptic curve at protocol execution time is not
      available, SPAKE2 is useful.</t>
    </section>
    <section anchor="notation" title="Requirements Notation">
      <t>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 <xref target="RFC2119" /> <xref target="RFC8174" />
        when, and only when, they
        appear in all capitals, as shown here.</t>
    </section>
    <section anchor="definition" title="Definition of SPAKE2">
                <section title="Protocol Flow" anchor="flow">
        <t>SPAKE2 is a two round protocol, wherein the first round establishes a shared secret
          between A and B, and the second round serves as key confirmation. Prior to invocation, A and B are provisioned with
          information such as the input password needed to run the protocol. We assume that the roles of A and B
	  are agreed upon by both sides: A goes first and uses M, and B goes second and uses N. If this assignment
	  of roles is not possible a symmetric variant MUST be used as described later <xref target="variants"/>. For instance A may be the client when
	  using TCP or TLS as an underlying protocol and B the server. Most protocols have such a distinction.

          During the first round, A sends a public value pA to B, and
          B responds with its own public value pB. Both A and B then
          derive a shared secret used to produce encryption and
          authentication keys. The latter are used during the second
          round for key confirmation. (<xref target="keys"/> details
          the key derivation and confirmation steps.) In particular, A
          sends a key confirmation message cA to B, and B responds
          with its own key confirmation message cB. A MUST NOT
          consider the protocol complete until it receives and
          verifies cB. Likewise, B MUST NOT consider the protocol
          complete until it receives and verifies cA.</t>

        <t>This sample flow is shown below.</t>
        <figure><artwork><![CDATA[
                A                  B
                | (setup protocol) |
                |                  |
  (compute pA)  |        pA        |
                |----------------->|
                |        pB        | (compute pB)
                |<-----------------|
                |                  |
                | (derive secrets) |
                |                  |
  (compute cA)  |        cA        |
                |----------------->|
                |        cB        | (compute cB)
                |                  | (check cA)
                |<-----------------|
  (check cB)    |                  |
        ]]></artwork></figure>
      </section>
      <section title="Setup" anchor="setup">
        <t>Let G be a group in which the gap Diffie-Hellman (GDH)
          problem is hard. Suppose G has order p*h where p is a large prime;
          h will be called the cofactor. Let I be the unit element in
          G, e.g., the point at infinity if G is an elliptic curve group. We denote the
          operations in the group additively. We assume there is a representation of
          elements of G as byte strings: common choices would be SEC1 <xref target="SEC1"/>
          uncompressed or compressed for elliptic curve groups or big
          endian integers of a fixed (per-group) length for prime field DH. Applications MUST
	  specify this encoding, typically by referring to the document defining the group.
          We fix two elements M and N in the prime-order subgroup of G as defined
          in the table in this document for common groups, as well as a generator P
          of the (large) prime-order subgroup of G. In the case of a composite order group
	        we will work in the quotient group. For common groups used in this document, 
          P is specified in the document defining the group, and so we do not repeat it here.</t>
	        
          <t> For elliptic curves other than the ones in this document the methods of 
          <xref target="I-D.irtf-cfrg-hash-to-curve"/> SHOULD be used to generate M and N, e.g. via 
          M = hash_to_curve("M SPAKE2 seed OID x") and N = hash_to_curve("N SPAKE2 seed OID x"), where 
          x is an OID for the curve. Applications MAY include a DST tag in this step, as specified
          in <xref target="I-D.irtf-cfrg-hash-to-curve"/>, though this is not required.</t>

        <t>|| denotes concatenation of byte strings. We also let len(S) denote the
          length of a string in bytes, represented as an eight-byte little-
          endian number. Finally, let nil represent an empty string, i.e.,
          len(nil) = 0. Text strings in double quotes are treated as their ASCII encodings
	        throughout this document.</t>

        <t>KDF(ikm, salt, info, L) is a key-derivation function that
        takes as input a salt, intermediate keying material (IKM),
        info string, and derived key length L to derive a
        cryptographic key of length L. MAC(key, message) is a Message
        Authentication Code algorithm that takes a secret key and
        message as input to produce an output.  Let Hash be a hash
        function from arbitrary strings to bit strings of a fixed
        length, at least 256 bits long. Common choices for Hash are
        SHA256 or SHA512 <xref target="RFC6234"/>. Let MHF be a
        memory-hard hash function designed to slow down brute-force
        attackers. Scrypt <xref target="RFC7914"/> is a common example
        of this function. The output length of MHF matches that of
        Hash. Parameter selection for MHF is out of scope for this
        document.  <xref target="Ciphersuites"/> specifies variants of
        KDF, MAC, and Hash suitable for use with the protocols
        contained herein.</t>

        <t>Let A and B be two parties. A and B may also have digital
          representations of the parties&apos; identities such as Media Access Control addresses
          or other names (hostnames, usernames, etc). A and B may share Additional
          Authenticated Data (AAD) of length at most 2^16 - 128 bits that is separate
          from their identities which they may want to include in the protocol execution.
          One example of AAD is a list of supported protocol versions if SPAKE2 were
          used in a higher-level protocol which negotiates use of a particular PAKE. Including
          this list would ensure that both parties agree upon the same set of supported protocols
          and therefore prevent downgrade attacks. We also assume A and B share an integer w;
          typically w = MHF(pw) mod p, for a user-supplied password pw.
          Standards such as NIST.SP.800-56Ar3 suggest taking mod p of a
          hash value that is 64 bits longer than that needed to represent p to remove
          statistical bias introduced by the modulation. Protocols using this specification MUST define
          the method used to compute w. In some cases, it may be necessary to carry out various
          forms of normalization of the password before hashing <xref target="RFC8265" />.
          The hashing algorithm SHOULD be a MHF so as to slow down brute-force
          attackers. </t>

      </section>

      <section title="SPAKE2" anchor="spake2">
        <t>To begin, A picks x randomly and uniformly from the integers in [0,p),
            and calculates X=x*P and pA=w*M+X, then transmits pA to B.</t>

        <t>B selects y randomly and uniformly from the integers in [0,p), and calculates
            Y=y*P, pB=w*N+Y, then transmits pB to A.</t>

        <t>Both A and B calculate a group element K. A calculates it
        as h*x*(pB-w*N), while B calculates it as h*y*(pA-w*M). A knows pB
        because it has received it, and likewise B knows pA. The
        multiplication by h prevents small subgroup confinement
        attacks by computing a unique value in the quotient
        group.</t>

        <t>K is a shared value, though it MUST NOT be used or output as a shared secret
          from the protocol. Both A and B must derive two additional shared secrets from 
          the protocol transcript, which includes K.
          This prevents man-in-the-middle attackers from inserting themselves into
          the exchange. The transcript TT is encoded as follows:</t>

        <figure><artwork><![CDATA[
        TT = len(A)  || A
          || len(B)  || B
          || len(pA) || pA
          || len(pB) || pB
          || len(K)  || K
          || len(w)  || w
        ]]></artwork></figure>

        <t> Here w is encoded as a big endian number padded to the length of p. This representation
        prevents timing attacks that otherwise would reveal the length of w. len(w) is thus a constant.
	      We include it for consistency.</t>
        <t>If an identity is absent, it is encoded as a zero-length string.
        This MUST only be done for applications in which identities are implicit. Otherwise,
        the protocol risks unknown key share attacks, where both sides of a connection disagree over who is authenticated.</t>

        <t>Upon completion of this protocol, A and B compute shared secrets Ke, KcA, and KcB as
            specified in <xref target="keys"/>. A MUST send B a key confirmation message
            so both parties agree upon these shared secrets. This confirmation message cA
            is computed as a MAC over the protocol transcript TT using KcA, as follows:
            cA = MAC(KcA, TT). Similarly, B MUST send A a confirmation message using a MAC
            computed equivalently except with the use of KcB. Key confirmation verification
            requires computing cB and checking for equality against that which was received.</t>
      </section>

    </section>
    <section title="Key Schedule and Key Confirmation" anchor="keys">
        <t>The protocol transcript TT, as defined in  <xref target="spake2"/>, is unique and secret to A and B.
            Both parties use TT to derive shared symmetric secrets Ke and Ka as Ke || Ka = Hash(TT), with |Ke| = |Ka|.
            The length of each key is equal to half of the digest output, e.g., 128 bits for SHA-256. Keys MUST be at least
	128 bits in length.</t>

        <t>Both endpoints use Ka to derive subsequent MAC keys for key confirmation messages.
            Specifically, let KcA and KcB be the MAC keys used by A and B, respectively.
            A and B compute them as KcA || KcB = KDF(Ka, nil, "ConfirmationKeys" || AAD, L), where AAD
            is the associated data each given to each endpoint, or nil if none was provided.
            The length of each of KcA and KcB is equal to half of the underlying hash output length, e.g.,
            |KcA| = |KcB| = 128 bits for HKDF(SHA256), with L=256 bits. </t>

        <t>The resulting key schedule for this protocol, given transcript TT and additional associated
            data AAD, is as follows.</t>

        <figure><artwork><![CDATA[
    TT  -> Hash(TT) = Ke || Ka
    AAD -> KDF(Ka, nil, "ConfirmationKeys" || AAD) = KcA || KcB
        ]]></artwork></figure>

        <t>A and B output Ke as the shared secret from the protocol. Ka and its derived keys are not
            used for anything except key confirmation.</t>
    </section>
    <section title="Per-User M and N and M=N" target="variants">
      <t>
      To avoid concerns that an attacker needs to solve a single ECDH instance to break the authentication of SPAKE2, it is
      possible to vary M and N using <xref target="I-D.irtf-cfrg-hash-to-curve"/> as follows:
        <figure><artwork><![CDATA[
    M = hash_to_curve(Hash("M SPAKE2" || len(A) || A || len(B) || B))
    N = hash_to_curve(Hash("N SPAKE2" || len(A) || A || len(B) || B))
    ]]></artwork></figure>
	
	There is also a symmetric variant where M=N. For this variant we set
	<figure><artwork><![CDATA[
    M = hash_to_curve(Hash("M AND N SPAKE2"))
    ]]></artwork></figure>

	This variant MUST be used when it is not possible to determine which of A and B
	should use M or N, due to asymmetries in the protocol flows or the desire to use only a single shared secret with
	nil identities for authentication. The security of these variants is examined in <xref
	target="MNVAR"/>. The variant with per-user M and N may not be
	suitable for protocols that require the initial messages to be
	generated by each party at the same time and do not know the
	exact identity of the parties before the flow begins.
     </t>
    </section>
    <section title="Ciphersuites" anchor="Ciphersuites">
        <t>
            This section documents SPAKE2 ciphersuite
            configurations. A ciphersuite indicates a group,
            cryptographic hash function, and pair of KDF and MAC
            functions, e.g., SPAKE2-P256-SHA256-HKDF-HMAC. This
            ciphersuite indicates a SPAKE2 protocol instance over
            P-256 that uses SHA256 along with HKDF <xref
            target="RFC5869"/> and HMAC <xref target="RFC2104"/> for
            G, Hash, KDF, and MAC functions, respectively. For Ed25519
            the compressed encoding is used <xref target="RFC8032"/>,
            all others use the uncompressed SEC1 encoding.</t>

            <texttable anchor="spake2_ciphersuites" title="SPAKE2 Ciphersuites">
                <ttcol align='center'>G</ttcol>
                <ttcol align='center'>Hash</ttcol>
                <ttcol align='center'>KDF</ttcol>
                <ttcol align='center'>MAC</ttcol>

                <!-- P256-SHA256-HKDF-HMAC -->
                <c>P-256</c>
                <c>SHA256 <xref target="RFC6234"/></c>
                <c>HKDF <xref target="RFC5869"/></c>
                <c>HMAC <xref target="RFC2104"/></c>

                <!-- P256-SHA512-HKDF-HMAC -->
                <c>P-256</c>
                <c>SHA512 <xref target="RFC6234"/></c>
                <c>HKDF <xref target="RFC5869"/></c>
                <c>HMAC <xref target="RFC2104"/></c>

                <!-- P384-SHA256-HKDF-HMAC -->
                <c>P-384</c>
                <c>SHA256 <xref target="RFC6234"/></c>
                <c>HKDF <xref target="RFC5869"/></c>
                <c>HMAC <xref target="RFC2104"/></c>

                <!-- P384-SHA512-HKDF-HMAC -->
                <c>P-384</c>
                <c>SHA512 <xref target="RFC6234"/></c>
                <c>HKDF <xref target="RFC5869"/></c>
                <c>HMAC <xref target="RFC2104"/></c>

                <!-- P521-SHA512-HKDF-HMAC -->
                <c>P-521</c>
                <c>SHA512 <xref target="RFC6234"/></c>
                <c>HKDF <xref target="RFC5869"/></c>
                <c>HMAC <xref target="RFC2104"/></c>

                <!-- edwards25519-SHA256-HKDF-HMAC -->
                <c>edwards25519 <xref target="RFC8032"/></c>
                <c>SHA256 <xref target="RFC6234"/></c>
                <c>HKDF <xref target="RFC5869"/></c>
                <c>HMAC <xref target="RFC2104"/></c>

                <!-- edwards448-SHA512-HKDF-HMAC -->
                <c>edwards448 <xref target="RFC7748"/></c>
                <c>SHA512 <xref target="RFC6234"/></c>
                <c>HKDF <xref target="RFC5869"/></c>
                <c>HMAC <xref target="RFC2104"/></c>

                <!-- P256-SHA256-HKDF-CMAC -->
                <c>P-256</c>
                <c>SHA256 <xref target="RFC6234"/></c>
                <c>HKDF <xref target="RFC5869"/></c>
                <c>CMAC-AES-128 <xref target="RFC4493"/></c>

                <!-- P256-SHA512-HKDF-CMAC -->
                <c>P-256</c>
                <c>SHA512 <xref target="RFC6234"/></c>
                <c>HKDF <xref target="RFC5869"/></c>
                <c>CMAC-AES-128 <xref target="RFC4493"/></c>
            </texttable>

        <t>The following points represent permissible point generation seeds
            for the groups listed in the Table <xref target="spake2_ciphersuites"/>,
            using the algorithm presented in <xref target="pointgen"/>.
            These bytestrings are compressed points as in <xref target="SEC1" />
            for curves from <xref target="SEC1" />.</t>

      <t>For P256:</t>

      <figure><artwork><![CDATA[
M =
02886e2f97ace46e55ba9dd7242579f2993b64e16ef3dcab95afd497333d8fa12f
seed: 1.2.840.10045.3.1.7 point generation seed (M)

N =
03d8bbd6c639c62937b04d997f38c3770719c629d7014d49a24b4f98baa1292b49
seed: 1.2.840.10045.3.1.7 point generation seed (N)
]]></artwork></figure>

      <t>For P384:</t>

      <figure><artwork><![CDATA[
M =
030ff0895ae5ebf6187080a82d82b42e2765e3b2f8749c7e05eba366434b363d3dc
36f15314739074d2eb8613fceec2853
seed: 1.3.132.0.34 point generation seed (M)

N =
02c72cf2e390853a1c1c4ad816a62fd15824f56078918f43f922ca21518f9c543bb
252c5490214cf9aa3f0baab4b665c10
seed: 1.3.132.0.34 point generation seed (N)
]]></artwork></figure>

      <t>For P521:</t>

      <figure><artwork><![CDATA[
M =
02003f06f38131b2ba2600791e82488e8d20ab889af753a41806c5db18d37d85608
cfae06b82e4a72cd744c719193562a653ea1f119eef9356907edc9b56979962d7aa
seed: 1.3.132.0.35 point generation seed (M)

N =
0200c7924b9ec017f3094562894336a53c50167ba8c5963876880542bc669e494b25
32d76c5b53dfb349fdf69154b9e0048c58a42e8ed04cef052a3bc349d95575cd25
seed: 1.3.132.0.35 point generation seed (N)
]]></artwork></figure>

      <t>For edwards25519:</t>
      <figure><artwork><![CDATA[
M =
d048032c6ea0b6d697ddc2e86bda85a33adac920f1bf18e1b0c6d166a5cecdaf
seed: edwards25519 point generation seed (M)

N =
d3bfb518f44f3430f29d0c92af503865a1ed3281dc69b35dd868ba85f886c4ab
seed: edwards25519 point generation seed (N)
]]></artwork></figure>

      <t>For edwards448:</t>
      <figure><artwork><![CDATA[
M =
b6221038a775ecd007a4e4dde39fd76ae91d3cf0cc92be8f0c2fa6d6b66f9a12
942f5a92646109152292464f3e63d354701c7848d9fc3b8880
seed: edwards448 point generation seed (M)

N =
6034c65b66e4cd7a49b0edec3e3c9ccc4588afd8cf324e29f0a84a072531c4db
f97ff9af195ed714a689251f08f8e06e2d1f24a0ffc0146600
seed: edwards448 point generation seed (N)
]]></artwork></figure>

    </section>
    <section title="Security Considerations">
      <t>A security proof of SPAKE2 for prime order groups is found in
      <xref target="REF" />, reducing the security of SPAKE2 to the
      gap Diffie-Hellman assumption.  Note that the choice of M and N
      is critical for the security proof.  The generation methods
      specified in this document are designed to eliminate concerns
      related to knowing discrete logs of M and N.</t>

       <t>Elements received from a peer MUST be checked for group
      membership: failure to properly deserialize and validate group 
      elements can lead to attacks. An endpoint MUST abort the protocol
      if any received public value is not a member of G.</t>

      <t>The choices of random numbers MUST BE uniform. Randomly
      generated values, e.g., x and y, MUST NOT be reused; such reuse
      may permit dictionary attacks on the password. It is RECOMMENDED
      to generate these uniform numbers using rejection sampling.</t>
      
      <t>Some implementations of elliptic curve multiplication may
      leak information about the length of the scalar. These MUST NOT
      be used. All operations on elliptic curve points must take time
      independent of the inputs. Hashing of the transcript may take
      time depending only on the length of the transcript, but not the
      contents.</t>

      <t>SPAKE2 does not support augmentation. As a result, the server
      has to store a password equivalent. This is considered a
      significant drawback in some use cases. Applications that need
      augmented PAKEs should use <xref target="I-D.irtf-cfrg-opaque"/>.</t>

      <t>The HMAC keys in this document are shorter than recommended
      in <xref target="RFC8032" />. This is appropriate as the
      difficulty of the discrete logarithm problem is comparable with
      the difficulty of brute forcing the keys.</t>

    </section>
    <section title="IANA Considerations">
      <t>No IANA action is required.</t>
    </section>
    <section title="Acknowledgments">
      <t>Special thanks to Nathaniel McCallum and Greg Hudson for
      generation of M and N, and Chris Wood for test vectors.  Thanks
      to Mike Hamburg for advice on how to deal with cofactors. Greg
      Hudson also suggested the addition of warnings on the reuse of x
      and y. Thanks to Fedor Brunner, Adam Langley, Liliya
      Akhmetzyanova, and the members of the CFRG for comments and
      advice. Thanks to Scott Fluhrer and those Crypto Panel experts
      involved in the PAKE selection process
      (https://github.com/cfrg/pake-selection) who have provided
      valuable comments. Chris Wood contributed substantial text and
      reformatting to address the excellent review comments from Kenny
      Paterson. </t>
    </section>
  </middle>
  <back>
    <references title="Normative References">
      &h2c;
      &RFC2104;
      &RFC2119;
      &RFC4493;
      &RFC5480;
      &RFC5869;
      &RFC6234;
      &RFC8032;
      &RFC7914;
      &RFC8032;
      &RFC8174;
    </references>
    <references title="Informative References">
      <reference anchor="SEC1">
	<front>
	  <title>SEC 1: Elliptic Curve Cryptography</title>
	  <author>
	    <organization> Standards for Efficient Cryptography Group</organization>
	  </author>
	  <date month="May" year="2009"/>
	</front>
      </reference>
      <reference anchor="MNVAR">
	<front>
	  <title> Universally Composable Relaxed Password Authentication</title>
	  <author initials="M." surname="Abdalla"/>
	  <author initials="M." surname="Barbosa"/>
	  <author initials="T." surname="Bradley"/>
	  <author initials="S." surname="Jarecki"/>
	  <author initials="J." surname="Katz"/>
	  <author initials="J." surname="Xu"/>
	  <date month="August" year="2020"/>
	</front>
	<annotation> Appears in Micciancio D., Ristenpart T. (eds)
	Advances in Cryptology -CRYPTO 2020. Crypto 2020. Lecture
	notes in Computer Science volume 12170. Springer.</annotation>
      </reference>
      <reference anchor="REF">
        <front>
          <title>Simple Password-Based Encrypted Key Exchange Protocols.</title>
          <author initials="M." surname="Abdalla" />
          <author initials="D." surname="Pointcheval" />
          <date month="Feb" year="2005" />
        </front>
        <annotation>Appears in A. Menezes, editor. Topics in
          Cryptography-CT-RSA 2005, Volume 3376 of Lecture Notes in Computer
          Science, pages 191-208, San Francisco, CA, US.  Springer-Verlag,
          Berlin, Germany.
        </annotation>
      </reference>
      <reference anchor="TDH">
        <front>
          <title>The Twin-Diffie Hellman Problem and Applications</title>
          <author initials="D." surname="Cash" />
          <author initials="E." surname="Kiltz" />
          <author initials="V." surname="Shoup" />
          <date year="2008" />
        </front>
        <annotation>EUROCRYPT 2008.  Volume 4965 of Lecture notes in Computer
          Science, pages 127-145.  Springer-Verlag, Berlin, Germany.
        </annotation>
      </reference>
      &RFC8265;
      &opaque;
    </references>
    <section anchor="pointgen" title="Algorithm used for Point Generation">
      <t>This section describes the algorithm that was used to generate
      the points M and N in the table in <xref target="Ciphersuites"/>.</t>

      <t>For each curve in the table below, we construct a string
      using the curve OID from <xref target="RFC5480" /> (as an ASCII
      string) or its name,
      combined with the needed constant, e.g., "1.3.132.0.35
      point generation seed (M)" for P-521.  This string is turned
      into a series of blocks by hashing with SHA256, and hashing that
      output again to generate the next 32 bytes, and so on.  This
      pattern is repeated for each group and value, with the string
      modified appropriately.</t>

      <t>A byte string of length equal to that of an encoded group
      element is constructed by concatenating as many blocks as are
      required, starting from the first block, and truncating to the
      desired length.  The byte string is then formatted as required
      for the group.  In the case of Weierstrass curves, we take the
      desired length as the length for representing a compressed point
      (section 2.3.4 of <xref target="SEC1" />),
      and use the low-order bit of the first byte as the sign bit.
      In order to obtain the correct format, the value of the first
      byte is set to 0x02 or 0x03 (clearing the first six bits
      and setting the seventh bit), leaving the sign bit as it was
      in the byte string constructed by concatenating hash blocks.
      For the <xref target="RFC8032" /> curves a different procedure is used.
      For edwards448 the 57-byte input has the least-significant 7 bits of the
      last byte set to zero, and for edwards25519 the 32-byte input is
      not modified.  For both the <xref target="RFC8032" /> curves the
      (modified) input is then interpreted
      as the representation of the group element.
      If this interpretation yields a valid group element with the
      correct order (p), the (modified) byte string is the output.  Otherwise,
      the initial hash block is discarded and a new byte string constructed
      from the remaining hash blocks. The procedure of constructing a
      byte string of the appropriate length, formatting it as
      required for the curve, and checking if it is a valid point of the correct
      order, is repeated
      until a valid element is found.</t>

      <t>The following python snippet generates the above points,
      assuming an elliptic curve implementation following the
      interface of Edwards25519Point.stdbase() and
      Edwards448Point.stdbase() in Appendix A of <xref target="RFC8032" />:</t>

      <figure><artwork><![CDATA[
def iterated_hash(seed, n):
    h = seed
    for i in range(n):
        h = hashlib.sha256(h).digest()
    return h

def bighash(seed, start, sz):
    n = -(-sz // 32)
    hashes = [iterated_hash(seed, i) for i in range(start, start + n)]
    return b''.join(hashes)[:sz]

def canon_pointstr(ecname, s):
    if ecname == 'edwards25519':
        return s
    elif ecname == 'edwards448':
        return s[:-1] + bytes([s[-1] & 0x80])
    else:
        return bytes([(s[0] & 1) | 2]) + s[1:]

def gen_point(seed, ecname, ec):
    for i in range(1, 1000):
        hval = bighash(seed, i, len(ec.encode()))
        pointstr = canon_pointstr(ecname, hval)
        try:
            p = ec.decode(pointstr)
            if p != ec.zero_elem() and p * p.l() == ec.zero_elem():
                return pointstr, i
        except Exception:
            pass
]]></artwork></figure>
    </section>

    <section anchor="testvectors" title="Test Vectors">
      <t>This section contains test vectors for SPAKE2 using 
        the P256-SHA256-HKDF-HMAC ciphersuite. (Choice of MHF is omitted
        and values for w, x, and y are provided directly.) All points are 
        encoded using the uncompressed format, i.e., with a 0x04 octet
        prefix, specified in <xref target="SEC1"/> A and B identity strings
        are provided in the protocol invocation.
      </t>

      <section title="SPAKE2 Test Vectors">
        <figure><artwork><![CDATA[
spake2: A='server', B='client'
w = 0x2ee57912099d31560b3a44b1184b9b4866e904c49d12ac5042c97dca461b1a5f
x = 0x43dd0fd7215bdcb482879fca3220c6a968e66d70b1356cac18bb26c84a78d729
pA = 0x04a56fa807caaa53a4d28dbb9853b9815c61a411118a6fe516a8798434751470
f9010153ac33d0d5f2047ffdb1a3e42c9b4e6be662766e1eeb4116988ede5f912c 
y = 0xdcb60106f276b02606d8ef0a328c02e4b629f84f89786af5befb0bc75b6e66be
pB = 0x0406557e482bd03097ad0cbaa5df82115460d951e3451962f1eaf4367a420676
d09857ccbc522686c83d1852abfa8ed6e4a1155cf8f1543ceca528afb591a1e0b7 
K = 0x0412af7e89717850671913e6b469ace67bd90a4df8ce45c2af19010175e37eed
69f75897996d539356e2fa6a406d528501f907e04d97515fbe83db277b715d3325 
TT = 0x06000000000000007365727665720600000000000000636c69656e744100000
00000000004a56fa807caaa53a4d28dbb9853b9815c61a411118a6fe516a8798434751
470f9010153ac33d0d5f2047ffdb1a3e42c9b4e6be662766e1eeb4116988ede5f912c4
1000000000000000406557e482bd03097ad0cbaa5df82115460d951e3451962f1eaf43
67a420676d09857ccbc522686c83d1852abfa8ed6e4a1155cf8f1543ceca528afb591a
1e0b741000000000000000412af7e89717850671913e6b469ace67bd90a4df8ce45c2a
f19010175e37eed69f75897996d539356e2fa6a406d528501f907e04d97515fbe83db2
77b715d332520000000000000002ee57912099d31560b3a44b1184b9b4866e904c49d1
2ac5042c97dca461b1a5f
Hash(TT) = 0x0e0672dc86f8e45565d338b0540abe6915bdf72e2b35b5c9e5663168e960a91b
Ke = 0x0e0672dc86f8e45565d338b0540abe69
Ka = 0x15bdf72e2b35b5c9e5663168e960a91b
KcA = 0x00c12546835755c86d8c0db7851ae86f
KcB = 0xa9fa3406c3b781b93d804485430ca27a
A conf = 0x58ad4aa88e0b60d5061eb6b5dd93e80d9c4f00d127c65b3b35b1b5281fee38f0
B conf = 0xd3e2e547f1ae04f2dbdbf0fc4b79f8ecff2dff314b5d32fe9fcef2fb26dc459b


spake2: A='', B='client'
w = 0x0548d8729f730589e579b0475a582c1608138ddf7054b73b5381c7e883e2efae
x = 0x403abbe3b1b4b9ba17e3032849759d723939a27a27b9d921c500edde18ed654b
pA = 0x04a897b769e681c62ac1c2357319a3d363f610839c4477720d24cbe32f5fd85f
44fb92ba966578c1b712be6962498834078262caa5b441ecfa9d4a9485720e918a 
y = 0x903023b6598908936ea7c929bd761af6039577a9c3f9581064187c3049d87065
pB = 0x04e0f816fd1c35e22065d5556215c097e799390d16661c386e0ecc84593974a6
1b881a8c82327687d0501862970c64565560cb5671f696048050ca66ca5f8cc7fc 
K = 0x048f83ec9f6e4f87cc6f9dc740bdc2769725f923364f01c84148c049a39a735e
bda82eac03e00112fd6a5710682767cff5361f7e819e53d8d3c3a2922e0d837aa6 
TT = 0x00000000000000000600000000000000636c69656e74410000000000000004a
897b769e681c62ac1c2357319a3d363f610839c4477720d24cbe32f5fd85f44fb92ba9
66578c1b712be6962498834078262caa5b441ecfa9d4a9485720e918a4100000000000
00004e0f816fd1c35e22065d5556215c097e799390d16661c386e0ecc84593974a61b8
81a8c82327687d0501862970c64565560cb5671f696048050ca66ca5f8cc7fc4100000
000000000048f83ec9f6e4f87cc6f9dc740bdc2769725f923364f01c84148c049a39a7
35ebda82eac03e00112fd6a5710682767cff5361f7e819e53d8d3c3a2922e0d837aa62
0000000000000000548d8729f730589e579b0475a582c1608138ddf7054b73b5381c7e
883e2efae
Hash(TT) = 0x642f05c473c2cd79909f9a841e2f30a70bf89b18180af97353ba198789c2b963
Ke = 0x642f05c473c2cd79909f9a841e2f30a7
Ka = 0x0bf89b18180af97353ba198789c2b963
KcA = 0xc6be376fc7cd1301fd0a13adf3e7bffd
KcB = 0xb7243f4ae60440a49b3f8cab3c1fba07
A conf = 0x47d29e6666af1b7dd450d571233085d7a9866e4d49d2645e2df975489521232b
B conf = 0x3313c5cefc361d27fb16847a91c2a73b766ffa90a4839122a9b70a2f6bd1d6df

spake2: A='server', B=''
w = 0x626e0cdc7b14c9db3e52a0b1b3a768c98e37852d5db30febe0497b14eae8c254
x = 0x07adb3db6bc623d3399726bfdbfd3d15a58ea776ab8a308b00392621291f9633
pA = 0x04f88fb71c99bfffaea370966b7eb99cd4be0ff1a7d335caac4211c4afd855e2
e15a873b298503ad8ba1d9cbb9a392d2ba309b48bfd7879aefd0f2cea6009763b0 
y = 0xb6a4fc8dbb629d4ba51d6f91ed1532cf87adec98f25dd153a75accafafedec16
pB = 0x040c269d6be017dccb15182ac6bfcd9e2a14de019dd587eaf4bdfd353f031101
e7cca177f8eb362a6e83e7d5e729c0732e1b528879c086f39ba0f31a9661bd34db 
K = 0x0445ee233b8ecb51ebd6e7da3f307e88a1616bae2166121221fdc0dadb986afa
f3ec8a988dc9c626fa3b99f58a7ca7c9b844bb3e8dd9554aafc5b53813504c1cbe 
TT = 0x06000000000000007365727665720000000000000000410000000000000004f
88fb71c99bfffaea370966b7eb99cd4be0ff1a7d335caac4211c4afd855e2e15a873b2
98503ad8ba1d9cbb9a392d2ba309b48bfd7879aefd0f2cea6009763b04100000000000
000040c269d6be017dccb15182ac6bfcd9e2a14de019dd587eaf4bdfd353f031101e7c
ca177f8eb362a6e83e7d5e729c0732e1b528879c086f39ba0f31a9661bd34db4100000
0000000000445ee233b8ecb51ebd6e7da3f307e88a1616bae2166121221fdc0dadb986
afaf3ec8a988dc9c626fa3b99f58a7ca7c9b844bb3e8dd9554aafc5b53813504c1cbe2
000000000000000626e0cdc7b14c9db3e52a0b1b3a768c98e37852d5db30febe0497b1
4eae8c254
Hash(TT) = 0x005184ff460da2ce59062c87733c299c3521297d736598fc0a1127600efa1afb
Ke = 0x005184ff460da2ce59062c87733c299c
Ka = 0x3521297d736598fc0a1127600efa1afb
KcA = 0xf3da53604f0aeecea5a33be7bddf6edf
KcB = 0x9e3f86848736f159bd92b6e107ec6799
A conf = 0xbc9f9bbe99f26d0b2260e6456e05a86196a3307ec6663a18bf6ac825736533b2
B conf = 0xc2370e1bf813b086dff0d834e74425a06e6390f48f5411900276dcccc5a297ec

spake2: A='', B=''
w = 0x7bf46c454b4c1b25799527d896508afd5fc62ef4ec59db1efb49113063d70cca
x = 0x8cef65df64bb2d0f83540c53632de911b5b24b3eab6cc74a97609fd659e95473
pA = 0x04a65b367a3f613cf9f0654b1b28a1e3a8a40387956c8ba6063e8658563890f4
6ca1ef6a676598889fc28de2950ab8120b79a5ef1ea4c9f44bc98f585634b46d66 
y = 0xd7a66f64074a84652d8d623a92e20c9675c61cb5b4f6a0063e4648a2fdc02d53
pB = 0x04589f13218822710d98d8b2123a079041052d9941b9cf88c6617ddb2fcc0494
662eea8ba6b64692dc318250030c6af045cb738bc81ba35b043c3dcb46adf6f58d 
K = 0x041a3c03d51b452537ca2a1fea6110353c6d5ed483c4f0f86f4492ca3f378d40
a994b4477f93c64d928edbbcd3e85a7c709b7ea73ee97986ce3d1438e135543772 
TT = 0x00000000000000000000000000000000410000000000000004a65b367a3f613
cf9f0654b1b28a1e3a8a40387956c8ba6063e8658563890f46ca1ef6a676598889fc28
de2950ab8120b79a5ef1ea4c9f44bc98f585634b46d66410000000000000004589f132
18822710d98d8b2123a079041052d9941b9cf88c6617ddb2fcc0494662eea8ba6b6469
2dc318250030c6af045cb738bc81ba35b043c3dcb46adf6f58d4100000000000000041
a3c03d51b452537ca2a1fea6110353c6d5ed483c4f0f86f4492ca3f378d40a994b4477
f93c64d928edbbcd3e85a7c709b7ea73ee97986ce3d1438e1355437722000000000000
0007bf46c454b4c1b25799527d896508afd5fc62ef4ec59db1efb49113063d70cca
Hash(TT) = 0xfc6374762ba5cf11f4b2caa08b2cd1b9907ae0e26e8d6234318d91583cd74c86
Ke = 0xfc6374762ba5cf11f4b2caa08b2cd1b9
Ka = 0x907ae0e26e8d6234318d91583cd74c86
KcA = 0x5dbd2f477166b7fb6d61febbd77a5563
KcB = 0x7689b4654407a5faeffdc8f18359d8a3
A conf = 0xdfb4db8d48ae5a675963ea5e6c19d98d4ea028d8e898dad96ea19a80ade95dca
B conf = 0xd0f0609d1613138d354f7e95f19fb556bf52d751947241e8c7118df5ef0ae175

    ]]></artwork></figure>
      </section>
    </section>
  </back>
</rfc>