path: root/doc/rfc/rfc9588.txt

Internet Engineering Task Force (IETF)                       N. McCallum
Request for Comments: 9588                                      S. Sorce
Category: Standards Track                                     R. Harwood
ISSN: 2070-1721                                            Red Hat, Inc.
                                                               G. Hudson
                                                                     MIT
                                                             August 2024


      Kerberos Simple Password-Authenticated Key Exchange (SPAKE)
                           Pre-authentication

Abstract

   This document defines a new pre-authentication mechanism for the
   Kerberos protocol.  The mechanism uses a password-authenticated key
   exchange (PAKE) to prevent brute-force password attacks, and it may
   incorporate a second factor.

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://www.rfc-editor.org/info/rfc9588.

Copyright Notice

   Copyright (c) 2024 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
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   publication of this document.  Please review these documents
   carefully, as they describe your rights and restrictions with respect
   to this document.  Code Components extracted from this document must
   include Revised BSD License text as described in Section 4.e of the
   Trust Legal Provisions and are provided without warranty as described
   in the Revised BSD License.

Table of Contents

   1.  Introduction
     1.1.  Properties of PAKE
     1.2.  PAKE Algorithm Selection
     1.3.  PAKE and Two-Factor Authentication
     1.4.  SPAKE Overview
   2.  Document Conventions
   3.  Prerequisites
     3.1.  PA-ETYPE-INFO2
     3.2.  Cookie Support
     3.3.  More Pre-authentication Data Required
   4.  SPAKE Pre-authentication Message Protocol
     4.1.  First Pass
     4.2.  Second Pass
     4.3.  Third Pass
     4.4.  Subsequent Passes
     4.5.  Reply Key Strengthening
     4.6.  Optimizations
   5.  SPAKE Parameters and Conversions
   6.  Transcript Hash
   7.  Key Derivation
   8.  Second-Factor Types
   9.  Hint for Authentication Sets
   10. Security Considerations
     10.1.  SPAKE Computations
     10.2.  Unauthenticated Plaintext
     10.3.  Side Channels
     10.4.  KDC State
     10.5.  Dictionary Attacks
     10.6.  Brute-Force Attacks
     10.7.  Denial-of-Service Attacks
     10.8.  Reflection Attacks
     10.9.  Reply Key Encryption Type
     10.10. KDC Authentication
   11. Assigned Constants
   12. IANA Considerations
     12.1.  Kerberos Second-Factor Types
       12.1.1.  Registration Template
       12.1.2.  Initial Registry Contents
     12.2.  Kerberos SPAKE Groups
       12.2.1.  Registration Template
       12.2.2.  Initial Registry Contents
   13. References
     13.1.  Normative References
     13.2.  Informative References
   Appendix A.  ASN.1 Module
   Appendix B.  SPAKE M and N Value Selection
   Appendix C.  Test Vectors
   Acknowledgements
   Authors' Addresses

1.  Introduction

   The Kerberos protocol [RFC4120] commonly uses password-derived long-
   term keys to secure the initial authentication exchange between a
   Kerberos client and a Key Distribution Center (KDC).  As noted in
   Section 10 of [RFC4120], an attacker can perform an offline
   dictionary attack against the password; this is performed either by
   initiating an authentication exchange to a principal for which the
   KDC does not require pre-authentication or after eavesdropping on a
   legitimate authentication exchange that uses encrypted timestamp pre-
   authentication (Section 5.2.7.2 of [RFC4120]).

   This document defines a pre-authentication mechanism that
   authenticates using long-term keys but is resistant to offline
   dictionary attacks.  The mechanism additionally enables the use of
   second-factor authentication without the need for a separately
   established secure channel, by leveraging the trust relationship
   established by the shared long-term key.

1.1.  Properties of PAKE

   Password-authenticated key exchange (PAKE) algorithms [RFC8125]
   provide several properties that defend against offline dictionary
   attacks and make them ideal for use in a Kerberos pre-authentication
   mechanism.

   1.  Each side of the exchange contributes entropy.

   2.  Passive attackers cannot determine the shared key.

   3.  Active attackers cannot perform a machine-in-the-middle attack.

   These properties of PAKE allow us to establish high-entropy
   encryption keys resistant to offline brute-force attacks, even when
   the passwords used are weak (low entropy).

1.2.  PAKE Algorithm Selection

   The SPAKE algorithm (defined in [SPAKE]) works by encrypting the
   public keys of a Diffie-Hellman (DH) key exchange with a shared
   secret.  SPAKE is selected for this pre-authentication mechanism for
   the following properties:

   1.  SPAKE's encryption method ensures that the result is a member of
       the underlying group, so it can be used with elliptic curve
       cryptography, which is believed to provide equivalent security
       levels to finite-field DH key exchange at much smaller key sizes.

   2.  It can compute the shared key after just one message from each
       party, minimizing the need for additional round trips and state.

   3.  It requires a small number of group operations; therefore, it can
       be implemented simply and efficiently.

1.3.  PAKE and Two-Factor Authentication

   Using PAKE in a pre-authentication mechanism also has another benefit
   when used as a component of two-factor authentication (2FA).  2FA
   methods often require the secure transfer of plaintext material to
   the KDC for verification.  This includes one-time passwords,
   challenge/response signatures, and biometric data.  Encrypting this
   data using the long-term secret results in packets that are
   vulnerable to offline brute-force attacks on the password, using
   either an authentication tag or statistical properties of the 2FA
   credentials to determine whether a password guess is correct.

   In "One-Time Password (OTP) Pre-Authentication" [RFC6560], this
   problem is mitigated using flexible authentication secure tunneling
   (FAST) (Section 5.4 of [RFC6113]), which uses a secondary trust
   relationship to create a secure encryption channel within which pre-
   authentication data can be sent.  However, the requirement for a
   secondary trust relationship has proven to be cumbersome to deploy
   and often introduces third parties into the trust chain (such as
   certification authorities).  These requirements make it difficult to
   enable FAST without manual configuration of client hosts.  In
   contrast, SPAKE pre-authentication, can create a secure encryption
   channel implicitly, using the key exchange to negotiate a high-
   entropy encryption key.  This key can then be used to securely
   encrypt 2FA plaintext data without the need for a secondary trust
   relationship.  Further, if the second-factor verifiers are sent at
   the same time as the first-factor verifier, and the KDC is careful to
   prevent timing attacks, then an online brute-force attack cannot be
   used to attack the factors separately.

   For these reasons, this document departs from the advice given in
   Section 1 of [RFC6113], which states: "Mechanism designers should
   design FAST factors, instead of new pre-authentication mechanisms
   outside of FAST."  However, the SPAKE pre-authentication mechanism
   does not intend to replace FAST and may be used with it to further
   conceal the metadata of the Kerberos messages.

1.4.  SPAKE Overview

   The SPAKE algorithm can be broadly described in a series of four
   steps:

   1.  Calculation and exchange of the public key

   2.  Calculation of the shared secret (K)

   3.  Derivation of an encryption key (K')

   4.  Verification of the derived encryption key (K')

   In this mechanism, key verification happens implicitly by a
   successful decryption of the 2FA data or of a placeholder value when
   no second factor is required.  This mechanism uses a tailored method
   of deriving encryption keys from the calculated shared secret K, for
   several reasons:

   *  to fit within the framework of [RFC3961],

   *  to ensure negotiation integrity using a transcript hash,

   *  to derive different keys for each use, and

   *  to bind the KDC-REQ-BODY to the pre-authentication exchange.

2.  Document Conventions

   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.

   This document refers to numerous terms and protocol messages defined
   in [RFC4120].

   The terms "encryption type", "key generation seed length", and
   "random-to-key" are defined in [RFC3961].

   The terms "FAST", "PA-FX-COOKIE", "KDC_ERR_PREAUTH_EXPIRED",
   "KDC_ERR_MORE_PREAUTH_DATA_REQUIRED", "KDC_ERR_PREAUTH_FAILED", "pre-
   authentication facility", and "authentication set" are defined in
   [RFC6113].

   [SPAKE] defines SPAKE as a family of two key-exchange algorithms
   differing only in derivation of the final key.  This mechanism uses a
   derivation similar to the second algorithm (SPAKE2).  For simplicity,
   this document refers to the algorithm as "SPAKE".

   The terms "Abstract Syntax Notation One (ASN.1)" and "Distinguished
   Encoding Rules (DER)" are defined in [ITU-T.X680.2021] and
   [ITU-T.X690.2021], respectively.

   When discussing operations within algebraic groups, this document
   uses additive notation (as described in Section 2.2 of [RFC6090]).
   Group elements are denoted with uppercase letters, while scalar
   multiplier values are denoted with lowercase letters.

3.  Prerequisites

3.1.  PA-ETYPE-INFO2

   This mechanism requires the initial KDC pre-authentication state to
   contain a singular reply key.  Therefore, a KDC that offers SPAKE
   pre-authentication as a stand-alone mechanism MUST supply a PA-ETYPE-
   INFO2 value containing a single ETYPE-INFO2-ENTRY, following the
   guidance in Section 2.1 of [RFC6113].  PA-ETYPE-INFO2 is specified in
   Section 5.2.7.5 of [RFC4120].

3.2.  Cookie Support

   KDCs that implement SPAKE pre-authentication MUST have some secure
   mechanism for retaining state between authentication service requests
   (AS-REQs).  For stateless KDC implementations, this method will most
   commonly be an encrypted PA-FX-COOKIE.  Clients that implement SPAKE
   pre-authentication MUST support PA-FX-COOKIE, as described in
   Section 5.2 of [RFC6113].

3.3.  More Pre-authentication Data Required

   Both KDCs and clients that implement SPAKE pre-authentication MUST
   support the use of KDC_ERR_MORE_PREAUTH_DATA_REQUIRED, as described
   in Section 5.2 of [RFC6113].

4.  SPAKE Pre-authentication Message Protocol

   This mechanism uses the reply key and provides the client
   authentication and strengthening reply key pre-authentication
   facilities (Section 3 of [RFC6113]).  When the mechanism completes
   successfully, the client will have proved knowledge of the original
   reply key and possibly a second factor, and the reply key will be
   strengthened to a more uniform distribution based on the PAKE
   exchange.  This mechanism also ensures the integrity of the KDC-REQ-
   BODY contents.  This mechanism can be used in an authentication set;
   no pa-hint value is required or defined.

   This mechanism negotiates a choice of group for the SPAKE algorithm.
   Groups are defined in the "Kerberos SPAKE Groups" registry created by
   this document (see Section 12.2).  Each group definition specifies an
   associated hash function, which will be used for transcript
   protection and key derivation.  Clients and KDCs MUST implement the
   edwards25519 group, but they MAY choose not to offer or accept it by
   default.

   The subsections that follow will describe the flow of messages when
   performing SPAKE pre-authentication.  We will begin by explaining the
   most verbose version of the protocol, which all implementations MUST
   support.  Then, we will describe several optional optimizations to
   reduce round trips.

   Mechanism messages are communicated using PA-DATA elements within the
   padata field of KDC-REQ messages or within the METHOD-DATA in the
   e-data field of KRB-ERROR messages.  All PA-DATA elements for this
   mechanism MUST use the following padata-type:

   PA-SPAKE  151

   The padata-value for all PA-SPAKE PA-DATA values MUST be empty or
   contain a DER encoding for the ASN.1 type PA-SPAKE.

   PA-SPAKE ::= CHOICE {
       support     [0] SPAKESupport,
       challenge   [1] SPAKEChallenge,
       response    [2] SPAKEResponse,
       encdata     [3] EncryptedData,
       ...
   }

4.1.  First Pass

   The SPAKE pre-authentication exchange begins when the client sends an
   initial authentication service request (AS-REQ) without pre-
   authentication data.  Upon receipt of this AS-REQ, a KDC that
   requires pre-authentication and supports SPAKE SHOULD (unless
   configuration indicates otherwise) reply with a
   KDC_ERR_PREAUTH_REQUIRED error, with METHOD-DATA containing an empty
   PA-SPAKE PA-DATA element (possibly in addition to other PA-DATA
   elements).  This message indicates to the client that the KDC
   supports SPAKE pre-authentication.

4.2.  Second Pass

   Once the client knows that the KDC supports SPAKE pre-authentication
   and the client wants to use it, the client will generate a new AS-REQ
   message containing a PA-SPAKE PA-DATA element using the support
   choice.  This message indicates to the KDC which groups the client
   prefers for the SPAKE operation.  The group numbers are defined in
   the "Kerberos SPAKE Groups" registry (see Section 12.2).  The group's
   sequence is ordered from the most preferred group to the least
   preferred group.

   SPAKESupport ::= SEQUENCE {
       groups      [0] SEQUENCE (SIZE(1..MAX)) OF Int32,
       ...
   }

   Upon receipt of the support message, the KDC will select a group.
   The KDC SHOULD choose a group from the groups provided by the support
   message.  However, if the support message does not contain any group
   that is supported by the KDC, the KDC MAY select another group in
   hopes that the client might support it.  Otherwise, the KDC MUST
   respond with a KDC_ERR_PREAUTH_FAILED error.

   The group selection determines the group order, which shall be a
   large prime p multiplied by a small cofactor h (possibly 1), a
   generator P of a prime-order subgroup, and two masking points M and
   N.  The KDC selects a random integer x in the range 0 <= x < h*p,
   which MUST be divisible by h.  The KDC computes a public key
   T=x*P+w*M, where w is computed from the initial reply key according
   to Section 5.

   The KDC will reply to the client with a
   KDC_ERR_MORE_PREAUTH_DATA_REQUIRED error containing a PA-SPAKE PA-
   DATA element using the challenge choice.

   SPAKEChallenge ::= SEQUENCE {
       group       [0] Int32,
       pubkey      [1] OCTET STRING,
       factors     [2] SEQUENCE (SIZE(1..MAX)) OF SPAKESecondFactor,
       ...
   }

   The group field indicates the KDC-selected group used for all SPAKE
   calculations as defined in the "Kerberos SPAKE Groups" registry (see
   Section 12.2).

   The pubkey field indicates the KDC's public key T, serialized
   according to Section 5.

   The factors field contains an unordered list of second factors, which
   can be used to complete the authentication.  Each second factor is
   represented by a SPAKESecondFactor.

   SPAKESecondFactor ::= SEQUENCE {
       type        [0] Int32,
       data        [1] OCTET STRING OPTIONAL
   }

   The type field is a unique integer that identifies the second-factor
   type.  The factors field of SPAKEChallenge MUST NOT contain more than
   one SPAKESecondFactor with the same type value.

   The data field contains optional challenge data.  The contents in
   this field will depend upon the second-factor type chosen.  Note that
   this challenge is initially transmitted as unauthenticated plaintext;
   see Section 10.2.

   The client and KDC will each initialize a transcript hash (Section 6)
   using the hash function associated with the chosen group and update
   it with the concatenation of the DER-encoded PA-SPAKE messages sent
   by the client and the KDC.

4.3.  Third Pass

   Upon receipt of the challenge message, the client observes which
   group was selected by the KDC and deserializes the KDC's public key
   T.  The client selects a random integer y in the range 0 <= x < h*p,
   which MUST be divisible by h.  The client computes a public key
   S=y*P+w*N, where w is computed from the initial reply key according
   to Section 5.  The client computes a shared group element
   K=y*(T-w*M).

   The client will then choose one of the second-factor types listed in
   the factors field of the challenge message and gather whatever data
   is required for the chosen second-factor type, possibly using the
   associated challenge data.  Finally, the client will send an AS-REQ
   containing a PA-SPAKE PA-DATA element using the response choice.

   SPAKEResponse ::= SEQUENCE {
       pubkey      [0] OCTET STRING,
       factor      [1] EncryptedData, -- SPAKESecondFactor
       ...
   }

   The client and KDC will update the transcript hash with the pubkey
   value and use the resulting hash for all encryption key derivations.

   The pubkey field indicates the client's public key S, serialized
   according to Section 5.

   The factor field indicates the client's chosen second-factor data.
   The key for this field is K'[1] (specified in Section 7).  The kvno
   field of the EncryptedData sequence is omitted.  The key usage number
   for the encryption is KEY_USAGE_SPAKE.  The plaintext inside the
   EncryptedData is an encoding of the SPAKESecondFactor.  Once decoded,
   the SPAKESecondFactor provides the type of the second factor and any
   optional data used.  The contents of the data field will depend on
   the second-factor type chosen.  The client MUST NOT send a response
   containing a second-factor type that was not listed in the factors
   field of the challenge message.

   When the KDC receives the response message from the client, it
   deserializes the client's public key S, and computes the shared group
   element K=x*(S-w*N).  The KDC derives K'[1] and decrypts the factors
   field.  If decryption is successful, the first factor is successfully
   validated.  The KDC then validates the second factor.  If either
   factor fails to validate, the KDC MUST respond with a
   KDC_ERR_PREAUTH_FAILED error.

   If validation of the second factor requires further round trips, the
   KDC MUST reply to the client with a
   KDC_ERR_MORE_PREAUTH_DATA_REQUIRED error containing a PA-SPAKE PA-
   DATA element using the encdata choice.  The kvno field of the
   EncryptedData sequence is omitted.  The key for the EncryptedData
   value is K'[2] (specified in Section 7), and the key usage number is
   KEY_USAGE_SPAKE.  The plaintext of this message contains a DER-
   encoded SPAKESecondFactor message.  As before, the type field of this
   message will contain the second-factor type and the data field will,
   optionally, contain data specific to the second-factor type.

4.4.  Subsequent Passes

   Any number of additional round trips may occur using the encdata
   choice.  The contents of the plaintexts are specific to the second-
   factor type.  If a client receives a PA-SPAKE PA-DATA element using
   the encdata choice from the KDC, it MUST reply with a subsequent AS-
   REQ with a PA-SPAKE PA-DATA element using the encdata choice or abort
   the AS exchange.

   The key for client-originated encdata messages in subsequent passes
   is K'[3] (specified in Section 7) for the first subsequent pass,
   K'[5] for the second, and so on.  The key for KDC-originated encdata
   messages is K'[4] for the first subsequent pass, K'[6] for the
   second, and so on.

4.5.  Reply Key Strengthening

   When the KDC has successfully validated both factors, the reply key
   is strengthened and the mechanism is complete.  The strengthening of
   the reply key is accomplished by the client and KDC replacing it with
   K'[0] (as specified in Section 7).  The KDC then replies with a KDC-
   REP message or continues on to the next mechanism in the
   authentication set.  There is no final PA-SPAKE PA-DATA message from
   the KDC to the client.

   Reply key strengthening occurs only once: at the end of the exchange.
   The client and KDC MUST use the initial reply key as the base key for
   all K'[n] derivations.

4.6.  Optimizations

   The full protocol has two possible optimizations.

   First, the KDC MAY reply to the initial AS-REQ (containing no pre-
   authentication data) with a PA-SPAKE PA-DATA element using the
   challenge choice instead of an empty padata-value.  In this case, the
   KDC optimistically selects a group that the client may not support.
   If the group chosen by the challenge message is supported by the
   client, the client MUST skip to the third pass by issuing an AS-REQ
   with a PA-SPAKE message using the response choice.  In this case, no
   SPAKESupport message is sent by the client, so the first update to
   the transcript hash contains only the KDC's optimistic challenge.  If
   the KDC's chosen group is not supported by the client, the client
   MUST continue to the second pass.  In this case, both the client and
   KDC MUST reinitialize the transcript hash for the client's support
   message.  Clients MUST support this optimization.

   Second, clients MAY skip the first pass and send an AS-REQ with a PA-
   SPAKE PA-DATA element using the support choice.  If the KDC accepts
   the support message and generates a challenge, it MUST include a PA-
   ETYPE-INFO2 value within the METHOD-DATA of the
   KDC_ERR_MORE_PREAUTH_DATA_REQUIRED error response, as the client may
   not otherwise be able to compute the initial reply key.  If the KDC
   cannot continue with SPAKE (either because the initial reply key type
   is incompatible with SPAKE or because it does not support any of the
   client's groups) but can offer other pre-authentication mechanisms,
   the KDC MUST respond with a KDC_ERR_PREAUTH_FAILED error containing
   METHOD-DATA for the available mechanisms.  A client supporting this
   optimization MUST continue after a KDC_ERR_PREAUTH_FAILED error as
   described in Section 2 of [RFC6113].  KDCs MUST support this
   optimization.

5.  SPAKE Parameters and Conversions

   Group elements are converted to and from octet strings using the
   serialization method defined in the "Kerberos SPAKE Groups" registry
   (see Section 12.2).

   The SPAKE algorithm requires constants M and N for each group.  These
   constants are defined in the "Kerberos SPAKE Groups" registry (see
   Section 12.2).

   The SPAKE algorithm requires a shared secret input w to be used as a
   scalar multiplier.  This value MUST be produced from the initial
   reply key as follows:

   1.  Determine the length of the multiplier octet string as defined in
       the "Kerberos SPAKE Groups" registry (see Section 12.2).

   2.  Compose a pepper string by concatenating the string "SPAKEsecret"
       and the group number as a big-endian four-byte two's complement
       binary number.

   3.  Produce an octet string of the required length using PRF+(K,
       pepper), where K is the initial reply key and PRF+ is as defined
       in Section 5.1 of [RFC6113].

   4.  Convert the octet string to a multiplier scalar using the
       multiplier conversion method defined in the "Kerberos SPAKE
       Groups" registry (see Section 12.2).

   The KDC chooses a secret scalar value x and the client chooses a
   secret scalar value y.  As required by the SPAKE algorithm, these
   values are chosen randomly and uniformly.  The KDC and client MUST
   NOT reuse x or y values for authentications involving different
   initial reply keys (see Section 10.4).

6.  Transcript Hash

   The transcript hash is an octet string of length equal to the output
   length of the hash function associated with the selected group.  All
   bits are set to zero in the initial value.

   When the transcript hash is updated with an octet string input, the
   new value is the hash function computed over the concatenation of the
   old value and the input.

   In the normal message flow or with the second optimization described
   in Section 4.6, the transcript hash is:

   1.  updated with the concatenation of the client's support message
       and the KDC's challenge, then

   2.  updated a second time with the client's pubkey value.

   Therefore, it incorporates the client's supported groups, the KDC's
   chosen group, the KDC's initial second-factor messages, and the
   client and KDC public values.  Once the transcript hash is finalized,
   it is used without change for all key derivations (Section 7).  In
   particular, encrypted second-factor messages are not included in the
   transcript hash.

   If the first optimization described in Section 4.6 is used
   successfully, the transcript hash is first updated with the KDC's
   challenge message and updated a second time with the client's pubkey
   value.

   If the first optimization is used unsuccessfully (i.e., the client
   does not accept the KDC's selected group), the transcript hash is
   computed as in the normal message flow, without including the KDC's
   optimistic challenge.

7.  Key Derivation

   Implementations MUST NOT use the shared group element (denoted by K)
   directly for any cryptographic operation.  Instead, the SPAKE result
   is used to derive keys K'[n] (defined in this section).

   First, compute the hash function associated with the selected group
   over the concatenation of the following values:

   *  The fixed string "SPAKEkey".

   *  The group number as a big-endian four-byte two's complement binary
      number.

   *  The encryption type of the initial reply key as a big-endian four-
      byte two's complement binary number.

   *  The PRF+ output used to compute the initial secret input w (as
      specified in Section 5).

   *  The SPAKE result K, converted to an octet string (as specified in
      Section 5).

   *  The transcript hash.

   *  The KDC-REQ-BODY encoding for the request being sent or responded
      to.  Within a FAST channel, the inner KDC-REQ-BODY encoding MUST
      be used.

   *  The value n as a big-endian, four-byte, and unsigned binary
      number.

   *  A single-byte block counter with the initial value 0x01.

   If the hash output is too small for the encryption type's key
   generation seed length, the block counter value is incremented and
   the hash function is recomputed to produce as many blocks as are
   required.  The result is truncated to the key generation seed length,
   and the random-to-key function is used to produce an intermediate key
   with the same encryption type as the initial reply key.

   The key K'[n] has the same encryption type as the initial reply key,
   and has the value KRB-FX-CF2(initial-reply-key, intermediate-key,
   "SPAKE", "keyderiv"), where KRB-FX-CF2 is defined in Section 5.1 of
   [RFC6113].

8.  Second-Factor Types

   This document defines one second-factor type:

   SF-NONE  1

   This second-factor type indicates that no second factor is used.
   Whenever a SPAKESecondFactor is used with SF-NONE, the data field
   MUST be omitted.  The SF-NONE second factor always successfully
   validates.

9.  Hint for Authentication Sets

   If a KDC offers SPAKE pre-authentication as part of an authentication
   set (Section 5.3 of [RFC6113]), it SHOULD provide a pa-hint value
   containing the DER encoding of the ASN.1 type PA-SPAKE-HINT.  This
   helps the client determine whether SPAKE pre-authentication is likely
   to succeed if the authentication set is chosen.

   PA-SPAKE-HINT ::= SEQUENCE {
       groups      [0] SEQUENCE (SIZE(1..MAX)) OF Int32,
       factors     [1] SEQUENCE (SIZE(1..MAX)) OF SPAKESecondFactor
   }

   The groups field indicates the KDC's supported groups.  The factors
   field indicates the KDC's supported second factors.  The KDC MAY omit
   the data field of values in the factors list.

   A KDC MUST NOT include a PA-SPAKE-HINT message directly in a pa-value
   field; hints must only be provided within authentication sets.  A KDC
   SHOULD include a hint if SPAKE pre-authentication is offered as the
   second or later element of an authentication set.

   The PA-SPAKE-HINT message is not part of the transcript, and it does
   not replace any part of the SPAKE message flow.

10.  Security Considerations

10.1.  SPAKE Computations

   The deserialized public keys S and T MUST be verified to be elements
   of the group to prevent invalid curve attacks.  It is not necessary
   to verify that they are members of the prime-order subgroup; the
   computation of K by both parties involves a multiplication by the
   cofactor h.

   The aforementioned cofactor multiplication is accomplished by
   choosing private scalars x and y, which are divisible by the
   cofactor.  If the client or KDC chooses a scalar that might not be
   divisible by the cofactor, an attacker might be able to coerce values
   of K that are not members of the prime-order subgroup and deduce a
   limited amount of information about w from the order of K.

   The scalars x and y MUST be chosen uniformly.  They MUST NOT be
   reused for different initial reply keys.  If an x or y value is
   reused for pre-authentications involving two different initial reply
   keys, an attacker who observes both authentications and knows one of
   the initial reply keys can conduct an offline dictionary attack to
   recover the other one.

   The M and N values for a group MUST NOT have known discrete logs.  An
   attacker who knows the discrete log of M or N can perform an offline
   dictionary attack on passwords.  Therefore, it is important to
   demonstrate that the M and N values for each group were computed
   without multiplying a known value by the generator P.

10.2.  Unauthenticated Plaintext

   This mechanism includes unauthenticated plaintext in the support and
   challenge messages.  Beginning with the third pass, the integrity of
   this plaintext is ensured by incorporating the transcript hash into
   the derivation of the final reply key and second-factor encryption
   keys.  Downgrade attacks on support and challenge messages will
   result in the client and KDC deriving different reply keys and
   EncryptedData keys.  The KDC-REQ-BODY contents are also incorporated
   into key derivation, ensuring their integrity.  The unauthenticated
   plaintext in the KDC-REP message is not protected by this mechanism.

   Unless FAST is used, the factors field of a challenge message is not
   integrity protected until the response is verified.  Second-factor
   types MUST account for this when specifying the semantics of the data
   field.  In particular, second-factor data in the challenge should not
   be included in user prompts: it could be modified by an attacker to
   contain misleading or offensive information.

   Unless FAST is used, the factors field of a challenge message is
   visible to an attacker, who can use it to determine whether a second
   factor is required for the client.

   Subsequent factor data, including the data in the response, are
   encrypted in a derivative of the shared secret K.  Therefore, it is
   not possible to exploit the untrustworthiness of the challenge to
   turn the client into an encryption or signing oracle for the second-
   factor credentials, unless the attacker knows the client's long-term
   key.

   Unless FAST is used, any PA-SPAKE-HINT messages are unauthenticated
   and are not protected by the transcript hash if they are included
   when SPAKE is advertised in authentication sets.  Since hints do not
   replace any part of the message flow, manipulation of hint messages
   can only affect the client's decision to use or not use an
   authentication set, which could more easily be accomplished by
   removing authentication sets entirely.

10.3.  Side Channels

   An implementation of the SPAKE pre-authentication mechanism can have
   the property of indistinguishability, meaning that an attacker who
   guesses a long-term key and a second-factor value cannot determine
   whether one of the factors was correct unless both are correct.
   Indistinguishability is only maintained if the second factor can be
   validated solely based on the data in the response; the use of
   additional round trips will reveal to the attacker whether the long-
   term key is correct.  Indistinguishability also requires that there
   are no side channels.  When the KDC processes a response message,
   whether or not it decrypts the factor field, it must reply with the
   same error fields, take the same amount of time, and make the same
   observable communications to other servers.

   Both the size of the EncryptedData and the number of EncryptedData
   messages used for second-factor data (including the factor field of
   the SPAKEResponse message and messages using the encdata PA-SPAKE
   choice) may reveal information about the second factor used in an
   authentication.  Care should be taken to keep second-factor messages
   as small and as few as possible.

   Any side channels in the creation of the shared secret input w, or in
   the multiplications wM and wN, could allow an attacker to recover the
   client long-term key.  Implementations MUST take care to avoid side
   channels, particularly timing channels.  Generation of the secret
   scalar values x and y need not take constant time, but the amount of
   time taken MUST NOT provide information about the resulting value.

   The conversion of the scalar multiplier for the SPAKE w parameter may
   produce a multiplier that is larger than the order of the group.
   Some group implementations may be unable to handle such a multiplier.
   Others may silently accept such a multiplier but proceed to perform
   multiplication that is not constant time.  This is only a minor risk
   in most commonly used groups, but it is a more serious risk for P-521
   due to the extra seven high bits in the input octet string.  A common
   solution to this problem is achieved by reducing the multiplier
   modulo the group order, taking care to ensure constant time
   operation.

10.4.  KDC State

   A stateless KDC implementation generally must use a PA-FX-COOKIE
   value to remember its private scalar value x and the transcript hash.
   The KDC MUST maintain confidentiality and integrity of the cookie
   value, perhaps by encrypting it in a key known only to the realm's
   KDCs.  Cookie values may be replayed by attackers, perhaps by
   splicing them into different SPAKE exchanges.  The KDC SHOULD limit
   the time window of replays using a timestamp, and it SHOULD prevent
   cookie values from being applied to other pre-authentication
   mechanisms or other client principals.  Within the validity period of
   a cookie, an attacker can replay the final message of a pre-
   authentication exchange to any of the realm's KDCs and make it appear
   that the client has authenticated.

   The SPAKE pre-authentication mechanism is not designed to provide
   forward secrecy.  Nevertheless, some measure of forward secrecy may
   result depending on implementation choices.  A passive attacker who
   determines the client long-term key after the exchange generally will
   not be able to recover the ticket session key; however, an attacker
   who also determines the PA-FX-COOKIE encryption key (if the KDC uses
   an encrypted cookie) will be able to recover the ticket session key.
   If the KDC or client retains the x or y value for reuse with the same
   client long-term key, an attacker who recovers the x or y value and
   the long-term key will be able to recover the ticket session key.

10.5.  Dictionary Attacks

   Although the SPAKE pre-authentication mechanism is designed to
   prevent an offline dictionary attack by an active attacker posing as
   the KDC, such an attacker can attempt to downgrade the client to the
   encrypted timestamp pre-authentication mechanism.  Client
   implementations SHOULD provide a configuration option to enable or
   disable the encrypted timestamp mechanism on a per-realm basis to
   mitigate this attack.

   If the user enters the wrong password, the client might fall back to
   the encrypted timestamp mechanism after receiving a
   KDC_ERR_PREAUTH_FAILED error from the KDC, if the encrypted timestamp
   mechanism is offered by the KDC and not disabled by client
   configuration.  This fallback will enable a passive attacker to mount
   an offline dictionary attack against the incorrect password, which
   may be similar to the correct password.  Client implementations
   SHOULD assume that the encrypted timestamp and encrypted challenge
   mechanisms are unlikely to succeed if SPAKE pre-authentication fails
   in the second pass and SF-NONE was used.

   Like any other pre-authentication mechanism using the client long-
   term key, the SPAKE pre-authentication mechanism does not prevent
   online password guessing attacks.  The KDC is made aware of
   unsuccessful guesses and can apply facilities such as rate limiting
   to mitigate the risk of online attacks.

10.6.  Brute-Force Attacks

   The selected group's resistance to offline brute-force attacks may
   not correspond to the size of the reply key.  For performance
   reasons, a KDC MAY select a group whose brute-force work factor is
   less than the reply key length.  A passive attacker who solves the
   group discrete logarithm problem after the exchange will be able to
   conduct an offline attack against the client long-term key.  Although
   the use of password policies and costly, salted string-to-key
   functions may increase the cost of such an attack, the resulting cost
   will likely not be higher than the cost of solving the group discrete
   logarithm.

10.7.  Denial-of-Service Attacks

   Elliptic curve group operations are more computationally expensive
   than secret-key operations.  As a result, the use of this mechanism
   may affect the KDC's performance under normal load and its resistance
   to denial-of-service attacks.

10.8.  Reflection Attacks

   The encdata choice of PA-SPAKE can be used in either direction; the
   factor-specific plaintext does not necessarily indicate a direction.
   However, each encdata message is encrypted using a derived key K'[n],
   with client-originated messages using only odd values of n and KDC-
   originated messages using only even values.  Therefore, an attempted
   reflection attack would result in a failed decryption.

10.9.  Reply Key Encryption Type

   This mechanism does not upgrade the encryption type of the initial
   reply key and relies on that encryption type for confidentiality,
   integrity, and pseudorandom functions.  If the client long-term key
   uses a weak encryption type, an attacker might be able to subvert the
   exchange, and the replaced reply key will also be of the same weak
   encryption type.

10.10.  KDC Authentication

   This mechanism does not directly provide the KDC Authentication pre-
   authentication facility because it does not send a key confirmation
   from the KDC to the client.  When used as a stand-alone mechanism,
   the preexisting KDC authentication provided by the KDC-REP enc-part
   still applies.

11.  Assigned Constants

   The following key usage values are assigned for this mechanism:

   KEY_USAGE_SPAKE  65

12.  IANA Considerations

   IANA has assigned the following number for PA-SPAKE in the "Pre-
   authentication and Typed Data" registry:

                     +==========+=======+===========+
                     | Type     | Value | Reference |
                     +==========+=======+===========+
                     | PA-SPAKE | 151   | RFC 9588  |
                     +----------+-------+-----------+

                                 Table 1

   This document establishes two registries (see Sections 12.1 and 12.2)
   with the following procedure, in accordance with [RFC8126]:

   Registry entries are to be evaluated using the Specification Required
   method.  All specifications must be published prior to entry
   inclusion in the registry.  Once published, they can be submitted
   directly to the krb5-spake-review@ietf.org mailing list, where there
   will be a three-week-long review period by designated experts.

   The designated experts ensure that the specification is publicly
   available.  They may provide additional in-depth reviews, but their
   approval should not be taken as endorsement of the specification.

   Prior to the end of the review period, the designated experts must
   approve or deny the request.  This decision is conveyed to both IANA
   and the submitter.  Since the mailing list archives are not public,
   it should include both a reasonably detailed explanation in the case
   of a denial as well as whether the request can be resubmitted.

   IANA must only accept registry updates from the designated experts
   and should direct all requests for registration to the review mailing
   list.

12.1.  Kerberos Second-Factor Types

   This section specifies the "Kerberos Second-Factor Types" registry,
   which records the number, name, and reference for each second-factor
   protocol.

12.1.1.  Registration Template

   ID Number:  A value that uniquely identifies this entry.  It is a
      signed integer in the range -2147483648 to 2147483647, inclusive.
      Positive values must be assigned only for algorithms specified in
      accordance with these rules for use with Kerberos and related
      protocols.  Negative values should be used for private and
      experimental algorithms only.  Zero is reserved and must not be
      assigned.  Values should be assigned in increasing order.

   Name:  A brief, unique, human-readable name for this algorithm.

   Reference:  A URI or otherwise unique identifier for where the
      details of this algorithm can be found.  It should be as specific
      as reasonably possible.

12.1.2.  Initial Registry Contents

   ID Number:  0
   Name:  Reserved
   Reference:  RFC 9588

   ID Number:  1
   Name:  SF-NONE
   Reference:  RFC 9588

12.2.  Kerberos SPAKE Groups

   This section specifies the "Kerberos SPAKE Groups" registry, which
   records the number, name, specification, serialization, multiplier
   length, multiplier conversion, SPAKE M and N constants, and
   associated hash function for each SPAKE Group.

12.2.1.  Registration Template

   ID Number:  A value that uniquely identifies this entry.  It is a
      signed integer in the range -2147483648 to 2147483647, inclusive.
      Positive values must be assigned only for algorithms specified in
      accordance with these rules for use with Kerberos and related
      protocols.  Negative values should be used for private and
      experimental use only.  Zero is reserved and must not be assigned.
      Values should be assigned in increasing order.

   Name:  A brief, unique, human-readable name for this entry.

   Specification:  A reference to the definition of the group parameters
      and operations.

   Serialization:  A reference to the definition of the method used to
      serialize and deserialize group elements.

   Multiplier Length:  The length of the input octet string to
      multiplication operations.

   Multiplier Conversion:  A reference to the definition of the method
      used to convert an octet string to a multiplier scalar.

   SPAKE M Constant:  The serialized value of the SPAKE M constant in
      hexadecimal notation.

   SPAKE N Constant:  The serialized value of the SPAKE N constant in
      hexadecimal notation.

   Hash Function:  The group's associated hash function.

12.2.2.  Initial Registry Contents

12.2.2.1.  Edwards 25519

   ID Number:  1
   Name:  edwards25519
   Specification:  Section 4.1 of [RFC7748] (edwards25519)
   Serialization:  Section 3.1 of [RFC8032]
   Multiplier Length:  32
   Multiplier Conversion:  Section 3.1 of [RFC8032]
   SPAKE M Constant:
      d048032c6ea0b6d697ddc2e86bda85a33adac920f1bf18e1b0c6d166a5cecdaf
   SPAKE N Constant:
      d3bfb518f44f3430f29d0c92af503865a1ed3281dc69b35dd868ba85f886c4ab
   Hash function:  SHA-256 [RFC6234]

12.2.2.2.  P-256

   ID Number:  2
   Name:  P-256
   Specification:  Section 2.4.2 of [SEC2]
   Serialization:  Section 2.3.3 of [SEC1] (compressed format)
   Multiplier Length:  32
   Multiplier Conversion:  Section 2.3.8 of [SEC1]
   SPAKE M Constant:
      02886e2f97ace46e55ba9dd7242579f2993b64e16ef3dcab95afd497333d8fa12f
   SPAKE N Constant:
      03d8bbd6c639c62937b04d997f38c3770719c629d7014d49a24b4f98baa1292b49
   Hash function:  SHA-256 [RFC6234]

12.2.2.3.  P-384

   ID Number:  3
   Name:  P-384
   Specification:  Section 2.5.1 of [SEC2]
   Serialization:  Section 2.3.3 of [SEC1] (compressed format)
   Multiplier Length:  48
   Multiplier Conversion:  Section 2.3.8 of [SEC1]
   SPAKE M Constant:
      030ff0895ae5ebf6187080a82d82b42e2765e3b2f8749c7e05eba366434b363d3d
      c36f15314739074d2eb8613fceec2853
   SPAKE N Constant:
      02c72cf2e390853a1c1c4ad816a62fd15824f56078918f43f922ca21518f9c543b
      b252c5490214cf9aa3f0baab4b665c10
   Hash function:  SHA-384 [RFC6234]

12.2.2.4.  P-521

   ID Number:  4
   Name:  P-521
   Specification:  Section 2.6.1 of [SEC2]
   Serialization:  Section 2.3.3 of [SEC1] (compressed format)
   Multiplier Length:  48
   Multiplier Conversion:  Section 2.3.8 of [SEC1]
   SPAKE M Constant:
      02003f06f38131b2ba2600791e82488e8d20ab889af753a41806c5db18d37d8560
      8cfae06b82e4a72cd744c719193562a653ea1f119eef9356907edc9b56979962d7
      aa
   SPAKE N Constant:
      0200c7924b9ec017f3094562894336a53c50167ba8c5963876880542bc669e494b
      2532d76c5b53dfb349fdf69154b9e0048c58a42e8ed04cef052a3bc349d95575cd
      25
   Hash function:  SHA-512 [RFC6234]

13.  References

13.1.  Normative References

   [ITU-T.X680.2021]
              ITU-T, "Information technology - Abstract Syntax Notation
              One (ASN.1): Specification of basic notation", ITU-T
              Recommendation X.680, ISO/IEC 8824-1:2021, February 2021.

   [ITU-T.X690.2021]
              ITU-T, "Information technology - ASN.1 encoding rules:
              Specification of Basic Encoding Rules (BER), Canonical
              Encoding Rules (CER) and Distinguished Encoding Rules
              (DER)", ITU-T Recommendation X.690, ISO/IEC 8825-1:2021,
              February 2021.

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

   [RFC3961]  Raeburn, K., "Encryption and Checksum Specifications for
              Kerberos 5", RFC 3961, DOI 10.17487/RFC3961, February
              2005, <https://www.rfc-editor.org/info/rfc3961>.

   [RFC4120]  Neuman, C., Yu, T., Hartman, S., and K. Raeburn, "The
              Kerberos Network Authentication Service (V5)", RFC 4120,
              DOI 10.17487/RFC4120, July 2005,
              <https://www.rfc-editor.org/info/rfc4120>.

   [RFC6113]  Hartman, S. and L. Zhu, "A Generalized Framework for
              Kerberos Pre-Authentication", RFC 6113,
              DOI 10.17487/RFC6113, April 2011,
              <https://www.rfc-editor.org/info/rfc6113>.

   [RFC6234]  Eastlake 3rd, D. and T. Hansen, "US Secure Hash Algorithms
              (SHA and SHA-based HMAC and HKDF)", RFC 6234,
              DOI 10.17487/RFC6234, May 2011,
              <https://www.rfc-editor.org/info/rfc6234>.

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

   [RFC8032]  Josefsson, S. and I. Liusvaara, "Edwards-Curve Digital
              Signature Algorithm (EdDSA)", RFC 8032,
              DOI 10.17487/RFC8032, January 2017,
              <https://www.rfc-editor.org/info/rfc8032>.

   [RFC8126]  Cotton, M., Leiba, B., and T. Narten, "Guidelines for
              Writing an IANA Considerations Section in RFCs", BCP 26,
              RFC 8126, DOI 10.17487/RFC8126, June 2017,
              <https://www.rfc-editor.org/info/rfc8126>.

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

   [SEC1]     Standards for Efficient Cryptography Group, "SEC 1:
              Elliptic Curve Cryptography", May 2009.

   [SEC2]     Standards for Efficient Cryptography Group, "SEC 2:
              Recommended Elliptic Curve Domain Parameters", January
              2010.

13.2.  Informative References

   [RFC6090]  McGrew, D., Igoe, K., and M. Salter, "Fundamental Elliptic
              Curve Cryptography Algorithms", RFC 6090,
              DOI 10.17487/RFC6090, February 2011,
              <https://www.rfc-editor.org/info/rfc6090>.

   [RFC6560]  Richards, G., "One-Time Password (OTP) Pre-
              Authentication", RFC 6560, DOI 10.17487/RFC6560, April
              2012, <https://www.rfc-editor.org/info/rfc6560>.

   [RFC8125]  Schmidt, J., "Requirements for Password-Authenticated Key
              Agreement (PAKE) Schemes", RFC 8125, DOI 10.17487/RFC8125,
              April 2017, <https://www.rfc-editor.org/info/rfc8125>.

   [SPAKE]    Abdalla, M. and D. Pointcheval, "Simple Password-Based
              Encrypted Key Exchange Protocols", CT-RSA 2005, Lecture
              Notes in Computer Science, Volume 3376, pages 191-208,
              Springer, DOI 10.1007/978-3-540-30574-3_14, February 2005,
              <https://doi.org/10.1007/978-3-540-30574-3_14>.

Appendix A.  ASN.1 Module

   KerberosV5SPAKE {
           iso(1) identified-organization(3) dod(6) internet(1)
           security(5) kerberosV5(2) modules(4) spake(8)
   } DEFINITIONS EXPLICIT TAGS ::= BEGIN

   IMPORTS
       EncryptedData, Int32
         FROM KerberosV5Spec2 { iso(1) identified-organization(3)
           dod(6) internet(1) security(5) kerberosV5(2) modules(4)
           krb5spec2(2) };
           -- as defined in RFC 4120.

   SPAKESupport ::= SEQUENCE {
       groups      [0] SEQUENCE (SIZE(1..MAX)) OF Int32,
       ...
   }

   SPAKEChallenge ::= SEQUENCE {
       group       [0] Int32,
       pubkey      [1] OCTET STRING,
       factors     [2] SEQUENCE (SIZE(1..MAX)) OF SPAKESecondFactor,
       ...
   }

   SPAKESecondFactor ::= SEQUENCE {
       type        [0] Int32,
       data        [1] OCTET STRING OPTIONAL
   }

   SPAKEResponse ::= SEQUENCE {
       pubkey      [0] OCTET STRING,
       factor      [1] EncryptedData, -- SPAKESecondFactor
       ...
   }

   PA-SPAKE ::= CHOICE {
       support     [0] SPAKESupport,
       challenge   [1] SPAKEChallenge,
       response    [2] SPAKEResponse,
       encdata     [3] EncryptedData,
       ...
   }

   PA-SPAKE-HINT ::= SEQUENCE {
       groups      [0] SEQUENCE (SIZE(1..MAX)) OF Int32,
       factors     [1] SEQUENCE (SIZE(1..MAX)) OF SPAKESecondFactor
   }

   END

Appendix B.  SPAKE M and N Value Selection

   The M and N values for the initial contents of the SPAKE group
   registry were generated using the following Python snippet, which
   assumes an elliptic curve implementation following the interface of
   Edwards25519Point.stdbase() and Edwards448Point.stdbase() in
   Appendix A of [RFC8032]:

   def iterhash(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 = [iterhash(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

   The initial seed strings are as follows:

   *  For group 1 M: edwards25519 point generation seed (M)

   *  For group 1 N: edwards25519 point generation seed (N)

   *  For group 2 M: 1.2.840.10045.3.1.7 point generation seed (M)

   *  For group 2 N: 1.2.840.10045.3.1.7 point generation seed (N)

   *  For group 3 M: 1.3.132.0.34 point generation seed (M)

   *  For group 3 N: 1.3.132.0.34 point generation seed (N)

   *  For group 4 M: 1.3.132.0.35 point generation seed (M)

   *  For group 4 N: 1.3.132.0.35 point generation seed (N)

Appendix C.  Test Vectors

   For the following text vectors:

   *  The key is the string-to-key of "password" with the salt
      "ATHENA.MIT.EDUraeburn" for the designated initial reply key
      encryption type.

   *  x and y were chosen randomly within the order of the designated
      group, then multiplied by the cofactor.

   *  The SPAKESupport message contains only the designated group's
      number.

   *  The SPAKEChallenge message offers only the SF-NONE second-factor
      type.

   *  The KDC-REQ-BODY message does not contain KDC options, but does
      contain the client principal name "raeburn@ATHENA.MIT.EDU", the
      server principal name "krbtgt/ATHENA.MIT.EDU", the realm
      "ATHENA.MIT.EDU", the till field "19700101000000Z", the nonce
      zero, and an etype list containing only the designated encryption
      type.

   des3-cbc-sha1 edwards25519
   key: 850bb51358548cd05e86768c313e3bfef7511937dcf72c3e
   w (PRF+ output): 686d84730cb8679ae95416c6567c6a63
                    f2c9cef124f7a3371ae81e11cad42a37
   w (reduced multiplier): a1f1a25cbd8e3092667e2fddba8ecd24
                           f2c9cef124f7a3371ae81e11cad42a07
   x: 201012d07bfd48ddfa33c4aac4fb1e229fb0d043cfe65ebfb14399091c71a723
   y: 500b294797b8b042aca1bedc0f5931a4f52c537b3608b2d05cc8a2372f439f25
   X: ec274df1920dc0f690c8741b794127233745444161016ef950ad75c51db58c3e
   Y: d90974f1c42dac1cd4454561ac2d49af762f2ac87bf02436d461e7b661b43028
   T: 18f511e750c97b592acd30db7d9e5fca660389102e6bf610c1bfbed4616c8362
   S: 5d10705e0d1e43d5dbf30240ccfbde4a0230c70d4c79147ab0b317edad2f8ae7
   K: 25bde0d875f0feb5755f45ba5e857889d916ecf7476f116aa31dc3e037ec4292
   SPAKESupport: a0093007a0053003020101
   SPAKEChallenge: a1363034a003020101a122042018f511e750c97b592acd30
                   db7d9e5fca660389102e6bf610c1bfbed4616c8362a20930
                   073005a003020101
   Transcript hash after challenge: 22bb2271e34d329d52073c70b1d11879
                                    73181f0bc7614266bb79ee80d3335175
   Final transcript hash after pubkey: eaaa08807d0616026ff51c849efbf35b
                                       a0ce3c5300e7d486da46351b13d4605b
   KDC-REQ-BODY: 3075a00703050000000000a1143012a003020101a10b3009
                 1b077261656275726ea2101b0e415448454e412e4d49542e
                 454455a3233021a003020102a11a30181b066b7262746774
                 1b0e415448454e412e4d49542e454455a511180f31393730
                 303130313030303030305aa703020100a8053003020110
   K'[0]: baf12fae7cd958cbf1a29bfbc71f89ce49e03e295d89dafd
   K'[1]: 64f73dd9c41908206bcec1f719026b574f9d13463d7a2520
   K'[2]: 0454520b086b152c455829e6baeff78a61dfe9e3d04a895d
   K'[3]: 4a92260b25e3ef94c125d5c24c3e5bced5b37976e67f25c4

   rc4-hmac edwards25519
   key: 8846f7eaee8fb117ad06bdd830b7586c
   w (PRF+ output): 7c86659d29cf2b2ea93bfe79c3cefb88
                    50e82215b3ea6fcd896561d48048f49c
   w (reduced multiplier): 2713c1583c53861520b849bfef0525cd
                           4fe82215b3ea6fcd896561d48048f40c
   x: c8a62e7b626f44cad807b2d695450697e020d230a738c5cd5691cc781dce8754
   y: 18fe7c1512708c7fd06db270361f04593775bc634ceaf45347e5c11c38aae017
   X: b0bcbbdd25aa031f4608d0442dd4924be7731d49c089a8301859d77343ffb567
   Y: 7d1ab8aeda1a2b1f9eab8d11c0fda60b616005d0f37d1224c5f12b8649f579a5
   T: 7db465f1c08c64983a19f560bce966fe5306c4b447f70a5bca14612a92da1d63
   S: 38f8d4568090148ebc9fd17c241b4cc2769505a7ca6f3f7104417b72b5b5cf54
   K: 03e75edd2cd7e7677642dd68736e91700953ac55dc650e3c2a1b3b4acdb800f8
   SPAKESupport: a0093007a0053003020101
   SPAKEChallenge: a1363034a003020101a12204207db465f1c08c64983a19f5
                   60bce966fe5306c4b447f70a5bca14612a92da1d63a20930
                   073005a003020101
   Transcript hash after challenge: 3cde9ed9b562a09d816885b6c225f733
                                    6d9e2674bb4df903dfc894d963a2af42
   Final transcript hash after pubkey: f4b208458017de6ef7f6a307d47d87db
                                       6c2af1d291b726860f68bc08bfef440a
   KDC-REQ-BODY: 3075a00703050000000000a1143012a003020101a10b3009
                 1b077261656275726ea2101b0e415448454e412e4d49542e
                 454455a3233021a003020102a11a30181b066b7262746774
                 1b0e415448454e412e4d49542e454455a511180f31393730
                 303130313030303030305aa703020100a8053003020117
   K'[0]: 770b720c82384cbb693e85411eedecba
   K'[1]: 621deec88e2865837c4d3462bb50a1d5
   K'[2]: 1cc8f6333b9fa3b42662fd9914fbd5bb
   K'[3]: edb4032b7fc3806d5211a534dcbc390c

   aes128-cts-hmac-sha1-96 edwards25519
   key: fca822951813fb252154c883f5ee1cf4
   w (PRF+ output): 0d591b197b667e083c2f5f98ac891d3c
                    9f99e710e464e62f1fb7c9b67936f3eb
   w (reduced multiplier): 17c2a9030afb7c37839bd4ae7fdfeb17
                           9e99e710e464e62f1fb7c9b67936f30b
   x: 50be049a5a570fa1459fb9f666e6fd80602e4e87790a0e567f12438a2c96c138
   y: b877afe8612b406d96be85bd9f19d423e95be96c0e1e0b5824127195c3ed5917
   X: e73a443c678913eb4a0cad5cbd3086cf82f65a5a91b611e01e949f5c52efd6dd
   Y: 473c5b44ed2be9cb50afe1762b535b3930530489816ea6bd962622cccf39f6e8
   T: 9e9311d985c1355e022d7c3c694ad8d6f7ad6d647b68a90b0fe46992818002da
   S: fbe08f7f96cd5d4139e7c9eccb95e79b8ace41e270a60198c007df18525b628e
   K: c2f7f99997c585e6b686ceb62db42f17cc70932def3bb4cf009e36f22ea5473d
   SPAKESupport: a0093007a0053003020101
   SPAKEChallenge: a1363034a003020101a12204209e9311d985c1355e022d7c
                   3c694ad8d6f7ad6d647b68a90b0fe46992818002daa20930
                   073005a003020101
   Transcript hash after challenge: 4512310282c01b39dd9aebd0cc2a5e53
                                    2ed077a6c11d4c973c4593d525078797
   Final transcript hash after pubkey: 951285f107c87f0169b9c918a1f51f60
                                       cb1a75b9f8bb799a99f53d03add94b5f
   KDC-REQ-BODY: 3075a00703050000000000a1143012a003020101a10b3009
                 1b077261656275726ea2101b0e415448454e412e4d49542e
                 454455a3233021a003020102a11a30181b066b7262746774
                 1b0e415448454e412e4d49542e454455a511180f31393730
                 303130313030303030305aa703020100a8053003020111
   K'[0]: 548022d58a7c47eae8c49dccf6baa407
   K'[1]: b2c9ba0e13fc8ab3a9d96b51b601cf4a
   K'[2]: 69f0ee5fdb6c237e7fcd38d9f87df1bd
   K'[3]: 78f91e2240b5ee528a5cc8d7cbebfba5

   aes256-cts-hmac-sha1-96 edwards25519
   key: 01b897121d933ab44b47eb5494db15e50eb74530dbdae9b634d65020ff5d88c1
   w (PRF+ output): e902341590a1b4bb4d606a1c643cccb3
                    f2108f1b6aa97b381012b9400c9e3f4e
   w (reduced multiplier): 35b35ca126156b5bf4ec8b90e9545060
                           f2108f1b6aa97b381012b9400c9e3f0e
   x: 88c6c0a4f0241ef217c9788f02c32d00b72e4310748cd8fb5f94717607e6417d
   y: 88b859df58ef5c69bacdfe681c582754eaab09a74dc29cff50b328613c232f55
   X: 23c48eaff2721051946313840723b38f563c59b92043d6ffd752f95781af0327
   Y: 3d51486ec1d9be69bc45386bb675c013db87fd0488f6a9cacf6b43e8c81a0641
   T: 6f301aacae1220e91be42868c163c5009aeea1e9d9e28afcfc339cda5e7105b5
   S: 9e2cc32908fc46273279ec75354b4aeafa70c3d99a4d507175ed70d80b255dda
   K: cf57f58f6e60169d2ecc8f20bb923a8e4c16e5bc95b9e64b5dc870da7026321b
   SPAKESupport: a0093007a0053003020101
   SPAKEChallenge: a1363034a003020101a12204206f301aacae1220e91be428
                   68c163c5009aeea1e9d9e28afcfc339cda5e7105b5a20930
                   073005a003020101
   Transcript hash after challenge: 23a5e72eb4dedd1ca860f43736c458f0
                                    775c3bb1370a26af8a9374d521d70ec9
   Final transcript hash after pubkey: 1c605649d4658b58cbe79a5faf227acc
                                       16c355c58b7dade022f90c158fe5ed8e
   KDC-REQ-BODY: 3075a00703050000000000a1143012a003020101a10b3009
                 1b077261656275726ea2101b0e415448454e412e4d49542e
                 454455a3233021a003020102a11a30181b066b7262746774
                 1b0e415448454e412e4d49542e454455a511180f31393730
                 303130313030303030305aa703020100a8053003020112
   K'[0]: a9bfa71c95c575756f922871524b6528
          8b3f695573ccc0633e87449568210c23
   K'[1]: 1865a9ee1ef0640ec28ac007391cac62
          4c42639c714767a974e99aa10003015f
   K'[2]: e57781513fefdb978e374e156b0da0c1
          a08148f5eb26b8e157ac3c077e28bf49
   K'[3]: 008e6487293c3cc9fabbbcdd8b392d6d
          cb88222317fd7fe52d12fbc44fa047f1

   aes256-cts-hmac-sha1-96 P-256
   key: 01b897121d933ab44b47eb5494db15e50eb74530dbdae9b634d65020ff5d88c1
   w (PRF+ output): eb2984af18703f94dd5288b8596cd369
                    88d0d4e83bfb2b44de14d0e95e2090bd
   w (reduced multiplier): eb2984af18703f94dd5288b8596cd369
                           88d0d4e83bfb2b44de14d0e95e2090bd
   x: 935ddd725129fb7c6288e1a5cc45782198a6416d1775336d71eacd0549a3e80e
   y: e07405eb215663abc1f254b8adc0da7a16febaa011af923d79fdef7c42930b33
   X: 03bc802165aea7dbd98cc155056249fe0a37a9c203a7c0f7e872d5bf687bd105e2
   Y: 0340b8d91ce3852d0a12ae1f3e82c791fc86df6b346006431e968a1b869af7c735
   T: 024f62078ceb53840d02612195494d0d0d88de21feeb81187c71cbf3d01e71788d
   S: 021d07dc31266fc7cfd904ce2632111a169b7ec730e5f74a7e79700f86638e13c8
   K: 0268489d7a9983f2fde69c6e6a1307e9d252259264f5f2dfc32f58cca19671e79b
   SPAKESupport: a0093007a0053003020102
   SPAKEChallenge: a1373035a003020102a1230421024f62078ceb53840d0261
                   2195494d0d0d88de21feeb81187c71cbf3d01e71788da209
                   30073005a003020101
   Transcript hash after challenge: 0a142afca77c2e92b066572a90389eac
                                    40a6b1f1ed8b534d342591c0e7727e00
   Final transcript hash after pubkey: 20ad3c1a9a90fc037d1963a1c4bfb15a
                                       b4484d7b6cf07b12d24984f14652de60
   KDC-REQ-BODY: 3075a00703050000000000a1143012a003020101a10b3009
                 1b077261656275726ea2101b0e415448454e412e4d49542e
                 454455a3233021a003020102a11a30181b066b7262746774
                 1b0e415448454e412e4d49542e454455a511180f31393730
                 303130313030303030305aa703020100a8053003020112
   K'[0]: 7d3b906f7be49932db22cd3463f032d0
          6c9c078be4b1d076d201fc6e61ef531e
   K'[1]: 17d74e36f8993841fbb7feb12fa4f011
          243d3ae4d2ace55b39379294bbc4db2c
   K'[2]: d192c9044081a2aa6a97a6c69e2724e8
          e5671c2c9ce073dd439cdbaf96d7dab0
   K'[3]: 41e5bad6b67f12c53ce0e2720dd6a988
          7f877bf9463c2d5209c74c36f8d776b7

   aes256-cts-hmac-sha1-96 P-384
   key: 01b897121d933ab44b47eb5494db15e50eb74530dbdae9b634d65020ff5d88c1
   w (PRF+ output): 0304cfc55151c6bbe889653db96dbfe0ba4acafc024c1e88
                    40cb3a486f6d80c16e1b8974016aa4b7fa43042a9b3825b1
   w (reduced multiplier): 0304cfc55151c6bbe889653db96dbfe0
                           ba4acafc024c1e8840cb3a486f6d80c1
                           6e1b8974016aa4b7fa43042a9b3825b1
   x: f323ca74d344749096fd35d0adf20806e521460637176e84d977e9933c49d76f
      cfc6e62585940927468ff53d864a7a50
   y: 5b7c709acb175a5afb82860deabca8d0b341facdff0ac0f1a425799aa905d750
      7e1ea9c573581a81467437419466e472
   X: 0211e3334f117b76635dd802d4022f601680a1fd066a56606b7f246493a10351
      7797b81789b225bd5bb1d9ae1da2962250
   Y: 0383dfa413496e5e7599fc8c6430f8d6910d37cf326d81421bc92c0939b555c4
      ca2ef6a993f6d3db8cb7407655ef60866e
   T: 02a1524603ef14f184696f854229d3397507a66c63f841ba748451056be07879
      ac298912387b1c5cdff6381c264701be57
   S: 020d5adfdb92bc377041cf5837412574c5d13e0f4739208a4f0c859a0a302bc6
      a533440a245b9d97a0d34af5016a20053d
   K: 0264aa8c61da9600dfb0beb5e46550d63740e4ef29e73f1a30d543eb43c25499
      037ad16538586552761b093cf0e37c703a
   SPAKESupport: a0093007a0053003020103
   SPAKEChallenge: a1473045a003020103a133043102a1524603ef14f184696f
                   854229d3397507a66c63f841ba748451056be07879ac2989
                   12387b1c5cdff6381c264701be57a20930073005a0030201
                   01
   Transcript hash after challenge: 4d4095d9f94552e15015881a3f2cf458
                                    1be83217cf7ad830d2f051dba3ec8caa
                                    6e354eaa85738d7035317ac557f8c294
   Final transcript hash after pubkey: 5ac0d99ef9e5a73998797fe64f074673
                                       e3952dec4c7d1aacce8b75f64d2b0276
                                       a901cb8539b4e8ed69e4db0ce805b47b
   KDC-REQ-BODY: 3075a00703050000000000a1143012a003020101a10b3009
                 1b077261656275726ea2101b0e415448454e412e4d49542e
                 454455a3233021a003020102a11a30181b066b7262746774
                 1b0e415448454e412e4d49542e454455a511180f31393730
                 303130313030303030305aa703020100a8053003020112
   K'[0]: b917d37c16dd1d8567fbe379f64e1ee3
          6ca3fd127aa4e60f97e4afa3d9e56d91
   K'[1]: 93d40079dab229b9c79366829f4e7e72
          82e6a4b943ac7bac69922d516673f49a
   K'[2]: bfc4f16f12f683e71589f9a888e23287
          5ef293ac9793db6c919567cd7b94bcd4
   K'[3]: 3630e2b5b99938e7506733141e8ec344
          166f6407e5fc2ef107c156e764d1bc20

   aes256-cts-hmac-sha1-96 P-521
   key: 01b897121d933ab44b47eb5494db15e50eb74530dbdae9b634d65020ff5d88c1
   w (PRF+ output): de3a095a2b2386eff3eb15b735398da1caf95bc8425665d8
                    2370aff58b0471f34a57bccddf1ebf0a2965b58a93ee5b45
                    e85d1a5435d1c8c83662999722d542831f9a
   w (reduced multiplier): 003a095a2b2386eff3eb15b735398da1
                           caf95bc8425665d82370aff58b0471f3
                           4cce63791cfed967f0c94c16054b3e17
                           03133681bece1e05219f5426bc944b0f
                           bfb3
   x: 017c38701a14b490b6081dfc83524562be7fbb42e0b20426465e3e37952d30bc
      ab0ed857010255d44936a1515607964a870c7c879b741d878f9f9cdf5a865306
      f3f5
   y: 003e2e2950656fa231e959acdd984d125e7fa59cec98126cbc8f3888447911eb
      cd49428a1c22d5fdb76a19fbeb1d9edfa3da6cf55b158b53031d05d51433ade9
      b2b4
   X: 03003e95272223b210b48cfd908b956a36add04a7ff443511432f94ddd87e064
      1d680ba3b3d532c21fa6046192f6bfae7af81c4b803aa154e12459d1428f8f2f
      56e9f2
   Y: 030064916687960df496557ecab08298bf075429eca268c6dabbae24e258d568
      c62841664dc8ecf545369f573ea84548faa22f118128c0a87e1d47315afabb77
      3bb082
   T: 02017d3de19a3ec53d0174905665ef37947d142535102cd9809c0dfbd0dfe007
      353d54cf406ce2a59950f2bb540df6fbe75f8bbbef811c9ba06cc275adbd9675
      6696ec
   S: 02004d142d87477841f6ba053c8f651f3395ad264b7405ca5911fb9a55abd454
      fef658a5f9ed97d1efac68764e9092fa15b9e0050880d78e95fd03abf5931791
      6822b5
   K: 03007c303f62f09282cc849490805bd4457a6793a832cbeb55df427db6a31e99
      b055d5dc99756d24d47b70ad8b6015b0fb8742a718462ed423b90fa3fe631ac1
      3fa916
   SPAKESupport: a0093007a0053003020104
   SPAKEChallenge: a1593057a003020104a145044302017d3de19a3ec53d0174
                   905665ef37947d142535102cd9809c0dfbd0dfe007353d54
                   cf406ce2a59950f2bb540df6fbe75f8bbbef811c9ba06cc2
                   75adbd96756696eca20930073005a003020101
   Transcript hash after challenge: 554405860f8a80944228f1fa2466411d
                                    cf236162aa385e1289131b39e1fd59f2
                                    5e58b4c281ff059c28dc20f5803b87c6
                                    7571ce64cea01b39a21819d1ef1cdc7f
   Final transcript hash after pubkey: 8d6a89ae4d80cc4e47b6f4e48ea3e579
                                       19cc69598d0d3dc7c8bd49b6f1db1409
                                       ca0312944cd964e213aba98537041102
                                       237cff5b331e5347a0673869b412302e
   KDC-REQ-BODY: 3075a00703050000000000a1143012a003020101a10b3009
                 1b077261656275726ea2101b0e415448454e412e4d49542e
                 454455a3233021a003020102a11a30181b066b7262746774
                 1b0e415448454e412e4d49542e454455a511180f31393730
                 303130313030303030305aa703020100a8053003020112
   K'[0]: 1eb3d10bee8fab483adcd3eb38f3ebf1
          f4feb8db96ecc035f563cf2e1115d276
   K'[1]: 482b92781ce57f49176e4c94153cc622
          fe247a7dbe931d1478315f856f085890
   K'[2]: a2c215126dd3df280aab5a27e1e0fb7e
          594192cbff8d6d8e1b6f1818d9bb8fac
   K'[3]: cc06603de984324013a01f888de6d43b
          410a4da2dea53509f30e433c352fb668

   aes256-cts-hmac-sha1-96 edwards25519, accepted optimistic challenge
   key: 01b897121d933ab44b47eb5494db15e50eb74530dbdae9b634d65020ff5d88c1
   w (PRF+ output): e902341590a1b4bb4d606a1c643cccb3
                    f2108f1b6aa97b381012b9400c9e3f4e
   w (reduced multiplier): 35b35ca126156b5bf4ec8b90e9545060
                           f2108f1b6aa97b381012b9400c9e3f0e
   x: 70937207344cafbc53c8a55070e399c584cbafce00b836980dd4e7e74fad2a64
   y: 785d6801a2490df028903ac6449b105f2ff0db895b252953cdc2076649526103
   X: 13841224ea50438c1d9457159d05f2b7cd9d05daf154888eeed223e79008b47c
   Y: d01fc81d5ce20d6ea0939a6bb3e40ccd049f821baaf95e323a3657309ef75d61
   T: 83523b35f1565006cbfc4f159885467c2fb9bc6fe23d36cb1da43d199f1a3118
   S: 2a8f70f46cee9030700037b77f22cec7970dcc238e3e066d9d726baf183992c6
   K: d3c5e4266aa6d1b2873a97ce8af91c7e4d7a7ac456acced7908d34c561ad8fa6
   SPAKEChallenge: a1363034a003020101a122042083523b35f1565006cbfc4f
                   159885467c2fb9bc6fe23d36cb1da43d199f1a3118a20930
                   073005a003020101
   Transcript hash after challenge: 0332da8ba3095ccd127c51740cb905ba
                                    c76e90725e769570b9d8338e6d62a7f2
   Final transcript hash after pubkey: 26f07f9f8965307434d11ea855461d41
                                       e0cbabcc0a1bab48ecee0c6c1a4292b7
   KDC-REQ-BODY: 3075a00703050000000000a1143012a003020101a10b3009
                 1b077261656275726ea2101b0e415448454e412e4d49542e
                 454455a3233021a003020102a11a30181b066b7262746774
                 1b0e415448454e412e4d49542e454455a511180f31393730
                 303130313030303030305aa703020100a8053003020112
   K'[0]: 4569ec08b5de5c3cc19d941725913ace
          8d74524b521a341dc746acd5c3784d92
   K'[1]: 0d96ce1a4ac0f2e280a0cfc31742b064
          61d83d04ae45433db2d80478dd882a4c
   K'[2]: 58018c19315a1ba5d5bb9813b58029f0
          aec18a6f9ca59e0847de1c60bc25945c
   K'[3]: ed7e9bffd68c54d86fb19cd3c03f317f
          88a71ad9a5e94c28581d93fc4ec72b6a

   aes256-cts-hmac-sha1-96 P-521, rejected edwards25519 challenge
   key: 01b897121d933ab44b47eb5494db15e50eb74530dbdae9b634d65020ff5d88c1
   w (PRF+ output): de3a095a2b2386eff3eb15b735398da1caf95bc8425665d8
                    2370aff58b0471f34a57bccddf1ebf0a2965b58a93ee5b45
                    e85d1a5435d1c8c83662999722d542831f9a
   w (reduced multiplier): 003a095a2b2386eff3eb15b735398da1
                           caf95bc8425665d82370aff58b0471f3
                           4cce63791cfed967f0c94c16054b3e17
                           03133681bece1e05219f5426bc944b0f
                           bfb3
   x: 01687b59051bf40048d7c31d5a973d792fa12284b7a447e7f5938b5885ca0bb2
      c3f0bd30291a55fea08e143e2e04bdd7d19b753c7c99032f06cab0d9c2aa8f83
      7ef7
   y: 01ded675ebf74fe30c9a53710f577e9cf84f09f6048fe245a4600004884cc167
      733f9a9e43108fb83babe8754cd37cbd7025e28bc9ff870f084c7244f536285e
      25b4
   X: 03001bed88af987101ef52db5b8876f6287eb49a72163876c2cf99deb94f4c74
      9bfd118f0f400833cc8daad81971fe40498e6075d8ba0a2acfac35eb9ec8530e
      e0edd5
   Y: 02007bd3bf214200795ea449852976f241c9f50f445f78ff2714fffe42983f25
      cd9c9094ba3f9d7adadd6c251e9dc0991fc8210547e7769336a0ac406878fb94
      be2f1f
   T: 02014cb2e5b592ece5990f0ef30d308c061de1598bc4272b4a6599bed466fd15
      21693642abcf4dbe36ce1a2d13967de45f6c4f8d0fa8e14428bf03fb96ef5f1e
      d3e645
   S: 02016c64995e804416f748fd5fa3aa678cbc7cbb596a4f523132dc8af7ce84e5
      41f484a2c74808c6b21dcf7775baefa6753398425becc7b838b210ac5daa0cb0
      b710e2
   K: 0200997f4848ae2e7a98c23d14ac662030743ab37fccc2a45f1c721114f40bcc
      80fe6ec6aba49868f8aea1aa994d50e81b86d3e4d3c1130c8695b68907c673d9
      e5886a
   Optimistic SPAKEChallenge: a1363034a003020102a122042047ca8c
                              24c3a4a70b6eca228322529dadcfa85c
                              f58faceecf5d5c02907b9e2deba20930
                              073005a003020101
   SPAKESupport: a0093007a0053003020104
   SPAKEChallenge: a1593057a003020104a145044302014cb2e5b592ece5990f
                   0ef30d308c061de1598bc4272b4a6599bed466fd15216936
                   42abcf4dbe36ce1a2d13967de45f6c4f8d0fa8e14428bf03
                   fb96ef5f1ed3e645a20930073005a003020101
   Transcript hash after challenge: cb925b8baeae5e2867ab5b10ae1c941c
                                    4ff4b58a4812c1f7bd1c862ad480a8e1
                                    c6fcd5e88d846a2045e385841c91a75a
                                    d2035f0ff692717608e2a5a90842eff2
   Final transcript hash after pubkey: d0efed5e3e2c39c26034756d92a66fec
                                       3082ad793d0197f3f89ad36026f146a3
                                       996e548aa3fc49e2e82f8cac5d132c50
                                       5aa475b39e7be79cded22c26c41aa777
   KDC-REQ-BODY: 3075a00703050000000000a1143012a003020101a10b3009
                 1b077261656275726ea2101b0e415448454e412e4d49542e
                 454455a3233021a003020102a11a30181b066b7262746774
                 1b0e415448454e412e4d49542e454455a511180f31393730
                 303130313030303030305aa703020100a8053003020112
   K'[0]: 631fcc8596e7f40e59045950d72aa0b7
          bac2810a07b767050e983841cf3a2d4c
   K'[1]: 881464920117074dbc67155a8f3341d1
          121ef65f78ea0380bfa81a134c1c47b1
   K'[2]: 377b72ac3af2caad582d73ae4682fd56
          b531ee56706200dd6c38c42b8219837a
   K'[3]: 35ad8e4d580ed3f0d15ad928329773c0
          81bd19f9a56363f3a5f77c7e66108c26

   There are currently no encryption types with a seed size large enough
   to require multiple hash blocks during key derivation with any of the
   assigned hash functions.  To exercise this possibility, the following
   test vector illustrates what keys would be derived if there were a
   copy of the edwards25519 group with group number -1 and associated
   hash function SHA-1:

   AES256 edwards25519 SHA-1 group number -1
   key: 01b897121d933ab44b47eb5494db15e50eb74530dbdae9b634d65020ff5d88c1
   w (PRF+ output): 26da6b118cee6fa5ea795ed32d61490d
                    82b2f11102312f3f2fc04fb01c93df91
   w (reduced multiplier): d166c7cc9e72ca8c61f6a9185a987251
                           81b2f11102312f3f2fc04fb01c93df01
   x: 606c1b668008ed78fe2eee942e8f08007f3f1dcbef66d37fd69033525bda2030
   y: 10fc4e0bb1a902e58f632a1ea0bceb366360ac985f46996d956a02572bfcf050
   X: 389621509665abad35eaab26eab3a0f593c7b4380562aa5513c1140fd78ce048
   Y: de3ed05986eeac518958b566f9bad065b321402025cd188f3d198dc55c6d6b8d
   T: 2289a4f3c613e6e1df95e94aaa3c127dc062b9fceec3f9b62378dc729d61d0e3
   S: f9a7fa352930dedb422d567700bfcd39ba221e7f9ac3e6b36f2b63b68b88642c
   K: 6f61d6b18323b6c3ddaac7c56712845335384f095d3e116f69feb926a04f1340
   SPAKESupport: a0093007a00530030201ff
   SPAKEChallenge: a1363034a0030201ffa12204202289a4f3c613e6e1df95e9
                   4aaa3c127dc062b9fceec3f9b62378dc729d61d0e3a20930
                   073005a003020101
   Transcript hash after challenge: f5c051eb75290f92142c
                                    bbe80557ec2c85902c94
   Final transcript hash after pubkey: 9e26a3b148400c8f9cb8
                                       545331e4e7dcab399cc0
   KDC-REQ-BODY: 3075a00703050000000000a1143012a003020101a10b3009
                 1b077261656275726ea2101b0e415448454e412e4d49542e
                 454455a3233021a003020102a11a30181b066b7262746774
                 1b0e415448454e412e4d49542e454455a511180f31393730
                 303130313030303030305aa703020100a8053003020112
   K'[0]: 40bceb51bba474fd29ae65950022b704
          17b80d973fa8d8d6cd39833ff89964ad
   K'[1]: c29a7315453dc1cce938fa12a9e5c0db
          2894b2194da14c9cd4f7bc3a6a37223d
   K'[2]: f261984dba3c230fad99d324f871514e
          5aad670e44f00daef3264870b0851c25
   K'[3]: d24b2b46bab7c4d1790017d9116a7eeb
          ca88b0562a5ad8989c826cb7dab715c7

Acknowledgements

   Nico Williams (Cryptonector)

   Taylor Yu (MIT)

Authors' Addresses

   Nathaniel McCallum
   Red Hat, Inc.
   Email: nathaniel@mccallum.life


   Simo Sorce
   Red Hat, Inc.
   Email: ssorce@redhat.com


   Robbie Harwood
   Red Hat, Inc.
   Email: rharwood@pm.me


   Greg Hudson
   MIT
   Email: ghudson@mit.edu