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RFC 5683

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Password-Authenticated Key (PAK) Diffie-Hellman Exchange


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Independent Submission                                    A. Brusilovsky
Request for Comments: 5683                                   I. Faynberg
Category: Informational                                       Z. Zeltsan
ISSN: 2070-1721                                           Alcatel-Lucent
                                                                S. Patel
                                                            Google, Inc.
                                                           February 2010

        Password-Authenticated Key (PAK) Diffie-Hellman Exchange


   This document proposes to add mutual authentication, based on a
   human-memorizable password, to the basic, unauthenticated Diffie-
   Hellman key exchange.  The proposed algorithm is called the Password-
   Authenticated Key (PAK) exchange.  PAK allows two parties to
   authenticate themselves while performing the Diffie-Hellman exchange.

   The protocol is secure against all passive and active attacks.  In
   particular, it does not allow either type of attacker to obtain any
   information that would enable an offline dictionary attack on the
   password.  PAK provides Forward Secrecy.

Status of This Memo

   This document is not an Internet Standards Track specification; it is
   published for informational purposes.

   This is a contribution to the RFC Series, independently of any other
   RFC stream.  The RFC Editor has chosen to publish this document at
   its discretion and makes no statement about its value for
   implementation or deployment.  Documents approved for publication by
   the RFC Editor are not a candidate for any level of Internet
   Standard; see Section 2 of RFC 5741.

   Information about the current status of this document, any errata,
   and how to provide feedback on it may be obtained at

Page 2 
Copyright Notice

   Copyright (c) 2010 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
   ( in effect on the date of
   publication of this document.  Please review these documents
   carefully, as they describe your rights and restrictions with respect
   to this document.

Table of Contents

   1. Introduction ....................................................3
   2. Conventions .....................................................3
   3. Password-Authenticated Key Exchange .............................4
   4. Selection of Parameters .........................................5
      4.1. General Considerations .....................................5
      4.2. Over-the-Air Service Provisioning (OTASP) and Wireless
           Local Area Network (WLAN) Diffie-Hellman Parameters and
           Key Expansion Functions ....................................5
   5. Security Considerations .........................................7
   6. Acknowledgments .................................................8
   7. References ......................................................8
      7.1. Normative References .......................................8
      7.2. Informative References .....................................8

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1.  Introduction

   PAK has the following advantages:

   -  It provides a secure, authenticated key-exchange protocol.
   -  It is secure against offline dictionary attacks when passwords are
   -  It ensures Forward Secrecy.
   -  It has been proven to be as secure as the Diffie-Hellman solution.

   The PAK protocol ([BMP00], [MP05], [X.1035]) has been proven to be as
   secure as the Diffie-Hellman ([RFC2631], [DH76]) in the random oracle
   model [BR93].  That is, PAK retains its security when used with low-
   entropy passwords.  Therefore, it can be seamlessly integrated into
   existing applications, requiring secure authentication based on such
   low-entropy shared secrets.

2.  Conventions

   -  A is an identity of Alice.

   -  B is an identity of Bob.

   -  Ra is a secret random exponent selected by A.

   -  Rb is a secret random exponent selected by B.

   -  Xab denotes a value (X presumably computed by A) as derived by B.

   -  Yba denotes a value (Y presumably computed by B) as derived by A.

   -  A mod b denotes the least non-negative remainder when a is divided
      by b.

   -  Hi(u) denotes an agreed-on function (e.g., based on SHA-1,
      SHA-256, etc.) computed over a string u; the various H() act as
      independent random functions.  H1(u) and H2(u) are the key
      derivation functions.  H3(u), H4(u), and H5(u) are the hash

   -  s|t denotes concatenation of the strings s and t.

   -  ^ denotes exponentiation.

   -  Multiplication, division, and exponentiation are performed over
      (Zp)*; in other words:

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      1) a*b always means a*b (mod p).

      2) a/b always means a * x (mod p), where x is the multiplicative
         inverse of b modulo p.

      3) a^b means a^b (mod p).

3.  Password-Authenticated Key Exchange

   Diffie-Hellman key agreement requires that both the sender and
   recipient of a message create their own secret, random numbers and
   exchange the exponentiation of their respective numbers.

   PAK has two parties, Alice (A) and Bob (B), sharing a secret password
   PW that satisfies the following conditions:

      H1(A|B|PW) != 0
      H2(A|B|PW) != 0

   The global Diffie-Hellman publicly known constants, a prime p and a
   generator g, are carefully selected so that:

   1.  A safe prime p is large enough to make the computation of
       discrete logarithms infeasible, and

   2.  Powers of g modulo p cover the entire range of p-1 integers from
       1 to p-1.  (References demonstrate working examples of

   Initially, Alice (A) selects a secret, random exponent Ra and
   computes g^Ra; Bob (B) selects a secret, random exponent Rb and
   computes g^Rb.  For efficiency purposes, short exponents could be
   used for Ra and Rb, provided they have a certain minimum size.  Then:

   A --> B: {A, X = H1(A|B|PW)*(g^Ra)}
            (The above precondition on PW ensures that X != 0)

        receives Q (presumably Q = X), verifies that Q != 0
          (if Q = 0, Bob aborts the procedure);
        divides Q by H1(A|B|PW) to get Xab, the recovered value of g^Ra

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   B --> A: {Y = H2(A|B|PW)*(g^Rb), S1 = H3(A|B|PW|Xab|g^Rb|(Xab)^Rb)}
            (The above precondition on PW ensures that Y != 0)

        verifies that Y != 0;
        divides Y by H2(A|B|PW) to get Yba, the recovered value of g^Rb,
        and computes S1' = H3(A|B|PW|g^Ra|Yba|(Yba)^Ra);
        authenticates Bob by checking whether S1' = S1;
        if authenticated, then sets key K = H5(A|B|PW|g^Ra|Yba|(Yba)^Ra)

   A --> B:  S2 = H4(A|B|PW|g^Ra|Yba|(Yba)^Ra)

        Computes S2' = H4(A|B|PW|Xab|g^Rb|(Xab)^Rb) and
        authenticates Alice by checking whether S2' = S2;
        if authenticated, then sets K = H5(A|B|PW|Xab|g^Rb|(Xab)^Rb)

   If any of the above verifications fails, the protocol halts;
   otherwise, both parties have authenticated each other and established
   the key.

4.  Selection of Parameters

   This section provides guidance on selection of the PAK parameters.
   First, it addresses general considerations, then it reports on
   specific implementations.

4.1.  General Considerations

   In general implementations, the parameters must be selected to meet
   algorithm requirements of [BMP00].

4.2.  Over-the-Air Service Provisioning (OTASP) and Wireless Local Area
      Network (WLAN) Diffie-Hellman Parameters and Key Expansion

   [OTASP], [TIA683], and [WLAN] pre-set public parameters p and g to
   their "published" values.  This is necessary to protect against an
   attacker sending bogus p and g values, tricking the legitimate user
   to engage in improper Diffie-Hellman exponentiation and leaking some
   information about the password.

   According to [OTASP], [TIA683], and [WLAN], g shall be set to
   00001101, and p to the following 1024-bit prime number (most
   significant bit first):

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   0xFFFFFFFF  0xFFFFFFFF  0xC90FDAA2  0x2168C234  0xC4C6628B
   0x80DC1CD1  0x29024E08  0x8A67CC74  0x020BBEA6  0x3B139B22
   0x514A0879  0x8E3404DD  0xEF9519B3  0xCD3A431B  0x302B0A6D
   0xF25F1437  0x4FE1356D  0x6D51C245  0xE485B576  0x625E7EC6
   0xF44C42E9  0xA637ED6B  0x0BFF5CB6  0xF406B7ED  0xEE386BFB
   0x5A899FA5  0xAE9F2411  0x7C4B1FE6  0x49286651  0xECE65381

   In addition, if short exponents [MP05] are used for Diffie-Hellman
   parameters Ra and Rb, then they should have a minimum size of 384
   bits.  The independent, random functions H1 and H2 should each output
   1152 bits, assuming prime p is 1024 bits long and session keys K are
   128 bits long.  H3, H4, and H5 each output 128 bits.  More
   information on instantiating random functions using hash functions
   can be found in [BR93].  We use the FIPS 180 SHA-1 hashing function
   [FIPS180] below to instantiate the random function as done in [WLAN];
   however, SHA-256 can also be used:

   SHA-1(1|1|z) mod 2^128 | SHA-1(1|2|z) mod 2^128 |...|
   | SHA-1(1|9|z) mod 2^128

   SHA-1(2|1|z) mod 2^128 | SHA-1(2|2|z) mod 2^128 |...|
   | SHA-1(2|9|z) mod 2^128

   H3(z): SHA-1(3|len(z)|z|z) mod 2^128
   H4(z): SHA-1(4|len(z)|z|z) mod 2^128
   H5(z): SHA-1(5|len(z)|z|z) mod 2^128

   In order to create 1152 output bits for H1 and H2, nine calls to
   SHA-1 are made and the 128 least significant bits of each output are
   used.  The input payload of each call to SHA-1 consists of:

   a) 32 bits of function type, which for H1 is set to 1 and for H2 is
      set to 2;
   b) a 32-bit counter value, which is incremented from 1 to 9 for each
      call to SHA-1;
   c) the argument z [for (A|B|PW)].

   The functions H3, H4, and H5 require only one call to the SHA-1
   hashing function and their respective payloads consist of:

   a) 32 bits of function type (e.g., 3 for H3);
   b) a 32-bit value for the bit length of the argument z;
   c) the actual argument repeated twice.

   Finally, the 128 least significant bits of the output are used.

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5.  Security Considerations

   Security considerations are as follows:

   -  Identifiers

      Any protocol that uses PAK must specify a method for producing a
      single representation of identity strings.

   -  Shared secret

      PAK involves the use of a shared secret.  Protection of the shared
      values and managing (limiting) their exposure over time is
      essential and can be achieved using well-known security policies
      and measures.  If a single secret is shared among more than two
      entities (e.g., Alice, Bob, and Mallory), then Mallory can
      represent himself as Alice to Bob without Bob being any the wiser.

   -  Selection of Diffie-Hellman parameters

      The parameters p and g must be carefully selected in order not to
      compromise the shared secret.  Only previously agreed-upon values
      for parameters p and g should be used in the PAK protocol.  This
      is necessary to protect against an attacker sending bogus p and g
      values and thus tricking the other communicating party in an
      improper Diffie-Hellman exponentiation.  Both parties also need to
      randomly select a new exponent each time the key-agreement
      protocol is executed.  If both parties re-use the same values,
      then Forward Secrecy property is lost.

      In addition, if short exponents Ra and Rb are used, then they
      should have a minimum size of 384 bits (assuming that 128-bit
      session keys are used).  Historically, the developers, who strived
      for 128-bit security (and thus selected 256-bit exponents), added
      128 bits to the exponents to ensure the security reduction proofs.
      This should explain how an "odd" length of 384 has been arrived

   -  Protection against attacks

      a) There is a potential attack, the so-called discrete logarithm
         attack on the multiplicative group of congruencies modulo p, in
         which an adversary can construct a table of discrete logarithms
         to be used as a "dictionary".  A sufficiently large prime, p,
         must be selected to protect against such an attack.  A proper
         1024-bit value for p and an appropriate value for g are
         published in [WLAN] and [TIA683].  For the moment, this is what
         has been implemented; however, a larger prime (i.e., one that

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         is 2048 bits long, or even larger) will definitely provide
         better protection.  It is important to note that once this is
         done, the generator must be changed too, so this task must be
         approached with extreme care.

      b) An online password attack can be launched by an attacker by
         repeatedly guessing the password and attempting to
         authenticate.  The implementers of PAK should consider
         employing mechanisms (such as lockouts) for preventing such

   -  Recommendations on H() functions

      The independent, random functions H1 and H2 should output 1152
      bits each, assuming prime p is 1024 bits long and session keys K
      are 128 bits long.  The random functions H3, H4, and H5 should
      output 128 bits.

   An example of secure implementation of PAK is provided in [Plan9].

6.  Acknowledgments

   The authors are grateful for the thoughtful comments received from
   Shehryar Qutub, Ray Perlner, and Yaron Sheffer.  Special thanks go to
   Alfred Hoenes, Tim Polk, and Jim Schaad for their careful reviews and
   invaluable help in preparing the final version of this document.

7.  References

7.1.  Normative References

   [X.1035]    ITU-T, "Password-authenticated key exchange (PAK)
               protocol", ITU-T Recommendation X.1035, 2007.

   [TIA683]    TIA, "Over-the-Air Service Provisioning of Mobile
               Stations in Spread Spectrum Systems", TIA-683-D, May

7.2.  Informative References

   [Plan9]     Alcatel-Lucent, "Plan 9 from Bell Labs",

   [BMP00]     Boyko, V., MacKenzie, P., and S. Patel, "Provably secure
               password authentication and key exchange using Diffie-
               Hellman", Proceedings of Eurocrypt 2000.

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   [BR93]      Bellare, M. and P. Rogaway, "Random Oracles are
               Practical: A Paradigm for Designing Efficient Protocols",
               Proceedings of the 5th Annual ACM Conference on Computer
               and Communications Security, 1998.

   [DH76]      Diffie, W. and M.E. Hellman, "New directions in
               cryptography", IEEE Transactions on Information Theory 22
               (1976), 644-654.

   [FIPS180]   NIST Federal Information Processing Standards,
               Publication FIPS 180-3, "Secure Hash Standard", 2008.

   [MP05]      MacKenzie, P. and S. Patel, "Hard Bits of the Discrete
               Log with Applications to Password Authentication", CT-RSA

   [OTASP]     3GPP2, "Over-the-Air Service Provisioning of Mobile
               Stations in Spread Spectrum Systems", 3GPP2 C.S0016-C v.
               1.0 5, October 2004.

   [RFC2631]   Rescorla, E., "Diffie-Hellman Key Agreement Method", RFC
               2631, June 1999.

   [WLAN]      3GPP2, "Wireless Local Area Network (WLAN) Interworking",
               3GPP2 X.S0028-0, v.1.0, April 2005.

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Authors' Addresses

   Alec Brusilovsky
   Room 9B-226, 1960 Lucent Lane
   Naperville, IL 60566-7217  USA
   Tel: +1 630 979 5490

   Igor Faynberg
   Room 2D-144, 600 Mountain Avenue
   Murray Hill, NJ 07974  USA
   Tel: +1 908 582 2626

   Sarvar Patel
   Google, Inc.
   76 Ninth Avenue
   New York, NY 10011  USA
   Tel: +1 212 565 5907

   Zachary Zeltsan
   Room 2D-150, 600 Mountain Avenue
   Murray Hill, NJ 07974  USA
   Tel: +1 908 582 2359