Independent Submission
Request for Comments: 5683
Category: Informational
ISSN: 2070-1721
A. Brusilovsky
I. Faynberg
Z. Zeltsan
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.

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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

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:
  1. a*b always means a*b (mod p).
  1. a/b always means a * x (mod p), where x is the multiplicative inverse of b modulo p.
  1. 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
  1. Powers of g modulo p cover the entire range of p-1 integers from 1 to p-1. (References demonstrate working examples of selections).

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
   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):

   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.

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 at.

  • 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 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 attacks.

   -  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.
   [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.

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