I. Faynberg
Z. Zeltsan
AlcatelLucent
S. Patel
Google, Inc.
February 2010
PasswordAuthenticated Key (PAK) DiffieHellman Exchange
Abstract

This document proposes to add mutual authentication, based on a humanmemorizable 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 DiffieHellman 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 http://www.rfceditor.org/info/rfc5683.
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Table of Contents

1. Introduction ....................................................3 2. Conventions .....................................................3 3. PasswordAuthenticated Key Exchange .............................4 4. Selection of Parameters .........................................5 4.1. General Considerations .....................................5 4.2. OvertheAir Service Provisioning (OTASP) and Wireless Local Area Network (WLAN) DiffieHellman 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 keyexchange protocol.  It is secure against offline dictionary attacks when passwords are used.  It ensures Forward Secrecy.  It has been proven to be as secure as the DiffieHellman solution.
The PAK protocol ([BMP00], [MP05], [X.1035]) has been proven to be as secure as the DiffieHellman ([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 lowentropy 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 nonnegative remainder when a is divided by b.
 Hi(u) denotes an agreedon function (e.g., based on SHA1, SHA256, 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 functions.
 st denotes concatenation of the strings s and t.
 ^ denotes exponentiation.
 Multiplication, division, and exponentiation are performed over (Zp)*; in other words:

 a*b always means a*b (mod p).
 a/b always means a * x (mod p), where x is the multiplicative inverse of b modulo p.
 a^b means a^b (mod p).
3. PasswordAuthenticated Key Exchange

DiffieHellman 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(ABPW) != 0 H2(ABPW) != 0
The global DiffieHellman publicly known constants, a prime p and a generator g, are carefully selected so that:
 A safe prime p is large enough to make the computation of discrete logarithms infeasible, and
 Powers of g modulo p cover the entire range of p1 integers from 1 to p1. (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(ABPW)*(g^Ra)} (The above precondition on PW ensures that X != 0)

Bob
receives Q (presumably Q = X), verifies that Q != 0 (if Q = 0, Bob aborts the procedure); divides Q by H1(ABPW) to get Xab, the recovered value of g^Ra B > A: {Y = H2(ABPW)*(g^Rb), S1 = H3(ABPWXabg^Rb(Xab)^Rb)} (The above precondition on PW ensures that Y != 0)

Alice
verifies that Y != 0; divides Y by H2(ABPW) to get Yba, the recovered value of g^Rb, and computes S1' = H3(ABPWg^RaYba(Yba)^Ra); authenticates Bob by checking whether S1' = S1; if authenticated, then sets key K = H5(ABPWg^RaYba(Yba)^Ra) A > B: S2 = H4(ABPWg^RaYba(Yba)^Ra)

Bob
Computes S2' = H4(ABPWXabg^Rb(Xab)^Rb) and authenticates Alice by checking whether S2' = S2; if authenticated, then sets K = H5(ABPWXabg^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. OvertheAir Service Provisioning (OTASP) and Wireless Local Area Network (WLAN) DiffieHellman Parameters and Key Expansion Functions

[OTASP], [TIA683], and [WLAN] preset 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 DiffieHellman 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 1024bit 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 0xFFFFFFFF 0xFFFFFFFF
In addition, if short exponents [MP05] are used for DiffieHellman 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 SHA1 hashing function [FIPS180] below to instantiate the random function as done in [WLAN]; however, SHA256 can also be used:
H1(z): SHA1(11z) mod 2^128  SHA1(12z) mod 2^128 ...  SHA1(19z) mod 2^128 H2(z): SHA1(21z) mod 2^128  SHA1(22z) mod 2^128 ...  SHA1(29z) mod 2^128 H3(z): SHA1(3len(z)zz) mod 2^128 H4(z): SHA1(4len(z)zz) mod 2^128 H5(z): SHA1(5len(z)zz) mod 2^128
In order to create 1152 output bits for H1 and H2, nine calls to SHA1 are made and the 128 least significant bits of each output are used. The input payload of each call to SHA1 consists of:
a) 32 bits of function type, which for H1 is set to 1 and for H2 is set to 2; b) a 32bit counter value, which is incremented from 1 to 9 for each call to SHA1; c) the argument z [for (ABPW)].
The functions H3, H4, and H5 require only one call to the SHA1 hashing function and their respective payloads consist of:
a) 32 bits of function type (e.g., 3 for H3);
b) a 32bit 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 wellknown 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 DiffieHellman parameters

The parameters p and g must be carefully selected in order not to compromise the shared secret. Only previously agreedupon 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 DiffieHellman exponentiation. Both parties also need to randomly select a new exponent each time the keyagreement protocol is executed. If both parties reuse 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 128bit session keys are used). Historically, the developers, who strived for 128bit security (and thus selected 256bit 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 socalled 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 1024bit 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] ITUT, "Passwordauthenticated key exchange (PAK) protocol", ITUT Recommendation X.1035, 2007. [TIA683] TIA, "OvertheAir Service Provisioning of Mobile Stations in Spread Spectrum Systems", TIA683D, May 2006.
7.2. Informative References

[Plan9] AlcatelLucent, "Plan 9 from Bell Labs", http://netlib.belllabs.com/plan9/. [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), 644654. [FIPS180] NIST Federal Information Processing Standards, Publication FIPS 1803, "Secure Hash Standard", 2008. [MP05] MacKenzie, P. and S. Patel, "Hard Bits of the Discrete Log with Applications to Password Authentication", CTRSA 2005. [OTASP] 3GPP2, "OvertheAir Service Provisioning of Mobile Stations in Spread Spectrum Systems", 3GPP2 C.S0016C v. 1.0 5, October 2004. [RFC2631] Rescorla, E., "DiffieHellman Key Agreement Method", RFC 2631, June 1999. [WLAN] 3GPP2, "Wireless Local Area Network (WLAN) Interworking", 3GPP2 X.S00280, v.1.0, April 2005.
Authors' Addresses

Alec Brusilovsky AlcatelLucent Room 9B226, 1960 Lucent Lane Naperville, IL 605667217 USA Tel: +1 630 979 5490 EMail: Alec.Brusilovsky@alcatellucent.com Igor Faynberg AlcatelLucent Room 2D144, 600 Mountain Avenue Murray Hill, NJ 07974 USA Tel: +1 908 582 2626 EMail: igor.faynberg@alcatellucent.com Sarvar Patel Google, Inc. 76 Ninth Avenue New York, NY 10011 USA Tel: +1 212 565 5907 EMail: sarvar@google.com Zachary Zeltsan AlcatelLucent Room 2D150, 600 Mountain Avenue Murray Hill, NJ 07974 USA Tel: +1 908 582 2359 EMail: zeltsan@alcatellucent.com