Password-Authenticated Key (PAK) Diffie-Hellman Exchange
Adding password-based mutual authentication to Diffie-Hellman key exchange.
Brusilovsky, A., Faynberg, I., Zeltsan, Z., and S. Patel, "Password-Authenticated Key (PAK) Diffie-Hellman Exchange", RFC 5683, DOI 10.17487/RFC5683, February 2010, <https://www.rfc-editor.org/info/rfc5683>.
@misc{rfc5683,
series = {Request for Comments},
number = 5683,
howpublished = {RFC 5683},
publisher = {RFC Editor},
doi = {10.17487/RFC5683},
url = {https://www.rfc-editor.org/info/rfc5683},
author = {A. Brusilovsky and I. Faynberg and Z. Zeltsan and S. Patel},
title = {{Password-Authenticated Key (PAK) Diffie-Hellman Exchange}},
pagetotal = 10,
year = 2010,
month = feb,
} RFC text
Contents
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
Abstract
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
http://www.rfc-editor.org/info/rfc5683.
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
(http:trustee.ietf.org/license-info) 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
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
used.
- 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
functions.
- 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).
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
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)
Bob
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)
Alice
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)
Bob
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
Functions
[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
0xFFFFFFFF 0xFFFFFFFF
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:
H1(z):
SHA-1(1|1|z) mod 2^128 | SHA-1(1|2|z) mod 2^128 |...|
| SHA-1(1|9|z) mod 2^128
H2(z):
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
2006.
7.2. Informative References
[Plan9] Alcatel-Lucent, "Plan 9 from Bell Labs",
http://netlib.bell-labs.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), 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
2005.
[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
Alcatel-Lucent
Room 9B-226, 1960 Lucent Lane
Naperville, IL 60566-7217 USA
Tel: +1 630 979 5490
EMail: Alec.Brusilovsky@alcatel-lucent.com
Igor Faynberg
Alcatel-Lucent
Room 2D-144, 600 Mountain Avenue
Murray Hill, NJ 07974 USA
Tel: +1 908 582 2626
EMail: igor.faynberg@alcatel-lucent.com
Sarvar Patel
Google, Inc.
76 Ninth Avenue
New York, NY 10011 USA
Tel: +1 212 565 5907
EMail: sarvar@google.com
Zachary Zeltsan
Alcatel-Lucent
Room 2D-150, 600 Mountain Avenue
Murray Hill, NJ 07974 USA
Tel: +1 908 582 2359
EMail: zeltsan@alcatel-lucent.com
AI ELI5
On this page
§1Why Authenticated Key Exchange Matters
The Diffie-Hellman key exchange is a classic way for two parties to agree on a secret number over an open channel, even when others are listening. However, it does not prove who you are talking to. An attacker could pretend to be the other party.
The Password-Authenticated Key (PAK) exchange adds a crucial step: mutual authentication. If both parties share a password (even a simple, short one that a person can remember), PAK lets them prove they both know it while they exchange keys. The password remains secret throughout. The RFC proves PAK is mathematically as secure as plain Diffie-Hellman, and importantly, it resists offline attacks where an attacker guesses passwords.
Normally, two people using Diffie-Hellman to create a secret can't be sure they are talking to each other—an impostor could cut in. PAK adds a password handshake first, so both sides can check they are really talking to the right person.
§2How PAK Works
PAK requires both parties to know a shared password. The protocol unfolds in three messages:
Round 1: Alice picks a secret random number, computes a Diffie-Hellman value from it, masks that value with a hash derived from the password, and sends it to Bob along with her identity.
Round 2: Bob receives Alice's message, unmasks it using the same password hash (which only he can compute if he knows the correct password), and sends back his own Diffie-Hellman value—also masked—along with a proof that he computed the shared secret correctly.
Round 3: Alice receives Bob's message, unmasks it, verifies Bob's proof, and sends her own proof back. Both parties now have the same session key and have authenticated each other.
If any verification fails, the protocol stops immediately. The shared session key is computed using a final hash function applied to the Diffie-Hellman result, Alice's identity, Bob's identity, the password, and both Diffie-Hellman values—ensuring that only the party who knows the correct password can compute it.
Alice sends Bob a masked version of her secret. Bob unmasks it (proving he knows the password), sends back his own masked secret with proof he knows the answer. Alice checks the proof, sends her own proof back. Now both have a secret key and know they are talking to the right person.
- A, H1(A|B|PW) * g^RaAlice's identity and masked exponent.
- H2(A|B|PW) * g^Rb, S1Bob's masked exponent and authentication proof.
- S2Alice's authentication proof; both now have key K.
§3Setting Up PAK Parameters
PAK requires careful setup. Both parties must agree on a large prime number p and a base number g beforehand. The RFC recommends a 1024-bit prime, with g set to 13. These fixed values protect against attacks where an adversary sends fake parameters to leak information about the password.
The password hashing functions use SHA-1 (or SHA-256). The RFC specifies exactly how to instantiate them: H1 and H2 each output 1152 bits (made from nine concatenated SHA-1 calls), while H3, H4, and H5 each output 128 bits. When short exponents are used, they must be at least 384 bits to maintain security.
Both people need to agree on a big number and a base number before they start. They also need to agree on how to scramble the password using a hash function. The RFC tells you exactly what numbers to use and how many times to run the hash.
§4Security and Practical Concerns
PAK is resistant to several attack classes. An eavesdropper cannot extract the password from the exchange, even over many sessions, because each session uses fresh random numbers (providing forward secrecy). The protocol is proven secure against offline dictionary attacks—an attacker cannot guess passwords and verify them without participating in a live exchange with the holder of the password.
However, several precautions are essential. The password must be shared between exactly two parties; if three parties share the same password, any one of them can impersonate another without the third party detecting it. The p and g values must not be modified by an attacker, so protocols using PAK should pre-distribute these values. An implementer should also rate-limit login attempts to prevent brute-force attacks where an attacker repeatedly guesses passwords.
The RFC notes that a 1024-bit prime is current practice, but larger primes (2048 bits or more) would provide better long-term protection. Any change to p requires careful coordination because g must change too.
An eavesdropper cannot steal the password by listening, and cannot guess it by trying many passwords offline. But both people must be the only ones who know the password, and they need to agree on the big number ahead of time. If someone tries to log in too many times with wrong passwords, you should lock them out.
§5Background and Status
The PAK algorithm was created by Boyko, MacKenzie, and Patel and published at Eurocrypt 2000. The ITU-T standardized it as Recommendation X.1035. The 3GPP2 standards for mobile networks (TIA-683 and related documents) and wireless LANs incorporate PAK for over-the-air provisioning and device activation.
This RFC is published as Informational, meaning it documents an existing algorithm rather than proposing a new Internet Standard. The RFC provides guidance for implementers in wireless and cellular systems, particularly in environments where devices need to authenticate using a password provisioned at the factory.
PAK was invented by researchers in 2000 and later adopted by standards bodies for activating phones and wireless devices. This RFC is an informational guide, not an official Internet Standard, but it explains how to implement PAK correctly.