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J-PAKE: Password Authenticated Key Exchange by Juggling
Newcastle University (UK)
Claremont Tower, School of Computing Science, Newcastle University
Newcastle Upon Tyne
United Kingdom
+44 (0)191-208-6384
feng.hao@ncl.ac.uk
Security
password authenticated key exchange
This document specifies a Password Authenticated Key
Exchange by Juggling (J-PAKE) protocol. This protocol allows
the establishment of a secure end-to-end communication channel
between two remote parties over an insecure network solely
based on a shared password, without requiring a Public Key
Infrastructure (PKI) or any trusted third party.
Password-Authenticated Key Exchange (PAKE) is a technique
that aims to establish secure communication between two
remote parties solely based on their shared password,
without relying on a Public Key Infrastructure or any
trusted third party . The first
PAKE protocol, called EKE, was proposed by Steven Bellovin
and Michael Merrit in 1992 . Other
well-known PAKE protocols include SPEKE (by David Jablon
in 1996) and SRP (by Tom Wu in
1998) . SRP has been revised
several times to address reported security and efficiency
issues. In particular, the version 6 of SRP, commonly
known as SRP-6, is specified in .
This document specifies a PAKE protocol called Password
Authenticated Key Exchange by Juggling (J-PAKE), which was
designed by Feng Hao and Peter Ryan in 2008
.
There are a few factors that may be considered in favor of
J-PAKE. First, J-PAKE has security proofs, while
equivalent proofs are lacking in EKE, SPEKE and
SRP-6. Second, J-PAKE follows a
completely different design approach from all other PAKE
protocols, and is built upon a well-established Zero
Knowledge Proof (ZKP) primitive: Schnorr NIZK proof
. Third, J-PAKE adopts novel
engineering techniques to optimize the use of ZKP so that overall the protocol is
sufficiently efficient for practical use. Fourth, J-PAKE
is designed to work generically in both the finite field
and elliptic curve settings (i.e., DSA and ECDSA-like
groups respectively). Unlike SPEKE, it does not require
any extra primitive to hash passwords onto a designated
elliptic curve. Unlike SPAKE2 and SESPAKE , it does
not require a trusted setup (i.e., the so-called common reference model) to
define a pair of generators whose discrete logarithm must be unknown.
Finally, J-PAKE has been used in
real-world applications at a relatively large scale, e.g.,
Firefox sync, Pale moon sync and Google Nest products ;
it has been included into widely distributed open source
libraries such as OpenSSL, Network Security Services (NSS)
and the Bouncy Castle; since 2015, it has been included
into Thread as a standard key agreement mechanism for IoT
(Internet of Things) applications, and also included into
ISO/IEC 11770-4.
The key words "MUST", "MUST NOT", "REQUIRED", "SHALL", "SHALL NOT",
"SHOULD", "SHOULD NOT", "RECOMMENDED", "MAY", and "OPTIONAL" in this document
are to be interpreted as described in RFC 2119 .
The following notations are used in this document:
Alice: the assumed identity of the prover in the protocol
Bob: the assumed identity of the verifier in the protocol
s: a low-entropy secret shared between Alice and Bob
a | b: a divides b
a || b: concatenation of a and b
[a, b]: the interval of integers between and including a and b
H: a secure cryptographic hash function
p: a large prime
q: a large prime divisor of p-1, i.e., q | p-1
Zp*: a multiplicative group of integers modulo p
Gq: a subgroup of Zp* with prime order q
g: a generator of Gq
g^x: g raised to the power of x
a mod b: a modulo b
Fp: a finite field of p elements where p is a prime
E(Fp): an elliptic curve defined over Fp
G: a generator of the subgroup over E(Fp) with prime order n
n: the order of G
h: the cofactor of the subgroup generated by G, which is equal to the order of the elliptic curve divided by n
P x [b]: multiplication of a point P with a scalar b over E(Fp)
KDF(a): Key Derivation Function with input a
MAC(MacKey, MacData): MAC function with MacKey as the key and MacData as the input data

When implemented over a finite field, J-PAKE may use the same group parameters as DSA .
Let p and q be two large primes such that q | p-1. Let Gq denote a subgroup of Zp* with
prime order q. Let g be a generator for Gq. Any non-identity element in Gq can be a generator. The two communicating parties, Alice and Bob, both agree on (p, q, g), which can be hard-wired in the software code. They can also
use the method in NIST FIPS 186-4, Appendix A to generate (p, q, g). Here DSA group parameters are used only as an example. Other multiplicative groups suitable for cryptography can also be used for the implementation, e.g., groups defined in . A group setting that provides 128-bit security or above is recommended. The security proof of J-PAKE depends on the Decisional Diffie-Hellman (DDH) problem being intractable in the considered group.
Let s be a secret value derived from a low-entropy password shared between Alice and Bob.
The value of s is required to fall within the range of [1, q-1]. (Note that s must not be 0 for any non-empty secret.)
This range is defined as a necessary condition in for proving the "on-line dictionary
attack resistance", since s, s+q, s+2q, ..., are all considered equivalent values as far as the protocol specification
is concerned. In a practical implementation, one may obtain s by taking a cryptographic hash of the password and wrapping the result with respect to modulo q. Alternatively, one may simply treat the password as an octet string and convert the string to an integer modulo q by following the method defined in Section 2.3.8 of . In either case, one must ensure s is not equal to 0 modulo q.
Round 1: Alice selects an ephemeral private key x1 uniformly at random from [0, q-1] and another ephemeral private key x2 uniformly at random from [1, q-1]. Similarly, Bob selects an ephemeral private key x3 uniformly at random from [0, q-1] and another ephemeral private key x4 uniformly at random from [1, q-1].
Alice -> Bob: g1 = g^x1 mod p, g2 = g^x2 mod p and ZKPs for x1 and x2
Bob -> Alice: g3 = g^x3 mod p, g4 = g^x4 mod p and ZKPs for x3 and x4

In this round, the sender must send zero knowledge proofs to demonstrate the knowledge of the ephemeral private keys.
A suitable technique is to use the Schnorr NIZK proof . As an example, suppose
one wishes to prove the knowledge of the exponent for X = g^x mod p. The generated Schnorr NIZK proof
will contain: {UserID, V = g^v mod p, r = v - x * c mod q}
where UserID is the unique identifier for the prover, v is a number chosen uniformly at random from [0, q-1]
and c = H(g || V || X || UserID). The "uniqueness" of UserID is defined
from the user's perspective -- for example, if Alice communicates with several parties, she shall
associate a unique identity with each party. Upon receiving a Schnorr NIZK proof, Alice shall check
the prover's UserID is a valid identity and is different from her own identity. During the
key exchange process using J-PAKE, each party shall ensure that the other
party has been consistently using the same identity throughout the protocol execution. Details
about the Schnorr NIZK proof, including the generation and the verification procedures,
can be found in .
When this round finishes, Alice verifies the received ZKPs
as specified in and also checks that g4 != 1 mod p.
Similarly, Bob verifies the received ZKPs and also checks that g2 != 1 mod p.
If any of these checks fails, this session should be aborted.
Round 2:
Alice -> Bob: A = (g1*g3*g4)^(x2*s) mod p and a ZKP for x2*s
Bob -> Alice: B = (g1*g2*g3)^(x4*s) mod p and a ZKP for x4*s

In this round, the Schnorr NIZK proof is computed in the same
way as in the previous round except that the generator is different. For Alice, the
generator used is (g1*g3*g4) instead of g; for Bob, the generator
is (g1*g2*g3) instead of g. Since any non-identity element in Gq
can be used as a generator, Alice and Bob just need to ensure g1*g3*g4 != 1 mod p
and g1*g2*g3 != 1 mod p. With overwhelming probability, these
inequalities are statistically guaranteed even when the user is communicating with an adversary
(i.e., in an active attack). Nonetheless, for absolute guarantee, the receiving party shall
explicitly check if these inequalities hold, and abort the session in case such a check fails.
When the second round finishes, Alice and Bob verify the received ZKPs. If the verification fails, the session is aborted. Otherwise, the two parties compute the common key material as follows:
Alice computes Ka = (B/g4^(x2*s))^x2 mod p
Bob computes Kb = (A/g2^(x4*s))^x4 mod p

Here Ka = Kb = g^((x1+x3)*x2*x4*s) mod p. Let K denote the same key material held by both parties. Using K as input, Alice and Bob then apply a Key Derivation Function (KDF) to derive a common session key k.
If the subsequent secure communication uses a symmetric cipher in an authenticated mode (say AES-GCM), then
one key is sufficient, i.e., k = KDF(K). Otherwise, the session key should comprise an
encryption key (for confidentiality) and a MAC key (for integrity), i.e., k = k_enc || k_mac,
where k_enc = KDF(K || "JPAKE_ENC") and k_mac = KDF(K || "JPAKE_MAC"). The exact choice of the KDF is left to
specific applications to define.
The computational cost is estimated based on counting the number of
modular exponentiations since they are the predominant cost factors.
Note that it takes one exponentiation to generate a Schnorr NIZK proof
and two to verify it . For Alice, she needs to
perform 8 exponentiations in the first round, 4 in the second round, and 2 in the final computation of the
session key. Hence, that is 14 modular exponentiations in total.
Based on the symmetry, the computational cost for Bob is exactly the same.
The J-PAKE protocol works basically the same in the elliptic curve (EC) setting, except
that the underlying multiplicative group over a finite field is replaced by
an additive group over an elliptic curve. Nonetheless, the EC
version of J-PAKE is specified here for completeness.
When implemented over an elliptic curve, J-PAKE may use the same EC parameters as ECDSA . The FIPS 186-4 standard defines three types of curves suitable for ECDSA: pseudo-random curves over prime fields, pseudo-random curves over binary fields and special curves over binary fields called Koblitz curves or anomalous binary curves. All these curves that are suitable for ECDSAA can also be used to implement J-PAKE. However, for the illustration purpose, only curves over prime fields are described in this document. Typically, such curves include NIST P-256, P-384 and P-521. When choosing a curve, a level of 128-bit security or above is recommended. Let E(Fp) be an elliptic curve defined over a finite field Fp where p is a large prime. Let G be a generator for the subgroup over E(Fp) of prime order n. Here the NIST curves are used only as an example. Other secure curves such as Curve25519 are also suitable for the implementation. The security proof of J-PAKE relies on the assumption that the DDH problem is intractable in the considered group.
As before, let s denote the shared secret between Alice and Bob. The value
of s falls within [1, n-1]. In particular, note that s must not be equal to 0 mod n.
Round 1: Alice selects ephemeral private keys x1 and x2 uniformly at random from [1, n-1]. Similarly, Bob selects ephemeral private keys x3 and x4 uniformly at random from [1, n-1].
Alice -> Bob: G1 = G x [x1], G2 = G x [x2] and ZKPs for x1 and x2
Bob -> Alice: G3 = G x [x3], G4 = G x [x4] and ZKPs for x3 and x4

When this round finishes, Alice and Bob verify the received ZKPs
as specified in . As an example, to
prove the knowledge of the discrete logarithm of X = G x [x] with respect to the base point G,
the ZKP contains: {UserID, V = G x [v], r = v - x * c mod n} where UserID is the unique identifier for the prover, v is a number chosen uniformly at random from [1, n-1] and c = H(G || V || X || UserID). The verifier shall check the prover's UserID is a valid identity and is different from its own identity.
If the verification of the ZKP fails, the session is aborted.
Round 2:
Alice -> Bob: A = (G1 + G3 + G4) x [x2*s] and a ZKP for x2*s
Bob -> Alice: B = (G1 + G2 + G3) x [x4*s] and a ZKP for x4*s

When the second round finishes, Alice and Bob verify the received ZKPs. The ZKPs are computed in the same way as in the previous round except that the generator is different. For Alice, the new generator is G1 + G3 + G4; for Bob, it is G1 + G2 + G3. Alice and Bob shall check that these new generators are not points at infinity.
If any of these checks fails, the session is aborted. Otherwise, the two parties compute the common key material as follows:
Alice computes Ka = (B - (G4 x [x2*s])) x [x2]
Bob computes Kb = (A - (G2 x [x4*s])) x [x4]

Here Ka = Kb = G x [(x1+x3)*(x2*x4*s)]. Let K denote the same key material held by both parties. Using K as input, Alice and Bob then apply a Key Derivation Function (KDF) to derive a common
session key k.
In the EC setting, the computational cost of J-PAKE is estimated based on counting the number of
scalar multiplications over the elliptic curve. Note that it takes one multiplication to generate a Schnorr NIZK proof
and one to verify it . For Alice, she has to
perform 6 multiplications in the first round, 3 in the second round, and 2 in the final computation of the
session key. Hence, that is 11 multiplications in total.
Based on the symmetry, the computational cost for Bob is exactly the same.
The two-round J-PAKE protocol is completely symmetric, which significantly simplifies the security analysis.
In practice, one party normally initiates the communication and the other party
responds. In that case, the protocol will be completed in three passes instead of two rounds.
The two-round J-PAKE protocol can be trivially changed to three passes
without losing security. Take the finite field setting as an example and assume Alice initiates the key exchange. The three-pass variant works as follows:
Alice -> Bob: g1 = g^x1 mod p, g2 = g^x2 mod p, ZKPs for x1 and x2.
Bob -> Alice: g3 = g^x3 mod p, g4 = g^x4 mod p, B = (g1*g2*g3)^(x4*s) mod p, ZKPs for x3, x4, and x4*s.
Alice -> Bob: A = (g1*g3*g4)^(x2*s) mod p and a ZKP for x2*s.

Both parties compute the session keys in exactly the same way as before.
The two-round J-PAKE protocol (or the three-pass variant) provides
cryptographic guarantee that only the authenticated party who used
the same password at the other end is able to compute the same
session key. So far the authentication is only implicit.
The key confirmation is also implicit .
The two parties may use the derived key straight-away to start secure communication by
encrypting messages in an authenticated mode. Only the party with the same derived
session key will be able to decrypt and read those messages.
For achieving explicit authentication, an additional key confirmation
procedure should be performed. This provides explicit assurance that
the other party has actually derived the same key.
In this case, the key confirmation is explicit .
In J-PAKE, explicit key confirmation is recommended whenever the network bandwidth allows it.
It has the benefit of providing explicit and immediate confirmation if the two parties have derived
the same key and hence are authenticated to each other. This allows a practical implementation of
J-PAKE to effectively detect online dictionary attacks (if any), and stop them accordingly by setting a threshold for the
consecutively failed connection attempts.
To achieve explicit key confirmation, there are several methods available. They are generically applicable to
all key exchange protocols, not just J-PAKE.
In general, it is recommended to use a different key from the session key for key confirmation, say
using k' = KDF(K || "JPAKE_KC"). The advantage of using a different
key for key confirmation is that the session key remains indistinguishable from random after the key
confirmation process (although this perceived advantage is actually
subtle and only theoretical).
Two explicit key confirmation methods are presented here.
The first method is based on the one used in the SPEKE protocol .
Suppose Alice initiates the key confirmation. Alice sends to Bob H(H(k')), which Bob will verify. If the verification is successful,
Bob sends back to Alice H(k'), which Alice will verify. This key confirmation procedure needs to be completed in two rounds, as shown below.
Alice -> Bob: H(H(k'))
Bob -> Alice: H(k')

The above procedure requires two rounds instead of one, because the second message depends on the first. If both parties attempt to send the first message at the same time without an agreed order, they cannot tell if the message that they receive is a genuine challenge or a replayed message, and consequently may enter a deadlock.
The second method is based on the unilateral key confirmation scheme specified in NIST SP 800-56A
Revision 1 . Alice and Bob send to each other a MAC tag, which they
will verify accordingly. This key confirmation procedure can be completed in one round.
In the finite field setting it works as follows.
Alice -> Bob: MacTagAlice = MAC(k', "KC_1_U" || Alice || Bob || g1 || g2 || g3 || g4)
Bob -> Alice: MacTagBob = MAC(k', "KC_1_U" || Bob || Alice || g3 || g4 || g1 || g2)

In the EC setting, the key confirmation works basically the same.
Alice -> Bob: MacTagAlice = MAC(k', "KC_1_U" || Alice || Bob || G1 || G2 || G3 || G4)
Bob -> Alice: MacTagBob = MAC(k', "KC_1_U" || Bob || Alice || G3 || G4 || G1 || G2)

The second method assumes an additional secure MAC function (e.g., one may use HMAC) and is slightly more complex than the first method. However, it can be completed within one round and it preserves the overall symmetry of the protocol implementation. For this reason, the second method is recommended.
A PAKE protocol is designed to provide two functions in one protocol execution.
The first one is to provide zero-knowledge authentication of a password. It is called "zero
knowledge" because at the end of the protocol, the two communicating parties will learn
nothing more than one bit information: whether the passwords supplied at two ends are equal.
Therefore, a PAKE protocol is naturally resistant against phishing attacks. The second
function is to provide session key establishment if the two passwords are equal. The session
key will be used to protect the confidentiality and integrity of the subsequent communication.
More concretely, a secure PAKE protocol shall satisfy the following security requirements .
Off-line dictionary attack resistance: It does not leak any information
that allows a passive/active attacker to perform off-line exhaustive
search of the password.
Forward secrecy: It produces session keys that remain secure even
when the password is later disclosed.
Known-key security: It prevents a disclosed session key from affecting
the security of other sessions.
On-line dictionary attack resistance: It limits an active attacker
to test only one password per protocol execution.

First, a PAKE protocol must resist off-line dictionary attacks.
A password is inherently weak. Typically, it has only about 20-30 bits entropy.
This level of security is subject to exhaustive search. Therefore,
in the PAKE protocol, the communication must not reveal any data that
allows an attacker to learn the password through off-line exhaustive search.
Second, a PAKE protocol must provide forward secrecy. The key exchange is authenticated
based on a shared password. However, there is no guarantee on the
long-term secrecy of the password. A secure PAKE scheme shall
protect past session keys even when the password is later disclosed. This
property also implies that if an attacker knows the password but only
passively observes the key exchange, he cannot learn the session key.
Third, a PAKE protocol must provide known key security. A session key lasts
throughout the session. An exposed session key must not cause
any global impact on the system, affecting the security of other
sessions.
Finally, a PAKE protocol must resist on-line dictionary attacks. If the attacker
is directly engaging in the key exchange, there is no way to
prevent such an attacker trying a random guess of the password.
However, a secure PAKE scheme should mitigate the effect of the
on-line attack to the minimum. In the best case, the attacker
can only guess exactly one password per impersonation attempt.
Consecutively failed attempts can be easily detected and the
subsequent attempts shall be thwarted accordingly. It is recommended
that the false authentication counter should be handled in such a
way that any error (which causes the session to fail during the key exchange or key confirmation)
would lead to incrementing the false authentication counter.
It has been proven in that J-PAKE satisfies all of the
four requirements based on the assumptions that the Decisional Diffie-Hellman
problem is intractable and the underlying Schnorr NIZK proof
is secure. An independent study that proves security of J-PAKE
in a model with algebraic adversaries and random oracles can be found in .
By comparison, it has been known that EKE has the problem
of leaking partial information about the password to a passive attacker,
hence not satisfying the first requirement . For SPEKE and
SRP-6, an attacker may be able to test more than one password in one on-line dictionary
attack (see and ), hence they do not
satisfy the fourth requirement in the strict theoretical sense. Furthermore,
SPEKE is found vulnerable to an impersonation attack and a key-malleability attack .
These two attacks affect the SPEKE protocol specified in Jablon's original 1996 paper as well in the D26 draft of IEEE P1363.2 and the latest published ISO/IEC 11770-4:2006 standard. As a result, the specification of SPEKE in ISO/IEC 11770-4:2006 has been revised to address the identified problems.
This document has no actions for IANA.
The editor would like to thank Dylan Clarke, Siamak Shahandashti, Robert Cragie, Stanislav Smyshlyaev and Russ Housley for many useful comments. This work is supported by EPSRC First Grant (EP/J011541/1) and ERC Starting Grant (No. 306994).
Standards for Efficient Cryptography. SEC 1: Elliptic Curve Cryptography
Security of the J-PAKE Password-Authenticated Key Exchange Protocol
Encrypted Key Exchange: Password-based Protocols Secure against Dictionary Attacks
Password Authenticated Key Exchange by Juggling
J-PAKE: Authenticated Key Exchange Without PKI
The SPEKE Protocol Revisited
Strong Password-Only Authenticated Key Exchange
Cryptography: Theory and Practice (3rd Edition)
The Secure Remote Password protocol
Schnorr NIZK proof: Non-interactive Zero Knowledge Proof for Discrete Logarithm
Recommendation for Pair-Wise Key Establishment Schemes Using Discrete Logarithm Cryptography (Revised)
Dual-Workfactor Encrypted Key Exchange: Efficiently Preventing Password Chaining and Dictionary Attacks
Analysis of The SPEKE Password-Authenticated Key Exchange Protocol
On Small Subgroup Non-Confinement Attacks
Simple Password-Based Encrypted Key Exchange Protocols
Federal Information Processing Standards Publication 186-4: Specifications for the Digital Signature Standard (DSS)
On the Security of One Password Authenticated Key Exchange Protocol