J-PAKE: Password Authenticated Key Exchange by Juggling

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 is efficient. It 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. 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 currently J-PAKE is being standardized by 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: concatenation of 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 Fq: a finite field of q elements where q is a prime E(Fq): an elliptic curve defined over Fq G: a generator of the subgroup over E(Fq) with prime order n n: the order of G h: the cofactor of the subgroup generated by G, as defined by h = |E(Fq)|/n P x [b]: multiplication of a point P with a scalar b over E(Fq) P.x: the x coordinate of a point P over E(Fq) KDF(a): Key Derivation Function with input a HMAC(MacKey, MacData): HMAC function with MacKey as the key and MacData as the input data

When implemented over a finite field, J-PAKE uses essentially the same group parameters as DSA. Choose two large primes, p and q, such that q | p-1. Let Gq denote a subgroup of Zp* with prime order q, in which the Decisional Diffie-Hellman problem (DDH) is intractable. 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). 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 0.

Round 1: Alice selects x1 uniformly at random from [0, q-1] and x2 from [1, q-1]. Similarly, Bob selects x3 uniformly at random from [0, q-1] and x4 from [1, q-1]. Alice -> Bob: g1 = g^x1 mod p, g2 = g^x2 mod p and knowledge proofs for x1 and x2 Bob -> Alice: g3 = g^x3 mod p, g4 = g^x4 mod p and knowledge proofs for x3 and x4 In this round, the sender must demonstrate the knowledge of the ephemeral private keys. A suitable technique is to use the Schnorr NIZK proof . The reference is an accompanying internet draft submission to IETF and it needs to be updated once it is accepted by IETF. 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 knowledge proofs as specified in and also checks that g4 != 1 mod p. Similarly, Bob verifies the received knowledge proofs and also checks that g2 != 1 mod p. Round 2: Alice -> Bob: A = (g1*g3*g4)^(x2*s) mod p and a knowledge proof for x2*s Bob -> Alice: B = (g1*g2*g3)^(x4*s) mod p and a knowledge proof 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 may want to explicitly check if these inequalities hold, and the cost of doing that is negligible. When the second round finishes, Alice and Bob verify the received knowledge proofs and then compute the 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. (In many cases, the KDF may simply be a cryptographic hash function, e.g., SHA-256.)

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 has 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 uses essentially the same EC parameters as ECDSA. Let E(Fq) be an elliptic curve defined over a finite field Fq where q is a large prime. Let G be a generator for the subgroup over E(Fq) of prime order n. As before, let s denote the shared secret between Alice and Bob. The value of s is required to fall within [1, n-1].

Round 1: Alice selects x1 and x2 uniformly at random from [1, n-1]. Similarly, Bob selects x3 and x4 uniformly at random from [1, n-1]. Alice -> Bob: G1 = G x [x1], G2 = G x [x2] and knowledge proofs for x1 and x2 Bob -> Alice: G3 = G x [x3], G4 = G x [x4] and knowledge proofs for x3 and x4 When this round finishes, Alice and Bob verify the received knowledge proofs as specified in . Round 2: Alice -> Bob: A = (G1 + G3 + G4) x [x2*s] and a knowledge proof for x2*s Bob -> Alice: B = (G1 + G2 + G3) x [x4*s] and a knowledge proof for x4*s When the second round finishes, Alice and Bob verify the received knowledge proofs and then compute the 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. Note that K is a point on E(Fq), consisting of the x and y coordinates. In practice, it is sufficient to use only the x coordinate as the input to KDF to derive the session key. The x coordinate, which is a field element in Fq, can be converted to an octet string, by following the method defined in section 2.3.3 in .

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, knowledge proofs for x1 and x2. Bob -> Alice: g3 = g^x3 mod p, g4 = g^x4 mod p, B = (g1*g2*g3)^(x4*s) mod p, knowledge proofs for x3, x4, and x4*s. Alice -> Bob: A = (g1*g3*g4)^(x2*s) mod p and a knowledge proof 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. For achieving explicit authentication, an additional key confirmation procedure shall be performed. This is to ensure that both parties have actually obtained the same session key. There are several explicit key confirmation 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 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 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 = HMAC(k', "KC_1_U" || Alice || Bob || g1 || g2 || g3 || g4) Bob -> Alice: MacTagBob = HMAC(k', "KC_1_U" || Bob || Alice || g3 || g4 || g1 || g2) In the EC setting it works basically the same. Let G1.x, G2.x, G3.x and G4.x be the x coordinates of G1, G2, G3 and G4 respectively. It is sufficient (and simpler) to include only the x coordinates in the HMAC function. Hence, the key confirmation works as follows. Alice -> Bob: MacTagAlice = HMAC(k', "KC_1_U" || Alice || Bob || G1.x || G2.x || G3.x || G4.x) Bob -> Alice: MacTagBob = HMAC(k', "KC_1_U" || Bob || Alice || G3.x || G4.x || G1.x || G2.x) The second method assumes an additional secure MAC function (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 can be thwarted accordingly. 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 latest IEEE P1363.2 standard draft D26 and the latest published ISO/IEC 11770-4:2006 standard. As a result, the specification of SPEKE in ISO/IEC 11770-4 is being revised to address the identified problems.

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The editor would like to thank Dylan Clarke, Siamak Shahandashti and Robert Cragie for useful comments. This work is supported by EPSRC First Grant (EP/J011541/1) and ERC Starting Grant (No. 306994).