Internet Research Task Force D. Harkins, Ed. Internet-Draft Aruba Networks Intended status: Informational February 24, 2013 Expires: August 28, 2013 Dragonfly Key Exchange draft-irtf-cfrg-dragonfly-01 Abstract This document specifies a key exchange using discrete logarithm cryptogprahy that is authenticated using a password or passphrase. It is resistant to active attack, passive attack, and off-line dictionary attack. Status of this Memo This Internet-Draft is submitted in full conformance with the provisions of BCP 78 and BCP 79. This document may not be modified, and derivative works of it may not be created, except to format it for publication as an RFC or to translate it into languages other than English. Internet-Drafts are working documents of the Internet Engineering Task Force (IETF). Note that other groups may also distribute working documents as Internet-Drafts. The list of current Internet- Drafts is at http://datatracker.ietf.org/drafts/current/. Internet-Drafts are draft documents valid for a maximum of six months and may be updated, replaced, or obsoleted by other documents at any time. It is inappropriate to use Internet-Drafts as reference material or to cite them other than as "work in progress." This Internet-Draft will expire on August 28, 2013. Copyright Notice Copyright (c) 2013 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. Code Components extracted from this document must include Simplified BSD License text as described in Section 4.e of Harkins Expires August 28, 2013 [Page 1]

Internet-Draft Dragonfly February 2013 the Trust Legal Provisions and are provided without warranty as described in the Simplified BSD License. Table of Contents 1. Introduction . . . . . . . . . . . . . . . . . . . . . . . . . 3 1.1. Requirements Language . . . . . . . . . . . . . . . . . . 3 1.2. Definitions . . . . . . . . . . . . . . . . . . . . . . . 3 1.2.1. Notations . . . . . . . . . . . . . . . . . . . . . . 3 1.2.2. Resistance to Dictionary Attack . . . . . . . . . . . 4 2. Discete Logarithm Cryptography . . . . . . . . . . . . . . . . 4 2.1. Elliptic Curve Cryptography . . . . . . . . . . . . . . . 4 2.2. Finite Field Cryptography . . . . . . . . . . . . . . . . 6 3. The Dragonfly Key Exchange . . . . . . . . . . . . . . . . . . 7 3.1. Assumptions . . . . . . . . . . . . . . . . . . . . . . . 7 3.2. Derviation of the Password Element . . . . . . . . . . . . 8 3.2.1. Hunting and Pecking with ECC Groups . . . . . . . . . 9 3.2.2. Hunting and Pecking with MODP Groups . . . . . . . . . 10 3.3. The Commit Exchange . . . . . . . . . . . . . . . . . . . 11 3.4. The Confirm Exchange . . . . . . . . . . . . . . . . . . . 12 4. Acknowledgements . . . . . . . . . . . . . . . . . . . . . . . 12 5. IANA Considerations . . . . . . . . . . . . . . . . . . . . . 13 6. Security Considerations . . . . . . . . . . . . . . . . . . . 13 7. References . . . . . . . . . . . . . . . . . . . . . . . . . . 14 7.1. Normative References . . . . . . . . . . . . . . . . . . . 14 7.2. Informative References . . . . . . . . . . . . . . . . . . 14 Author's Address . . . . . . . . . . . . . . . . . . . . . . . . . 14 Harkins Expires August 28, 2013 [Page 2]

Internet-Draft Dragonfly February 2013 1. Introduction Passwords and passphrases are the predominant way of doing authentication in the Internet today. Many protocols that use passwords and passphrases for authentication exchange password- derived data as a proof-of-knowledge of the password (for example, [RFC5996], and [RFC5433]). This opens the exchange up to an off-line dictionary attack where the attacker gleans enough knowledge from either an active or passive attack on the protocol to run through a pool of potential passwords and compute verifiers until it is able to match the password-derived data. This protocol employs discrete logarithm cryptography to perform an efficient exchange in a way that performs mutual authentication using a password but is resistant to an off-line dictionary attack. 1.1. Requirements Language 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 [RFC2119]. 1.2. Definitions 1.2.1. Notations The following notations are used in this memo password A shared, secret and potentially low-entropy word, phrase, code or key used as a credential to mutually authenticate the peers. It is not restricted to characters in a human language. a | b denotes concatenation of string "a" with string "b". len(a) indicates the length in bits of the string "a". LSB(a) returns the least-significant bit of the bitstring "a". min(a,b) returns the lexicographical minimum of strings a and b, or zero (0) if a equals b. Harkins Expires August 28, 2013 [Page 3]

Internet-Draft Dragonfly February 2013 max(a,b) returns the lexicographical maximum of strings a and b, or zero (0) if a equals b. The convention for this memo to represent an element in a finite cyclic group is to use an upper-case letter or acronym, while a scalar is indicated with a lower-case letter or acronym. An element that represents a point on an elliptic curve has an implied composite nature-- i.e. it has both an x- and y-coordinate. 1.2.2. Resistance to Dictionary Attack Resistance to dictionary attack means that any advantage an adversary can gain must be directly related to the number of interactions she makes with an honest protocol participant and not through computation. The adversary will not be able to obtain any information about the password except whether a single guess from a protocol run is correct or incorrect. 2. Discete Logarithm Cryptography Dragonfly uses discrete logarithm cryptography to achieve authentication and key agreement (see [SP800-56A]). Each party to the exchange derives ephemeral keys with respect to a particular set of domain parameters (referred to here as a "group"). A group can be based on Finite Field Cryptography (FFC) or Elliptic Curve Cryptography (ECC). Three operations are defined for both types of groups: o "scalar operation"-- takes a scalar and an element in the group to produce another element-- Z = scalar-op(x, Y). o "element operation"-- takes two elements in the group to produce a third-- Z = element-op(X, Y). o "inverse operation"-- takes an element and returns another element such that the element operation on the two produces the identity element of the group-- Y = inverse(X). 2.1. Elliptic Curve Cryptography Domain parameters for the ECC groups used by dragonfly are: o A prime, p, determining a prime field GF(p). The cryptographic group will be a subgroup of the full elliptic curve group that consists of points on an elliptic curve -- elements from GF(p) Harkins Expires August 28, 2013 [Page 4]

Internet-Draft Dragonfly February 2013 that satisfy the curve's equation -- together with the "point at infinity" that serves as the identity element. The group operation for ECC groups is addition of points on the elliptic curve. o Elements a and b from GF(p) that define the curve's equation. The point (x, y) in GF(p) x GF(p) is on the elliptic curve if and only if (y^2 - x^3 - a*x - b) mod p equals zero (0). o A point, G, on the elliptic curve, which serves as a generator for the ECC group. G is chosen such that its order, with respect to elliptic curve addition, is a sufficiently large prime. o A prime, q, which is the order of G, and thus is also the size of the cryptographic subgroup that is generated by G. An (x,y) pair is a valid ECC element if: 1) the x- and y-coordinates are both greater than zero (0) and less than the prime defining the underlying field; and, 2) the x- and y-coordinates satisfy the equation for the curve and produce a valid point on the curve that is not the point at infinity. If either one of those conditions do not hold the (x,y) pair is not a valid element. The scalar operation is multiplication of a point on the curve by itself a number of times. The point Y is multiplied x-times to produce another point Z: Z = scalar-op(x, Y) = x*Y The element operation is addition of two points on the curve. Points X and Y are summed to produce another point Z: Z = element-op(X, Y) = X + Y The inverse function is defined such that the sum of an element and its inverse is "0", the point-at-infinity of an elliptic curve group: R + inverse(R) = "0" Elliptic curve groups require a mapping function, q = F(Q), to convert a group element to an integer. The mapping function used in this memo returns the x-coordinate of the point it is passed. scalar-op(x, Y) can be viewed as x iterations of element-op() by defining: Y = scalar-op(1, Y) Harkins Expires August 28, 2013 [Page 5]

Internet-Draft Dragonfly February 2013 Y = scalar-op(x, Y) = element-op(Y, scalar-op(x-1, Y)), for x > 1 A definition of how to add two points on an elliptic curve (i.e. element-op(X, Y)) can be found in [RFC6090]. Note: There is another elliptic curve domain parameter, a co-factor, h, that is defined by the requirement that the size of the full elliptic curve group (including "0") be the product of h and q. Elliptic curve groups used with dragonfly authentication MUST have a co-factor of one (1). 2.2. Finite Field Cryptography Domain parameters for the FFC groups used in dragonfly are: o A prime, p, determining a prime field GF(p), the integers modulo p. The FFC group will be a subgroup of GF(p)*, the multiplicative group of non-zero elements in GF(p). The group operation for FFC groups is multiplication modulo p. o An element, G, in GF(p)* which serves as a generator for the FFC group. G is chosen such that its multiplicative order is a sufficiently large prime divisor of ((p-1)/2). o A prime, q, which is the multiplicative order of G, and thus also the size of the cryptographic subgroup of GF(p)* that is generated by G. A number is a valid element in an FFC group if: 1) it is between one (1) and one (1) less than the the prime, p, exclusive (i.e. 1 < element < p-1); and, 2) if modular exponentiation of the element by the group order, q, equals one (1). If either one of those conditions do not hold the number is not a valid element. The scalar operation is exponentiation of a generator modulo a prime. An element Y is taken to the x-th power modulo the prime returning another element, Z: Z = scalar-op(x, Y) = Y^x mod p The element operation is modular multiplication. Two elements, X and Y, are multiplied modulo the prime returning another element, Z: Z = element-op(X, Y) = (X * Y) mod p The inverse function for a MODP group is defined such that the product of an element and its inverse modulo the group prime equals one (1). In other words, Harkins Expires August 28, 2013 [Page 6]

Internet-Draft Dragonfly February 2013 (R * inverse(R)) mod p = 1 3. The Dragonfly Key Exchange There are two parties to the Dragonfly exchange named, for convenience and by convention, Alice and Bob. The two parties have a shared password that was established in an out-of-band mechanism and they both agree to use a particular domain parameter set (either ECC or FFC). Prior to beginning the Dragonfly exchange, the two peers MUST derive a secret element in the chosen domain parameter set. Two "hunting- and-pecking" techniques to determine a secret element, one for ECC and one for FFC, are described in Section 3.2 but any secure, determinstic method that is agreed up on can be used. For instance, the technique described in [hash2ec] can be used for ECC groups. The Dragonfly exchange consists of two message exchanges, a "Commit Exchange" in which both sides commit to a single guess of the password, and a "Confirm Exchange" in which both sides confirm knowledge of the password. A side effect of running the Dragonfly exchange is an authenticated, shared, and secret key whose cryptographic strength is set by the agreed-upon group. Dragonfly uses a random function, H(), a mapping function, F(), and a key derivation function, KDF(). 3.1. Assumptions The Dragonfly protocol achieves security under some basic assumptions: 1. Function H is a "random oracle" (see [RANDOR]) that maps a binary string of indeterminate length onto a fixed binary string that is x bits in length. H: {0,1}^* --> {0,1}^x Given knowledge of the input to H, an adversary is unable to distinguish the output of H from a random data source. 2. Function F is an injective mapping function that takes an element in a group and returns an integer. For ECC groups function F() returns the x-coordinate of the element (which is a point on the elliptic curve), for FFC groups function F() is the identity function (since all elements in an FFC group are already integers less than the prime): Harkins Expires August 28, 2013 [Page 7]

Internet-Draft Dragonfly February 2013 ECC: x = F(P), where P=(x,y) FFC: x = F(x) 3. Function KDF is a key derivation function (see, for instance, [SP800-108]) that takes a key to stretch, k, a label to bind to the key, label, and an indication of the desired output, n: stretch = KDF-n(k, label) so that len(stretch) equals n. 4. The discrete logarithm problem for the chosen group is hard. That is, given g, p, and y = g^x mod p, it is computationally infeasible to determine x. Similarly, for an ECC group given the curve definition, a generator G, and Y = x * G, it is computationally infeasible to determine x. 5. There exists a pool of passwords from which the password shared by the two peers is drawn. This pool can consist of words from a dictionary, for example. Each password in this pool has an equal probability of being the shared password. All potential attackers have access to this pool of passwords. 6. The peers have to ability to produce quality random numbers. 3.2. Derviation of the Password Element Prior to beginning the exchange of information, the peers MUST derive a secret element, called the Password Element (PE), in the group defined by the chosen domain parameter set. From the point-of-view of an attacker who does not know the password, PE will be a random element in the negotiated group. Two examples are described here for completeness but any method of deterministically mapping a secret string into an element in a selected group can be used, for instance the technique in [hash2ec] for ECC groups. If a different technique than the ones described here is used, the secret string SHOULD include the identities of the peers. To fix PE, both peers MUST have a common view of the password. If either side wants to store a salted version of the password it will be necessary to convey the salt to the other side prior to commensing the exchange. The determinstic process to select the PE begins with choosing a secret seed and then performing a group-specific hunting-and-pecking technique-- one for FFC groups and another for ECC groups. Harkins Expires August 28, 2013 [Page 8]

Internet-Draft Dragonfly February 2013 To thwart side channel attacks which attempt to determine the number of iterations of the "hunting-and-pecking" loop are used to find PE for a given password, a security parameter, k, is used to ensure that at least k iterations are always performed. The probability that one requires more than "n" iterations of the "hunting-and-pecking" loop to find an ECC PE is roughly (q/2p)^n and to find an FFC PE is roughly (q/p)^n, both of which rapidly approach zero (0) as "n" increases. The security parameter, k, SHOULD be set sufficiently large such that the probability that finding PE would take more than k iterations is sufficiently small (see Section 6). First, an 8-bit counter is set to one (1) and a secret base is computed using the negotiated one-way function with the identities of the two participants, Alice and Bob, the secret password and the counter: base = H((max(Alice,Bob) | min(Alice,Bob) | password | counter) The base is then stretched using the key derivation function, KDF, to the length of the prime from the group's domain parameter set: seed = KDF-n(base, "Dragonfly Hunting and Pecking") where n = len(p). The string bound to the derived seed is for illustrative purposes only. Implementations of the dragonfly key exchange SHOULD use a usage specific label with the KDF. The seed is then passed to the group-specific hunting and pecking technique. Note that if the protocol performing the dragonfly exchange has the ability to exchange random nonces those SHOULD be added to the computation of base to ensure that each run of the protocol produces a different PE. 3.2.1. Hunting and Pecking with ECC Groups The ECC specific hunting and pecking technique entails looping until a valid point on the elliptic curve has been found. If the seed is not a quadratic residue modulo p, then the counter is incremented, a new base and new seed are generated and the hunting and pecking continues. If the seed is a quadratic residue modulo p, then the seed is used as the x-coordinate with the equation of the curve to solve for a y-coordinate. An ambiguity exists since two values for the y-coordinate would be valid and the low-order bit of the base is used to unambiguously determine the correct y-coordinate. The resulting (x,y) pair becomes the Password Element, PE. Harkins Expires August 28, 2013 [Page 9]

Internet-Draft Dragonfly February 2013 Algorithmically, the process looks like this: found = 0 counter = 1 n = len(p) do { base = H(max(Alice,Bob) | min(Alice,Bob) | password | counter) seed = KDF-n(seed, "Dragonfly Hunting And Pecking") if (seed < p) then x = seed if ( (x^3 + ax + b) is a quadratic residue mod p) then y = sqrt(x^3 + ax + b) if (LSB(y) == LSB(base)) then PE = (x,y) else PE = (x,p-y) fi found = 1 fi fi counter = counter + 1 } while ((found == 0) || (counter <= k)) Figure 1: Fixing PE for ECC Groups 3.2.2. Hunting and Pecking with MODP Groups The MODP specific hunting and pecking technique entails finding a random element which, when used as a generator, will create a group with the same order as the group created by the generator from the domain parameter set. The secret generator is found by exponentiating the seed to the value ((p-1)/q), where p is the prime and q is the order from the domain parameter set. If that value is greater than one (1) it becomes PE, otherwise the counter is incremented, a new base and seed are generated, and the hunting and pecking continues. Algorithmically, the process looks like this: Harkins Expires August 28, 2013 [Page 10]

Internet-Draft Dragonfly February 2013 found = 0 counter = 1 n = len(p) do { base = H(max(Alice,Bob) | min(Alice,Bob) | password | counter) seed = KDF-n(base, "Dragonfly Hunting And Pecking") if (seed < p) then PE = seed ^ ((p-1)/q) mod p if (PE > 1) then found = 1 fi fi counter = counter + 1 } while ((found == 0) || (counter <= k)) Figure 2: Fixing PE for MODP Groups 3.3. The Commit Exchange In the Commit Exchange both sides commit to a single guess of the password. The peers generate a scalar and an element, exchange them with each other, and process the other's scalar and element to generate a common and shared secret. First each peer generates two random numbers, private and mask. These two secrets, the Password Element, and the order from the selected domain parameter set are then used to construct the scalar and element: scalar = (private + mask) modulo q Element = inverse(scalar-op(mask, PE)) If the scalar is less than two (2), the private and mask MUST be thrown away and new values generated. Once a valid scalar and Element are generated, the mask is no longer needed and MUST be irretrievably destroyed. The peers exchange their scalar and Element and validate the peer's scalar and Element, deemed peer-scalar and Peer-Element. For the peer-scalar to be valid, it MUST be between 1 and q exclusive. Validation of the Peer-Element depends on the type of cryptosystem-- validation of an (x,y) pair as an ECC element is specified in Section 2.1 and validation of a number as an FFC element is specified in Section 2.2. If either the peer-scalar or Peer-Element fail validation then the exchange MUST be terminated and authentication Harkins Expires August 28, 2013 [Page 11]

Internet-Draft Dragonfly February 2013 fails. If both the peer-scalar and Peer-Element are valid, they are used with the Password Element to derive a shared secret, ss: ss = F(scalar-op(private, element-op(peer-Element, scalar-op(peer-scalar, PE)))) To enforce key separation and crypto hygiene, the shared secret is stretched into two subkeys, a key confirmation key, KCK, and a master key, MK. Each of the subkeys SHOULD be at least the length of the prime used in the selected group. KCK | MK = KDF-n(ss, "Dragonfly Key Derivation") where n = len(p)*2. 3.4. The Confirm Exchange In the Confirm Exchange both sides confirm that they derived the same secret, and therefore, are in possession of the same password. The Commit Exchange consists of an exchange of data that is the output of the random function, H(), the key confirmation key, and the two scalars and two elements exchanged in the Commit Exchange. The order of the scalars and elements are, scalars before elements, and sender's value before recipient's value. So from each peer's perspective, it would generate: confirm = H(KCK | scalar | peer-scalar | Element | Peer-Element) The two peers exchange these confirmations and verify the correctness of the other peer's confirmation that they receive. If the other peer's confirmation is valid, authentication succeeds; if the other peer's confirmation is not valid, authentication fails. If authentication fails, all ephemeral state created as part of the particular run of the dragonfly exchange MUST be irretrievabley destroyed. If authentication does not fail, MK can be exported as an authenticated and secret key that can be used by another protocol, for instance IPsec, to protect other data. 4. Acknowledgements The author would like to thank Kevin Igoe and David McGrew, chairmen of the Crypto Forum Research Group (CFRG) for agreeing to accept this memo as a CFRG work item. Additional thanks go to Scott Fluhrer and Harkins Expires August 28, 2013 [Page 12]

Internet-Draft Dragonfly February 2013 Hideyuki Suzuki for discovering attacks against earlier versions of this key exchange and suggesting fixes to address them. Lily Chen provided helpful discussions on hashing into an elliptic curve. Rich Davis suggested the validation steps used on received elements. Dylan Clarke and Feng Hao discovered a dictionary attack against dragonfly if those validation steps are not performed. Martin Rex, Peter Gutmann, Marsh Ray, Rene Struik and Kevin Igoe discussed the possibility of a side-channel attack against the hunting-and-pecking loop and motivated the addition of the security parameter, k. 5. IANA Considerations This memo includes no request to IANA. 6. Security Considerations The dragonfly exchange requires both participants to have an identical representation of the password. Salting of the password merely generates a new credential-- the salted password-- which must be identically represented on both sides. If an adversary is able to gain access to the database of salted passwords, she would be able to impersonate one side to the other, even if she was unable to determine the underlying, unsalted, password. Resistance to dictionary attack means that an adversary must launch an active attack to make a single guess at the password. If the size of the dictionary from which the password was extracted was "d", and each password in the dictionary has an equal probability of being chosen, then the probability of success after a single guess is 1/d. After "x" guesses, and removal of failed guesses from the pool of possible passwords, the probability becomes 1/(d-x). As "x" grows, so does the probability of success. Therefore, it is possible for an adversary to determine the password through repeated brute-force, active, guessing attacks. Users of the dragonfly key exchange SHOULD ensure that the size of the pool from which the password was drawn, "d", is sufficiently large to make this attack preventable. Implementations of dragonfly SHOULD support countermeasures to deal with this attack-- for instance, by refusing authentication attempts for a certain amount of time, after the number of failed authentication attempts reaches a certain threshold. No such threshold or amount of time is recommended in this memo. It is RECOMMENDED that an implementation set the security parameter, k, to a value of at least forty (40) which will put the probability that more than forty iterations are needed in the order of one in one trillion (1:1,000,000,000,000). Harkins Expires August 28, 2013 [Page 13]

Internet-Draft Dragonfly February 2013 7. References 7.1. Normative References [RFC2119] Bradner, S., "Key words for use in RFCs to Indicate Requirement Levels", BCP 14, RFC 2119, March 1997. 7.2. Informative References [RANDOR] Bellare, M. and P. Rogaway, "Random Oracles are Practical: A Paradigm for Designing Efficient Protocols", Proceedings of the 1st ACM Conference on Computer and Communication Security, ACM Press, 1993, <http://www.cs.ucsd.edu/~mihir/papers/ro.pdf>. [RFC5433] Clancy, T. and H. Tschofenig, "Extensible Authentication Protocol - Generalized Pre-Shared Key (EAP-GPSK) Method", RFC 5433, February 2009. [RFC5996] Kaufman, C., Hoffman, P., Nir, Y., and P. Eronen, "Internet Key Exchange Protocol Version 2 (IKEv2)", RFC 5996, September 2010. [RFC6090] McGrew, D., Igoe, K., and M. Salter, "Fundamental Elliptic Curve Cryptography Algorithms", RFC 6090, February 2011. [SP800-108] Chen, L., "Recommendations for Key Derivation Using Pseudorandom Functions", NIST Special Publication 800-108, April 2008. [SP800-56A] Barker, E., Johnson, D., and M. Smid, "Recommendations for Pair-Wise Key Establishment Schemes Using Discrete Logarithm Cryptography", NIST Special Publication 800-56A, March 2007. [hash2ec] Coron, J-B. and T. Icart, "An indifferentiable hash function into elliptic curves", Cryptology ePrint Archive Report 2009/340, 2009. Harkins Expires August 28, 2013 [Page 14]

Internet-Draft Dragonfly February 2013 Author's Address Dan Harkins (editor) Aruba Networks 1322 Crossman Avenue Sunnyvale, CA 94089-1113 United States of America Email: dharkins@arubanetworks.com Harkins Expires August 28, 2013 [Page 15]