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rfc:rfc7664

Internet Research Task Force (IRTF) D. Harkins, Ed. Request for Comments: 7664 Aruba Networks Category: Informational November 2015 ISSN: 2070-1721

                       Dragonfly Key Exchange

Abstract

 This document specifies a key exchange using discrete logarithm
 cryptography that is authenticated using a password or passphrase.
 It is resistant to active attack, passive attack, and offline
 dictionary attack.  This document is a product of the Crypto Forum
 Research Group (CFRG).

Status of This Memo

 This document is not an Internet Standards Track specification; it is
 published for informational purposes.
 This document is a product of the Internet Research Task Force
 (IRTF).  The IRTF publishes the results of Internet-related research
 and development activities.  These results might not be suitable for
 deployment.  This RFC represents the individual opinion(s) of one or
 more members of the Crypto Forum Research Group of the Internet
 Research Task Force (IRTF).  Documents approved for publication by
 the IRSG 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/rfc7664.

Copyright Notice

 Copyright (c) 2015 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.

Harkins Informational [Page 1] RFC 7664 Dragonfly November 2015

Table of Contents

 1.  Introduction  . . . . . . . . . . . . . . . . . . . . . . . .   2
   1.1.  Requirements Language . . . . . . . . . . . . . . . . . .   2
   1.2.  Definitions . . . . . . . . . . . . . . . . . . . . . . .   3
     1.2.1.  Notations . . . . . . . . . . . . . . . . . . . . . .   3
     1.2.2.  Resistance to Dictionary Attack . . . . . . . . . . .   3
 2.  Discrete Logarithm Cryptography . . . . . . . . . . . . . . .   4
   2.1.  Elliptic Curve Cryptography . . . . . . . . . . . . . . .   4
   2.2.  Finite Field Cryptography . . . . . . . . . . . . . . . .   5
 3.  The Dragonfly Key Exchange  . . . . . . . . . . . . . . . . .   6
   3.1.  Assumptions . . . . . . . . . . . . . . . . . . . . . . .   7
   3.2.  Derivation of the Password Element  . . . . . . . . . . .   8
     3.2.1.  Hunting and Pecking with ECC Groups . . . . . . . . .  10
     3.2.2.  Hunting and Pecking with MODP Groups  . . . . . . . .  12
   3.3.  The Commit Exchange . . . . . . . . . . . . . . . . . . .  13
   3.4.  The Confirm Exchange  . . . . . . . . . . . . . . . . . .  14
 4.  Security Considerations . . . . . . . . . . . . . . . . . . .  15
 5.  References  . . . . . . . . . . . . . . . . . . . . . . . . .  16
   5.1.  Normative References  . . . . . . . . . . . . . . . . . .  16
   5.2.  Informative References  . . . . . . . . . . . . . . . . .  16
 Acknowledgements  . . . . . . . . . . . . . . . . . . . . . . . .  18
 Author's Address  . . . . . . . . . . . . . . . . . . . . . . . .  18

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,
 [RFC7296] and [RFC5433]).  This opens the exchange up to an offline
 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 that is provably resistant to an offline dictionary
 attack.  Consensus of the CFRG for this document was rough.

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].

Harkins Informational [Page 2] RFC 7664 Dragonfly November 2015

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 bit string "a" with bit string "b".
 len(a)
    indicates the length in bits of the bit string "a".
 lsb(a)
    returns the least-significant bit of the bit string "a".
 lgr(a,b)
    takes "a" and a prime, "b", and returns the Legendre symbol (a/b).
 min(a,b)
    returns the lexicographical minimum of strings "a" and "b", or
    zero (0) if "a" equals "b".
 max(a,b)
    returns the lexicographical maximum of strings "a" and "b", or
    zero (0) if "a" equals "b".
 The convention for this memo is to represent an element in a finite
 cyclic group with an uppercase letter or acronym, while a scalar is
 indicated with a lowercase 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.

Harkins Informational [Page 3] RFC 7664 Dragonfly November 2015

2. Discrete 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)
    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

Harkins Informational [Page 4] RFC 7664 Dragonfly November 2015

 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 addition of a point on the curve with 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)
    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 cofactor,
 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
 cofactor 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.

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 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 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,
    (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).  In the Dragonfly exchange, both Alice and Bob share an
 identical view of the shared password -- i.e., it is not "augmented",
 where one side holds a password and the other side holds a non-
 invertible verifier.  This allows Dragonfly to be used in traditional
 client-server protocols and also in peer-to-peer applications in
 which there are not fixed roles and either party may initiate the
 exchange (and both parties may implement it simultaneously).
 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

Harkins Informational [Page 6] RFC 7664 Dragonfly November 2015

 and one for FFC, are described in Section 3.2, but any secure,
 deterministic method that is agreed upon 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

 In order to avoid attacks on the Dragonfly protocol, some basic
 assumptions are made:
 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
 2.  Function F is a 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).
        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.

Harkins Informational [Page 7] RFC 7664 Dragonfly November 2015

 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 the ability to produce quality random numbers.

3.2. Derivation 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, the 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 the PE, both peers MUST have a common view of the password.
 If there is any password processing necessary (for example, to
 support internationalization), the processed password is then used as
 the shared credential.  If either side wants to store a hashed
 version of the password (hashing the password with random data called
 a "salt"), it will be necessary to convey the salt to the other side
 prior to commencing the exchange, and the hashed password is then
 used as the shared credential.
 Note: Only one party would be able to maintain a salted password, and
 this would require that the Dragonfly key exchange be used in a
 protocol that has strict roles for client (that always initiates) and
 server (that always responds).  Due to the symmetric nature of
 Dragonfly, salting passwords does not prevent an impersonation attack
 after compromise of a database of salted passwords.
 The deterministic 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 Informational [Page 8] RFC 7664 Dragonfly November 2015

 To thwart side-channel attacks that attempt to determine the number
 of iterations of the hunting-and-pecking loop used to find the PE for
 a given password, a security parameter, k, is used that ensures 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 the PE would take more than k iterations is
 sufficiently small (see Section 4).
 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 identities are passed to the max() and min() functions to provide
 the necessary ordering of the inputs to H() while still allowing for
 a peer-to-peer exchange where both Alice and Bob each view themselves
 as the "initiator" of the exchange.
 The base is then stretched using the technique from Section B.5.1 of
 [FIPS186-4].  The key derivation function, KDF, is used to produce a
 bitstream whose length is equal to the length of the prime from the
 group's domain parameter set plus the constant sixty-four (64) to
 derive a temporary value, and the temporary value is modularly
 reduced to produce a seed:
 n = len(p) + 64
 temp = KDF-n(base, "Dragonfly Hunting and Pecking")
 seed = (temp mod (p - 1)) + 1
 The string bound to the derived temporary value is for illustrative
 purposes only.  Implementations of the Dragonfly key exchange SHOULD
 use a usage-specific label with the KDF.
 Note: The base is stretched to 64 more bits than are needed so that
 the bias from the modular reduction is not so apparent.
 The seed is then passed to the group-specific hunting-and-pecking
 technique.

Harkins Informational [Page 9] RFC 7664 Dragonfly November 2015

 If the protocol performing the Dragonfly exchange has the ability to
 exchange random nonces, those SHOULD be added to the computation of
 the 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.  The seed is used
 as an x-coordinate with the equation of the curve to check whether
 x^3 + a*x + b is a quadratic residue modulo p.  If it is not, then
 the counter is incremented, a new base and new seed are generated,
 and the hunting and pecking continues.  If it is a quadratic residue
 modulo p, then the x-coordinate is assigned the value of seed and the
 current base is stored.  When the hunting-and-pecking loop
 terminates, the x-coordinate is used 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
 stored base is used to unambiguously determine the correct
 y-coordinate.  The resulting (x,y) pair becomes the Password Element,
 PE.

Harkins Informational [Page 10] RFC 7664 Dragonfly November 2015

 Algorithmically, the process looks like this:
      found = 0
      counter = 1
      n = len(p) + 64
      do {
        base = H(max(Alice,Bob) | min(Alice,Bob) | password | counter)
        temp = KDF-n(base, "Dragonfly Hunting And Pecking")
        seed = (temp mod (p - 1)) + 1
        if ( (seed^3 + a*seed + b) is a quadratic residue mod p)
        then
          if ( found == 0 )
          then
            x = seed
            save = base
            found = 1
          fi
        fi
        counter = counter + 1
      } while ((found == 0) || (counter <= k))
      y = sqrt(x^3 + ax + b)
      if ( lsb(y) == lsb(save) )
      then
        PE = (x,y)
      else
        PE = (x,p-y)
      fi
                  Figure 1: Fixing PE for ECC Groups
 Checking whether a value is a quadratic residue modulo a prime can
 leak information about that value in a side-channel attack.
 Therefore, it is RECOMMENDED that the technique used to determine if
 the value is a quadratic residue modulo p blind the value with a
 random number so that the blinded value can take on all numbers
 between 1 and p-1 with equal probability while not changing its
 quadratic residuosity.  Determining the quadratic residue in a
 fashion that resists leakage of information is handled by flipping a
 coin and multiplying the blinded value by either a random quadratic
 residue or a random quadratic nonresidue and checking whether the
 multiplied value is a quadratic residue (qr) or a quadratic
 nonresidue (qnr) modulo p, respectively.  The random residue and
 nonresidue can be calculated prior to hunting and pecking by
 calculating the Legendre symbol on random values until they are
 found:

Harkins Informational [Page 11] RFC 7664 Dragonfly November 2015

    do {
      qr = random() mod p
    } while ( lgr(qr, p) != 1)
    do {
      qnr = random() mod p
    } while ( lgr(qnr, p) != -1)
 Algorithmically, the masking technique to find out whether or not a
 value is a quadratic residue looks like this:
    is_quadratic_residue (val, p) {
        r = (random() mod (p - 1)) + 1
        num = (val * r * r) mod p
        if ( lsb(r) == 1 )
           num = (num * qr) mod p
           if ( lgr(num, p) == 1)
           then
              return TRUE
           fi
        else
           num = (num * qnr) mod p
           if ( lgr(num, p) == -1)
           then
              return TRUE
           fi
        fi
        return FALSE
    }

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 the PE; otherwise, the counter is
 incremented, a new base and seed are generated, and the hunting and
 pecking continues.

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 Algorithmically, the process looks like this:
    found = 0
    counter = 1
    n = len(p) + 64
    do {
      base = H(max(Alice,Bob) | min(Alice,Bob) | password | counter)
      temp = KDF-n(seed, "Dragonfly Hunting And Pecking")
      seed = (temp mod (p - 1)) + 1
      temp = seed ^ ((p-1)/q) mod p
      if (temp > 1)
      then
        if (not found)
          PE = temp
          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 that
 are each greater than one (1) and less than the order from the
 selected domain parameter set:
    1 < private < q
    1 < mask < q
 These two secrets and the Password Element 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.

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 The peers exchange their scalar and Element and check the peer's
 scalar and Element, deemed peer-scalar and Peer-Element.  If the peer
 has sent an identical scalar and Element -- i.e., if scalar equals
 peer-scalar and Element equals Peer-Element -- it is sign of a
 reflection attack, and the exchange MUST be aborted.  If the values
 differ, peer-scalar and Peer-Element must be validated.  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 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 cryptographic 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 | <sender-id>)
 Where <sender-id> is the identity of the sender of the confirm
 message.  This identity SHALL be that contributed by the sender of
 the confirm message in generation of the base in Section 3.2.

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 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 irretrievably
 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. 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 -- that 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.
 Due to the problems with using groups that contain a small subgroup,
 it is RECOMMENDED that implementations of Dragonfly not allow for the
 specification of a group's complete domain parameter to be sent
 in-line, but instead use a common repository and pass an identifier
 to a domain parameter set whose strength has been rigorously proven
 and that does not have small subgroups.  If a group's complete domain
 parameter set is passed in-line, it SHOULD NOT be used with Dragonfly
 unless it directly matches a known good group.

Harkins Informational [Page 15] RFC 7664 Dragonfly November 2015

 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).
 The technique used to obtain the Password Element in Section 3.2.1
 addresses side-channel attacks in a manner deemed controversial by
 some reviewers in the CFRG.  An alternate method, such as the one
 defined in [hash2ec], can be used to alleviate concerns.
 This key exchange protocol has received cryptanalysis in [clarkehao].
 [lanskro] provides a security proof of Dragonfly in the random oracle
 model when both identities are included in the data sent in the
 Confirm Exchange (see Section 3.4).

5. References

5.1. Normative References

 [RFC2119]   Bradner, S., "Key words for use in RFCs to Indicate
             Requirement Levels", BCP 14, RFC 2119,
             DOI 10.17487/RFC2119, March 1997,
             <http://www.rfc-editor.org/info/rfc2119>.

5.2. Informative References

 [clarkehao] Clarke, D. and F. Hao, "Cryptanalysis of the Dragonfly
             Key Exchange Protocol", IET Information Security, Volume
             8, Issue 6, DOI 10.1049/iet-ifs.2013.0081, November 2014.
 [FIPS186-4] NIST, "Digital Signature Standard (DSS)", Federal
             Information Processing Standard (FIPS) 186-4,
             DOI 10.6028/NIST.FIPS.186-4, July 2013.
 [hash2ec]   Brier, E., Coron, J-S., Icart, T., Madore, D., Randriam,
             H., and M. Tibouchi, "Efficient Indifferentiable Hashing
             into Ordinary Elliptic Curves", Cryptology ePrint Archive
             Report 2009/340, 2009.
 [lanskro]   Lancrenon, J. and M. Skrobot, "On the Provable Security
             of the Dragonfly Protocol", Proceedings of 18th
             International Information Security Conference (ISC
             2015), pp 244-261, DOI 10.1007/978-3-319-23318-5_14,
             September 2015.

Harkins Informational [Page 16] RFC 7664 Dragonfly November 2015

 [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,
             DOI 10.1145/168588.168596, 1993.
 [RFC5433]   Clancy, T. and H. Tschofenig, "Extensible Authentication
             Protocol - Generalized Pre-Shared Key (EAP-GPSK) Method",
             RFC 5433, DOI 10.17487/RFC5433, February 2009,
             <http://www.rfc-editor.org/info/rfc5433>.
 [RFC6090]   McGrew, D., Igoe, K., and M. Salter, "Fundamental
             Elliptic Curve Cryptography Algorithms", RFC 6090,
             DOI 10.17487/RFC6090, February 2011,
             <http://www.rfc-editor.org/info/rfc6090>.
 [RFC7296]   Kaufman, C., Hoffman, P., Nir, Y., Eronen, P., and T.
             Kivinen, "Internet Key Exchange Protocol Version 2
             (IKEv2)", STD 79, RFC 7296, DOI 10.17487/RFC7296, October
             2014, <http://www.rfc-editor.org/info/rfc7296>.
 [SP800-108] Chen, L., "Recommendation for Key Derivation Using
             Pseudorandom Functions", NIST Special
             Publication 800-108, October 2009.
 [SP800-56A] Barker, E., Johnson, D., and M. Smid, "Recommendation for
             Pair-Wise Key Establishment Schemes Using Discrete
             Logarithm Cryptography (Revised)", NIST Special
             Publication 800-56A, March 2007.

Harkins Informational [Page 17] RFC 7664 Dragonfly November 2015

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
 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 to
 prevent a small subgroup attack.  Dylan Clarke and Feng Hao
 discovered a dictionary attack against Dragonfly if those checks are
 not made and a group with a small subgroup is used.  And finally, a
 very heartfelt thanks to Jean Lancrenon and Marjan Skrobot for
 developing a proof of the security of Dragonfly.
 The blinding scheme to prevent side-channel attacks when determining
 whether a value is a quadratic residue modulo a prime was suggested
 by Scott Fluhrer.  Kevin Igoe suggested addition of the security
 parameter k to hide the amount of time taken hunting and pecking for
 the password element.

Author's Address

 Dan Harkins (editor)
 Aruba Networks
 1322 Crossman Avenue
 Sunnyvale, CA  94089-1113
 United States
 Email: dharkins@arubanetworks.com

Harkins Informational [Page 18]

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