[curves] SPAKE2 and SPAKE2 Elligator Edition

Michael Scott mike.scott at certivox.com
Thu Feb 19 02:23:39 PST 2015

That's very nice. But there is a lot of hashing going on...

The importance of introducing hashing was motivated in the Abdalla paper.
The authors point out that an attacker who can launch a man-in-the-middle
attack, and who learns the value of the two session keys, can carry out an
off-line dictionary attack and hence determine the password.
But wait a minute. I see how a man in the middle can insert and modify
transmitted values, but he can also learn the value of session keys (as
calculated by Alice and Bob, but not by himself)? How is that to be done
exactly? And in the presence of an attacker who can grab session keys out
of process memory, then there is no security anyway. And if this attacker
can grab session keys, then how does hashing them help - surely the same
attacker with only a little more effort can grab the keys out of memory
before they are hashed?

Maybe I am missing something..


On Thu, Feb 12, 2015 at 9:53 AM, Mike Hamburg <mike at shiftleft.org> wrote:

> Hello curves,
> It's been suggested that I explain Elligator and SPAKE2-EE in more
> detail.  Get ready for a wall of text.
> TL;DR: SPAKE2-EE is probably slightly better overall than SPAKE2. It's
> slightly faster, has better security assumptions, and makes one or two
> cases that can arise in SPAKE2 slightly easier.  It's also more complicated
> to spec and implement, since you need Elligator. SPAKE2-vanilla needs more
> complicated optimizations to achieve its top speed, but at least you don't
> have to spec them.
> Elligator 2:
> Elligator 2 is a map from the base field F to any elliptic curve E (over a
> large characteristic field) with a point of order 2. There's also an
> Elligator 1 but it's more complicated and has more restrictions, so let's
> stick with Elligator 2.  If the curve doesn't have a point of order 2 (eg,
> the NIST curves), then the Shallue-Woestijne-Ulas (SWU) algorithm has very
> similar characteristics, and is only slightly more complicated.
> Any curve with a point of order 2 is isomorphic to one of the form y^2 =
> f(x) = x (x^2 + Ax + B), i.e. Weierstrass form with the point of order 2 at
> (0,0).  From there, the idea is to come up with two values x1 and x2 where
> one of them has a y on the curve and the other does not.  To do this, it
> suffices that:
> x1^2 + Ax1 + B = x2^2 + Ax2 + B, and
> x1 / x2 is not square in F
> The first condition is equivalent to x1 + x2 = -A.  The second condition
> is equivalent to x1 = ur^2 x2, where r is arbitrary and u is a fixed
> nonsquare value, eg -1 if p==3 mod 4.  (If r=0 then actually x1/x2 = 0 will
> be square, but that's good enough because (0,0) is on the curve.)  Solving
> these two equations gives x2 = -A/(1+ur^2), and x1 = ur^2 * that.  Notice
> that if you plug 1/ur into the equation for x1 you get x2 and vice-versa.
> So Elligator is conceptually pretty simple:
> * Map your input value to a field element r.
> * Compute whether f(-A/(1+ur^2)) is square.  If so, compute its square
> root.
> * If not, compute the square root of f(-Aur^2/(1+ur^2)) = ur^2
> f(-A/(1+ur^2)).
> There are two important optimizations here.  The first is that you can
> compute f(-A/(1+ur^2)) projectively, so you don't have to divide.  If it
> ends up as C/D, then you can compute s = sqrt(CD) and then have y = s/D,
> again projectively; or if you use the isqrt trick you can compute the
> actual value of y.  This latter technique is better, because you can easily
> enforce the spec that y is "positive" if you chose x1 and "negative" if you
> chose x2, for your favorite definition of those terms (even, Legendre
> symbol, whatever).
> The other important optimization is to compute both square roots in one
> step.  For example, if p == 3 mod 4, then u=-1 and x^((p+1)/4) =
> sqrt(+-x).  So compute f(...)^((p+1)/4).  If it comes out as sqrt(f(...)),
> good.  If not, it's sqrt(-f(...)) but you want sqrt(-r^2f(...)).  So
> multiply it by r and you're good.  In 5-mod-8 world, sqrt(-1) takes the
> place of -1 here.
> But you didn't want to hash to a Weierstrass curve.  You wanted to hash to
> an Edwards curve.  No problem.  Just hash to the isomorphic Montgomery
> curve, and apply the isomorphism, all in projective coordinates.  Or, in
> cases like SPAKE2, you'll need to multiply by 4 so that you get something
> in the main subgroup.  You can hash to Edwards and multiply by 4, or you
> can hash to the isogenous Montgomery curve instead of the isomorphic one,
> and apply the isogeny.  Your pick.
> All in all, this means you can compute Elligator 2 in projective
> coordinates with maybe one isqrt, a dozen muls, 35 lines of code, a couple
> cond-selects (or multiplies by values you know in advance to be +-1), and
> some careful testing.
> Elligator 2 is implemented in C as one of the many available point ops in:
> http://sourceforge.net/p/ed448goldilocks/code/ci/decaf/tree/src/ec_point.c
> This is an unnecessarily nasty implementation, because it maps to affine
> (using the inverse square root trick) instead of to projective.  Not worth
> it.  The Decaf version is simpler.
> In the case of Decaf, the main spec is in terms of a Jacobi quartic,
> rather than a Weierstrass or Edwards curve, but all the actual math takes
> place in twisted Edwards.  The Jacobi quartic is isomorphic to L_d : y^2 =
> x(x-1)(x-d).  So you can do Elligator to that curve -- remember that
> Elligator 2 doesn't require a Montgomery curve -- and then use the
> 2-isogeny from there to the twisted Edwards curve where it does all its
> math.  Same deal as with the Montgomery/Edwards combination, slightly
> different formulas.
> This is implemented in C as decaf_point_from_hash_nonuniform in
> http://sourceforge.net/p/ed448goldilocks/code/ci/decaf/tree/src/decaf.c
> There's also a function decaf_point_from_hash_uniform which calls
> Elligator twice and adds the results.  This is indifferentiable from a
> random oracle, but twice as slow.  It's not necessary in most protocols,
> because they really only need an argument that looks like "for all you
> know, the hashed point on the curve could secretly have an embedded CDH
> challenge" and that doesn't require uniformity, only almost-uniformity and
> almost-invertibility.
> For an RFC-style spec, you'd probably want to turn one of these C
> functions into a few lines of Python and specify that exact computation.
> OK, now on to PAKE.
> Elligator is 1:1 on [0,(p-1)/2], and it's easy to invert the map for those
> points in E where it has an inverse.  This matters for EKE, though, and
> we're talking about SPAKE2.  Remember that SPAKE2 works like so:
> Let g, M and N be random generators of G, with no known relationship in
> the group.  Let "password" be a suitably stretched salted hash of the
> actual password.  Either the client has to retrieve the salt, or it's equal
> to her username and the server's name.
> Alice -> Bob: A := g^x M^password, ID_Alice
> Bob -> Alice: B := g^y N^password, ID_Bob
> This can be optimized by a double-exp, possibly with fixed bases if you
> precomputed a table for M and one for N.  Also, the server can just store
> M,N^password in the database.
> Both sides can now compute g^xy = (A/M^password)^y = (B/N^password)^x.
> They can implement the division as written, or they can use a double-exp,
> i.e. A^y M^(-password*y).
> Then both sides compute validators and session key vA,vB,sk =
> H(A,B,ID_Alice,ID_Bob,g^xy,password).  The session messages are included
> to prevent tampering.  The password is included as an artifact of the
> security proof, and is probably technically unnecessary.  For augmented
> versions of SPAKE2, a hash of password is also a good enough artifact.
> Anyway, the two sides exchange validators and then start using the session
> key.
> The two DH flows and the two validation flows can each happen
> simultaneously, or more likely Bob's two flows will be merged but Alice's
> will remain separate.
> SPAKE2-EE ("Elligator Edition") is a simple variant of this protocol, in
> which M^password is replaced by h*Elligator(H("M",password)) where h is the
> cofactor, and likewise for N.  It's very important to clear the cofactor
> here, or to use a g which generates the entire group and not just the main
> subgroup; otherwise the cofactor components of the password hash will leak.
> Decaf's Elligator variant clears the cofactor for you; Goldilocks' does not.
> Putting Elligator in clearly doesn't hurt the security of the protocol in
> the ROM, because for all the attacker knows h*Elligator(H("M",password)) is
> actually equal to, or at least morally equivalent to, M^password.  The
> rigorous proof proceeds almost exactly the same as SPAKE2 proof, but with
> an extra invocation of the random oracle model.
> As far as I see it, SPAKE2-EE's advantage is that it's slightly tidier
> than SPAKE2 proper, because it removes the static-DH assumption.  In
> SPAKE2, you have to convince people that you generated M and N with no
> relationship to g, because an adversary knowing the discrete log of M base
> g can mount a dictionary attack against everyone.  (I believe that the same
> is only true for N if the flow ordering allows Alice to authenticate
> first.)  It shouldn't be hard to generate such an M,N, but it's even better
> if you don't have to.
> SPAKE2-EE has a small performance advantage over SPAKE2 proper, at least
> when Elligator is implemented as described above.  That implementation
> takes under 1M/bit since the expensive part is the square root, but even if
> you precompute tables for M,N, it will cost at least 1.8M/bit to compute
> M^password and again 1.8M/bit for N.
> The cost is that you have to spec and implement Elligator, which is
> annoying but not terrible.  At the same time, there won't be any temptation
> to implement double-fixed-base-comb-exp routines to improve performance.
> More-symmetric and less-symmetric PAKE:
> It's not very important that M != N.  I'm pretty sure the proof Trevor
> cited of this is wrong, but I'm also pretty sure that a proof exists under
> slightly worse assumptions than the original.
> Alternatively, N^password can be dropped, so long as Bob sends his
> validator first.  (If Alice sends hers first without N^password, then a
> fake Bob clearly has a dictionary attack because his first password-related
> computation is to check the validator.)  This doesn't harm the proof IIRC,
> it just adds the ordering restriction.
> These are both provable under some CDH variant (eg, GapDH if you like
> shorter proofs with unrealistic assumptions, otherwise straight CDH with
> tightness loss) in the random oracle model.
> SPAKE2-EE has the same properties as SPAKE2 with respect to these two
> options.  It also has another option, which is that Alice and Bob use
> either temporary identities (IP address, etc) or even just random nonces,
> and replace M with h*Elligator(H(nonce_M,password)) and likewise for N.
> Augmentation:
> SPAKE2 can be augmented, where the server's credentials (i.e. Bob's
> credentials) are not sufficient to log in.  This is accomplished by storing
> g^password2, where (password,password2) is the output of the key
> strengthening function.  Alice can compute g^(y*password2) from Bob's flow
> because she knows password2 and g^y.  Bob can compute it because he knows y
> and g^password2.  So they can throw g^(y*password2) into the hash function
> when they compute the session key.  Dan and Victor's book calls this PAKE2+
> and contains a security proof, but then again that book isn't published.
> I suspect that many PAKEs can be augmented in a similar way, but they
> usually aren't specified with augmentation.
> SPAKE2 can even be doubly augmented, where Alice does the same thing, but
> I'm not sure if there's any application to that.
> Anyway, because SPAKE2EE only affects M^password and N^password, it
> doesn't interfere with this augmentation scheme.
> Questions?  Comments?  Corrections?
> Cheers,
> -- Mike
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Michael Scott
Chief Cryptographer
CertiVox Ltd
Tel (353) 86 3888746
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