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Date: Sat, 13 Dec 2014 01:22:58 +0100
From: Thomas Pornin <pornin@...et.org>
To: discussions@...sword-hashing.net
Cc: gmaxwell@...il.com
Subject: Re: [PHC] Some KDF stumbling blocks, plus Common "memory-hard"
 approaches and amortized attack costs.

On Fri, Dec 12, 2014 at 10:38:05PM +0000, Gregory Maxwell wrote:
> I am saddened that none of the proposals supported delegation scheme
> with information-theoretic security ( which are possible,
> https://bitcointalk.org/index.php?topic=311000.0 )

Well, there _is_ a PHC candidate that supports delegation: Makwa. Like
the scheme suggested in that page, the core of Makwa is a sequence of
squarings modulo a RSA modulus. The delegation mechanism is a bit
different, though.

Actually, the delegation system described in that page may leak some
information. It blinds a message m to hash with:

   r = (g^b)*m

where g is a known generator, and b is chosen randomly. The value b
tends to be reused because the unblinding value y = g^e is expensive to
compute, as much as doing the work yourself instead of delegating
(therefore not reusing the value b would make little sense). Therefore,
if the same system performs two external delegations for messages
(passwords) m1 and m2, then eavesdroppers will see (g^b)*m1 and
(g^b)*m2, and thus be able to compute m1/m2, which can yield quite a lot
of information if both m1 and m2 are human-compatible passwords (e.g.
ASCII only).

As another remark, application of the Jacobi symbol should reveal one bit
of information on m, thereby halving the attack effort.


Makwa's delegation mechanism avoids both problems by using the following:

 - The modulus n = pq is a Blum integer (i.e. p = 3 mod 4 and q = 3 mod 4).

 - One squaring is performed locally, so that only quadratic residues
   ever appear in the rest of the protocol.

 - There is not one secret blinding pair (x,y) (where y=x^e), but 300
   public blinding pairs. The client generates a new pair for each
   delegation instance by selecting randomly half (or so) of the pairs
   and multiplying them together. Compared to the blinding described
   previously, the cost for the client is higher (an average of 300
   modular multiplications) but still acceptable (it is less than that
   of a single RSA private key operation).

Note that the blinding pairs used by Makwa need not be secret; they can
be shared; thus, they can be published on some server or even hardwired
in the application code.


Of course, Makwa is not memory-hard. I did not find a construction that
supports delegation AND is memory-hard (just like you experienced
yourself). Though memory hardness is, as you say, "fashionable", it is
not the Alpha and Omega of password hashing and its virtues should be
assessed in the light of economics, as you do.


As for parallelism: _any_ password hashing KDF can be parallelized by
computing hashes with several distinct salts, and XORing the results
together. If you envision delegation, then you might want to account for
failing servers; the "XOR" method can be extended to tolerate servers
not responding in time (or at all) with a construction reminiscent of
Shamir's Secret Sharing. E.g. you split the work into 10 shares, that
you delegate to 10 servers, so that if any 7 of them respond you can
complete the computation. I have not yet taken the time to write it down
formally, but it is described in the presentation I made at Passwords14
Las Vegas on August 2014. Slides can be found at the bottom of the Makwa
Web page: http://www.bolet.org/makwa/
(there is also a video: https://www.youtube.com/watch?v=9j3WfvOj-IQ
if you don't mind hearing me torturing English language.)

This kind of parallelizing works for about any KDF, but surviving
loss of some shares really makes sense only in a delegation setup.


(Minor terminology nitpick: none of the above is "information-theoretic
security". That term designates algorithms that remain secure against
attackers with unbounded computing abilities. By definition, this cannot
apply to password hashing, where exhaustive search always succeeds in
the long run. Also, anything involving a non-prime modulus can be broken
by factoring the modulus.)


	--Thomas Pornin

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