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Date: Wed, 2 Apr 2014 17:06:19 -0700
From: Andy Lutomirski <luto@...capital.net>
To: discussions <discussions@...sword-hashing.net>, Justin Cappos <jcappos@....edu>
Cc: santiago torres <sat417@...dents.poly.edu>
Subject: A weak password attack against PolyPassHash

On Mon, Mar 24, 2014 at 2:34 PM, Justin Cappos <jcappos@....edu> wrote:
> I would like to solicit the community's feedback about an submission to the
> PHC called PolyPassHash.
>
> This scheme is different than most PHC entries in that it uses a
> threshold-based storage technique to prevent passwords from being
> individually cracked.   To validate a password, one must recover a share in
> a Shamir Secret Store, which necessitates knowing a threshold of correct
> passwords.   (There are extensions to allow passwords to be securely
> validated by a server upon setup and also to support accounts that do not
> count toward the threshold.)
>
> PolyPassHash gives an exponential increase in the search space an attacker
> needs to explore while only increasing the server's time by a small linear
> factor.   If you take the three passwords that are composed of six random
> characters each and protect them with PolyPassHash, to search the key space
> would take every computer on the planet working together longer than the
> universe is estimated to have existed.   PolyPassHash is about as efficient
> in terms of memory, disk, and CPU time as existing salted secure hash
> techniques.   In fact, PolyPassHash is orthogonal to the secure hashing
> technique and should integrate with (any?) other submission.
>
> More information about the scheme (including both technical documentation
> and information for a more general audience) is available at:
> https://github.com/JustinCappos/PolyPassHash
>
> There is also a Python implementation available in that repository and a
> link to the C implementation (by Santiago Torres) which will be submitted to
> the contest.
>
> I welcome any comments or feedback.


First, a question: how do you verify that you've correctly recovered
the global secret?  If you can't check that, then won't *any* set of
passwords appear valid?  I'll assume that the server stores a hash of
the constant term.


Second, an attack, based on the observation that the distribution of
passwords is, in practice, far from uniform:

Suppose that k shares are needed to unlock the database.  Select, at
random, k users.  For each of them, calculate H(salt, "123456").  Then
try unlocking the database.  Repeat until you succeed.

As a rough estimate, let p be the probability that a given user
chooses the password "123456".  Each attempt succeeds w.p. 1/p^k.
Ignoring the dependence between attempts, the cost of the attack is
O(p^k) secret share computations.

If the hash function used is expensive, then this can be improved:
memoize the results of the hash calculations and sample k-tuples of
users from a distribution with an appropriate bias toward reusing
users.  I'll let actual attackers do the optimization :)

Mitigating this type of attack may be difficult, unless defenders are
willing to choose a rather large value for k.

Given that Shamir secret sharing is based on linear algebra, I
wouldn't be surprised if this attack can be improved, possibly to the
point that the complexity becomes polynomial in k.  As far as I know,
the information theoretic argument for the security of Shamir's scheme
says that, if you don't know k shares, you learn nothing about the
secret.  In PolyPassHash's case, with reasonably high probability, you
can guess considerably more than k shares, but they're just hidden
among a bunch of incorrect guesses.

This sounds awfully like a linear coding problem.  You have a whole
bunch of guesses of randomish linear combinations of some vectors with
a special form.  Quite a few of those guesses are correct, but most
are wrong.  The problem is to identify the ones that are correct.  I
don't know off the top of my head whether this particular instance is
in an NP-hard regime or in a polynomial-time regime.  If it's
polynomial-time, then the attacker has a huge advantage.

--Andy

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