Re: Algorimic Complexity Attacks

[Nicholas Weaver <nweaver () CS berkeley edu>]

Pardon me if this is obvious to everyone already...

On Sat, Jun 07, 2003 at 07:01:06PM +0200, Pavel Kankovsky composed:
> > Changing the secret once in a while reduces this attack and may well
> > make it impractical with many particular applications.  Note that one
> > doesn't have to use any additional true randomness (and possibly
> > exhaust the randomness pool) for each new secret to be used with the
> > keyed hash.  If the secret itself is not leaked in the attack (and it
> > shouldn't be), something as simple as secret++ could suffice.
> > However, this does have its difficulty: maintaining existing entries.
>
> First, let us observe the attacker needs no less than O(h) inserts (where
> h is the size of the hash table) to find a collision of an unknown hash
> function with a non-negligible probability of success.

> This means the attack will be thwarted if the secret hash function (e.g.
> a universal hash function using a secret parameter) is changed every
> O(h) inserts.

Actually, thanks to the Birthday paradox, it is O(sqrt(h)) when
collisions start to appear.

Possibly a better solution is to simply make the buckets themselves
use (lg(n)) search behavior rather than linear search behavior.  EG,
if the hash buckets are based on skiplists rather than linked lists,
this provides expected O(lg(n)) search time in the bucket instead of
O(n), with O(lg(n)) instert, unless the adversary can predict the
random sequence used by the skiplist.


Likewise, another solution is to simply use a GOOD cryptographic
function for your hash.  If, for an attacker to create h(y) == h(x),
requires the attacker to discover the key used in the hash function or
otherwise break the hash function, simply make sure that the key is
well created and use a strong cypher as the basis of the hash
function.

In that case, for a smaller table size, collisions WILL occur, but as
long as they can't be maliciously generated by an adversary, there
isn't a problem as the attacker can't turn the expected O(1) search
into expected O(n) search on a single bucket.  The problem is not that
hash-tables have collisions, but that a malicious attacker can
generate collisions at-will.

-- 
Nicholas C. Weaver                                 nweaver () cs berkeley edu