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Date: Mon, 06 Jan 2014 10:39:38 +0100
From: Christian Forler <>
Subject: Re: [PHC] Security concern for Catena

On 05.01.2014 21:47, Bill Cox wrote:
> I just skimmed through the paper on Catena.  Not that my opinion counts,
> but it's a very nice piece of work!  Many ideas in Catena are very close
> to what I was looking for in password security.
> That said, I see a potential security flaw.  Let's say an attacker has a
> goal of 100X reduction in the memory cost.  Let's assume the hash
> function is SHA-256, and that V(i+1) = SHA-256(V(i)).  In the first
> loop, we fill memory with the chained SHA-256 output.  An attacker has
> to compute it all.  Instead of storing every memory location as he goes,
> he only stores 1 in 100 locations, saving 100X the memory area and
> bandwidth.

This observation is the reason why we came up with Catena-\lambda. :-)

It is true for a single BRG (\lambda=1) since you have T= O(N^2/S). This
does not work for (\lambda > 1) due to the fact that we have
T = O(N^{\lambda+1}/S^\lambda). To compute a node in a lowerish
row, you need a bunch of memory or a vast computational effort.

Note that infinity-memory hardness is comparable to sequential-memory
hardness. :-)

>  Again, he must have the values at each of these
> random-ish locations and is forced to recompute them from the saved
> SHA-256 data, which on average will be 50 rounds of SHA-256 to provide
> the missing data.  So far, so good, because our area*time has not
> changed by more than a small constant.

Our countermeasure to the attack above is the stacking of multiple
(\lambda) BRGs, This dramatically increase the re-computation cost.

Therefore, your observations are valid for Catena-1, but we will either
submit Catena-3 or Catena-4. :-)

BTW. Just try to implement your attack for Catena-2.

> This whole custom ASIC uses 1/100th of the external memory and
> bandwidth, but has 100 on-chip SHA-256 hashing cores.  Maybe that many
> cores costs a bit of die area, but it's tiny compared to the multi-GB of
> RAM that the attacker may have saved.

We recommend to use Catena with 1-8 MB of RAM. A lot of devices can not
afford to allocate multi-GB or cores for a single login process,
especially when you have a multi-user system. Therefore, we designed
Catena in a way to be deployable in almost all environments. You
can even deploy Catena on a web 2.0 service (such as twitter), embedded
system or a network-router by using the server-relief version of Catena.

> I don't want to sound down on Catena.

No problem. I rather like tough reviews than uncritically approval. :-)

Best regards,

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