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Date: Mon, 30 Mar 2015 14:58:45 +0200
From: Dmitry Khovratovich <>
To: "" <>
Subject: Re: [PHC] Tradeoff cryptanalysis of Catena, Lyra2, and generic
 memory-hard functions

Hi Alexander,

you're certainly right, the stored indices must be taken into account
when counting memory for tradeoffs. For the default parameters of
Catena and Lyra2 that we considered, these indices occupy less than
1/1000 of total memory, and we just (explicitly or implicitly) ignored
this small addend in the computation. The same happened to other
"service" information that we use in the attacks (such as storing the
access complexities, which accounted to 1/6000 of memory, cf. page 9
of the report).

Indeed, smaller blocks would increase the proportion of indices in the
memory. To have them occupy, say, 1/10 of memory, let us consider 1 GB
memory size. 100 MB of 3-byte indices would imply 2^{25} blocks, i.e.
32-byte blocks. However, on the x86 architecture random access to
32-byte blocks  would increase latency dramatically (my estimate would
be 3-5x at least), and make the scheme much slower and thus weaker.

On the architectures where the memory latency is only  a few CPU
cycles (AVR?), smaller blocks can be useful.


On Mon, Mar 30, 2015 at 2:31 PM, Solar Designer <> wrote:
> On Fri, Feb 13, 2015 at 04:45:35PM +0100, Dmitry Khovratovich wrote:
>> The report is permanently available at
>>  and will be soon added to ePrint
>> as well.
> Sorry for stating the (maybe) obvious, but here's an additional comment,
> which I don't recall seeing on this mailing list yet:
> Indices of or some other kind of references to vertices cost memory too,
> a few bytes each.  They will be needed to actually mount the attacks,
> and their number (and thus total size) may be significant.
> For example, Table 6 seems to imply that a massively-parallel machine
> with essentially free computing cores would compute the hash with a time
> penalty equal to the depth penalty, which is only 26.6 for 1/10 memory.
> (While this feels like a bad tradeoff, it could make sense for some
> attackers who e.g. only have 1/10 memory in a chip.)  However, with a
> small enough block size, storing that tree's structure as needed for
> such optimal computation may take a comparable amount of memory to that
> for the actual data being stored.
> Thus, decreasing block size is a way to make TMTO less efficient.  In an
> extreme case, the block size may be comparable to one index size (e.g.,
> 4 bytes), and this would defeat even mild TMTO attacks (except on
> TMTO-friendly schemes like the original scrypt, where it is not
> necessary to record any metadata).  However, to significantly discourage
> TMTO attacks needing a depth like the 26.6 figure above, block size of a
> few hundred bytes would likely work as well (and this is a reasonable
> setting for some PHC schemes).
> BTW, the extra/sim-tmto.c simple TMTO attack implementation included in
> the yescrypt submission accounts for the metadata required to mount the
> attack, and reports it like this:
> scrypt ircmaxell:
> B' = 24efce90
> t_cost = 512
> m_cost = 256
> scrypt ircmaxell TMTO1 = 5, TMTO2 = 2:
> B' = 24efce90
> t_cost = 978
> m_cost = 180 elements + 256 indices (363 alloc + 256 ptrs) + 512 counters
> This shows that while the number of "elements" decreased from 256
> without TMTO to 180 with TMTO, we also added "256 indices (363 alloc +
> 256 ptrs) + 512 counters" for this algorithm's metadata.  At small
> element size, this would be non-negligible, up to the point where the
> TMTO doesn't reduce total memory usage at all despite of reducing the
> number of elements significantly.
> Here's what I wrote in a private e-mail reply a year ago:
> ---
> Someone wrote:
>> [...] I seem to recall from
>> my theory of computation class that TMTO is a property of anything
>> that can run on a Turing machine, so it seems hopeless to defeat it
>> completely,
> Sort of.  You can in fact always recompute instead of storing and
> retrieving intermediate results, but the theory that the memory needs
> can always be reduced, up to the point of needing no memory for any
> intermediate results, fails when you realize that referring to
> intermediate results that you don't store has non-zero memory cost -
> e.g., it can be the size of a set of array indices.
> Suppose at some point in computation your algorithm finds out it needs a
> past intermediate result it didn't store.  It can initiate recomputation
> of that intermediate result, but for that it needs to store some
> identifier of that intermediate result somewhere (e.g. in a variable or
> in a recursive call's stack frame or on a Turing machine's tape), so
> that the re-invoked algorithm can store this one intermediate result
> this time.  OK, it's only one such identifier, not a big deal - but what
> happens if the same issue had already occurred (and thus will occur
> again) during the recomputation of that one intermediate result?  You
> now have two identifiers to store, and so on.
> Given this theory, it's _not_ hopeless to defeat TMTO completely.  This
> may be done by keeping the size of intermediate result identifiers on
> par with or larger than the size of those intermediate results
> themselves - in other words, making them tiny but numerous - then coming
> up with an algorithm that computes the function in the least amount of
> memory possible.
> However, this might be impractical, for other reasons.
>> but I definitely see where there is room for improvement.
> Yes, the current focus is on discouraging TMTO (unless it's specifically
> desired in a given use case), not on making it impossible (even if that
> could be done).
> ---
> Once again, sorry for stating the (maybe) obvious.
> Alexander

Best regards,
Dmitry Khovratovich

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