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Date: Wed, 22 Jan 2014 18:54:53 -0500
From: Bill Cox <>
Subject: Computational comparison of 3 schemes...

I've compared Catena-3, NoelKDF, and what I think Alexander is doing
in escrypt.  I wrote two programs.  One pebbles DAGs somewhat
efficiently, and is meant to provide an upper bound.  The second
program tests how strongly using password-dependent random reads
punishes an attacker.

Summary: Catena3 wins my pebbling benchmarks, for timing-resistant
KDFs.  Noel-KDF wins the non-timing-resistant benchmarks for
sequential-memory-hard KDFs.  Alexander's power-of-two sliding window
does respectably in both.  Catena-3 graphs seem quite difficult to
pebble efficiently with a generic algorithm.  However, it does not
significantly punish an attacker when doing password-dependent reads.
This makes it the best timing-attack resistant algorithm, but a poor
choice as the starting point for a sequential-memory-hard KDF.  This
should be no surprise, as Catena is designed for timing-attack

There are very interesting trade-offs.  First, all three algorithms
fail completely in timing-attack-resistant mode for an attacker using
1/4 of memory.  All three required no significant recalculations, and
an attacker gets to use 1/4 of memory basically for free.  Catena-3
only uses 1/4 of the memory, because it reuses the row memory, and
there are 4 rows.  Both attacker and defender benefit from the 1/4th

All three algorithms completely defeat my pebbling algorithm for an
attack using 1/5th of the memory rather than 1/4th, but Catena-3 did
it much sooner.  Even 1 pebble less than 1/4 punished my pebbling
algorithm quite a bit, leading to a 24X recomputation cost for
pebbling a 2048 node graph (G == 512).

In sequential-memory-hard mode, NoelKDF seems to devastate TMTO
attacks at the 1/4 memory mode, making it look practical to have the
final password-dependent loop run only 1% of the total runtime.
Escrypt did plenty well, and for large numbers of blocks (>
1,000,000), it does well enough even at 1% coverage, but there may be
an argument for running longer.  TMTO attacks of < 1/2 seem
impractical for these sequential-memory-hard algorithms.  However,
with memory timing data, a NoelKDF attacker can probably use 1/8th or
possibly less of memory, as it relatively easy to pebble.  Escrypt
would probably still require closer to 1/4th, while I suspect
attackers wont even bother trying to save even one memory location
when attacking Catena-3.

In short, Catena-3 is the natural timing-resistant choice, but it only
uses 1/4 of the memory of an equivalent sequential-memory-hard KDF.
However, the sequential memory-hard KDFs can lose most of their memory
requirements if cache timing is known (they still have to run for a
long time per guess).

That's all I have for now.  Does anyone know of a good generic
pebbling algorithm?  Mine works like this:

1 Put all the initial pebbles down over the first N locations if there
are N pebbles
2 Add all non-fixed, non-in-use pebbles that have no edges pointing at
them from beyond the current runner pebble position to the "pebble
    2.5 If there are no such pebbles, use a heuristic to add the
"best" pebble to the group.
3 When pebbling, take virtual pebbles from the pebble group.  Reduce
the "num available" pebbles on the group by 1, but alloc a new pebble
rather than removing any from the group
4 When num available reaches 0, delete all the pebbles in the group, and goto 2


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