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Date: Mon, 25 Nov 2013 11:21:43 +0400
From: Solar Designer <solar@...nwall.com>
To: discussions@...sword-hashing.net
Subject: Re: [PHC] A (naively?) simple PHC submission using hash chains

On Wed, Aug 07, 2013 at 10:11:28AM +0200, CodesInChaos wrote:
> I don't know any sources for this technique, but I suspect it can be used
> on all hashing schemes with predictable memory access.
> 
> * You can use it to accelerate scrypt's first phase, but not the second
> phase. Since scrypt only claims that the second phase is memory-hard, this
> doesn't break scrypt.
>    Discussed this with solardiz on twitter:
> https://twitter.com/solardiz/status/362644103546142721
> * You can use it on yarrkov's high AT proof-of-work scheme:
>    Discussed on the crypto mailing list a few months ago
> http://www.mail-archive.com/cryptography@randombit.net/msg03689.html
> 
> The general idea is that you don't keep everything in memory you're
> supposed to, but only an occasional checkpoint. From this checkpoint you
> can recompute memory near it. Often it's a good choice to keep sqrt(n)
> checkpoints from which you can recompute any value you're interested in in
> time n/sqrt(n)=sqrt(n).

The total AT cost, disregarding area consumed by computation cores, is:

N^(1/2 area + 1 time) + N^(1 area + 1/2 time) = 2*N^1.5

> On specialized hardware it's often cheap to recompute it in parallel to
> your normal computation without delaying the normal computation.
> Unpredictable memory access often breaks this, since the recomputation
> delays the normal computation instead of running in parallel.
> 
> It's often possible to improve upon the sqrt(n) technique by adding more
> intermediate steps. But I didn't work out the details.

I gave this some thought today.  Here's what I am getting with one extra
intermediate step (and N^(3/4) parallel cores in the last step):

N^(1/2 + 1) + N^(3/4 + 1/2) + N^(1 + 1/4) = N^1.5 + 2*N^1.25

So it's still O(N^1.5) due to the first step, regardless of how many
more steps we add, but there is an up to a factor of 2 improvement over
the 2-step algorithm.

Alexander

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