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Date: Tue, 28 Jul 2015 08:32:06 -0700
From: Bill Cox <waywardgeek@...il.com>
To: "discussions@...sword-hashing.net" <discussions@...sword-hashing.net>
Subject: Re: [PHC] [OT] Improvement to Momentum PoW

On Tue, Jul 28, 2015 at 6:57 AM, Dmitry Khovratovich <khovratovich@...il.com
> wrote:

>
>
> On 28 Jul 2015, at 15:13, Bill Cox <waywardgeek@...il.com> wrote:
>
> On Tue, Jul 28, 2015 at 4:36 AM, Dmitry Khovratovich <
> khovratovich@...il.com> wrote:
>
>> I am not sure what you're calling by "parallel rho", but the parallel
>> collision search by van Oorschot-Wiener deals exactly with this problem.
>> They call it "golden collision" search in Section 4 of
>> http://people.scs.carleton.ca/~paulv/papers/JoC97.pdf .
>>
>> The tradeoff in that paper is very favourable to the attacker: when the
>> memory is reduced by P^2, the time grows by the factor of P only, so it is
>> the sublinear growth. Storing only some small range of Hb(i) gives  a much
>> worse, linear tradeoff only.
>>
>> Dmitry
>>
>
> This is incorrect.  This is covered in the paper in the section "Run-Time
> Analysis for Finding a Large Number of Collisions" under finding golden
> collisions on page 9.
>
> The hash table algorithm uses O(sqrt(n)) memory, for n points, and runs in
> time O(n) to generate all collisions.  Decreasing it by a factor of T,
> causes it to run T times longer to generate all collisions, so memory*time
> = O(n^1.5), regardless of T.  The paper's algorithm runs with w
> distinguished points, using O(w) memory, and runs in time O(sqrt(n^3/w)).
> Memory*time is O(n^1.5*sqrt(w)).  This is far worse than the hash table,
> for all values of w >> 1.
>
>
> I understand it differently. The regular algorithm uses n memory for n
>> points and needs n time, so time*memory is n^2. It is equal to that of
>> dist. points algorithm for w=n, but the latter wins when you decrease w.
>> What is the hash table algorithm that needs sqrt(n) memory and n time?
>
>
>
> Ah, you are right.  I forgot the basics :-p  However, the dist. point
algorithm is still slower than the regular algorithm for a given amount of
memory.

The regular algorithm uses O(n) memory, and generates O(n) collisions in 1
pass.  It's memory*time is O(n^2) and it generates collisions in average
time of one call to Hb.  The dist. points algorithm if w = n/16, for
example, will find a dist. point on average every 16 calls to Hb, and then
resolve that collision on average with another 8 calls to Hb, with a 1/16
chance of failure, to generate 1 collision.  Using a hash table with only
n/16 entries requires n/16 calls to Hb to fill it, and then generates a
collision every 16 calls to Hb on average.  The dist. point algorithm is
slower for any memory size.

The reason the dist. point algorithm works well normally is we are normally
searching for a very rare collisions.  This is simply not the case here.
The paper did not even cover this case.

I know we don't like worrying about small constant factors much, but when
talking about ASIC defense, we need every 10%.  The regular Momentum
algorithm leaves itself open to parallel filtering on GPUs, and ASICs would
be even stronger.  I prefer this version.

In reality, we would not use a simple hash table.  It's too slow due to
cache-miss penalties.  Instead we would create maybe 1024 4KiB buffers,
which fit in L3 cache, and do a 10-bit radix sort of hashes into them,
storing the 26-bit input hash and 16-bit result hash because 10 bits are
determined by the buffer.  The speed of this radix sort is still pretty
bad.  I'll see if I can speed it up further with a cache-row sized
mini-buffer to improve write speeds to the main buffers.

Bill

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