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Date: Tue, 28 Jul 2015 15:57:26 +0200
From: Dmitry Khovratovich <>
To: "" <>
Subject: Re: [PHC] [OT] Improvement to Momentum PoW

> On 28 Jul 2015, at 15:13, Bill Cox <> wrote:
>> On Tue, Jul 28, 2015 at 4:36 AM, Dmitry Khovratovich <> 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 .
>> 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?

> For constant memory, both algorithms converge to O(n^2) runtime, no better than comparing random points.  This isn't in the paper, but it's clear from the algorithm if you imagine storing 1 distinguished point.  You have to take O(n) steps to hit it once, and the paper's algorithm requires hitting it twice.  Since random points collide with probability 1/n, you only need O(n) random comparisons to find a collision.  The same goes for the hash table algorithm.
> So, distinguished point algorithms fail at every memory size compared to using a hash table.
> Parallel Pollard Rho is where we share the distinguished points among a large number of workers.  This algorithm fails vs a shared hash table for the same reasons as above.
> Anyway, thanks for taking some time to examine it.  For now, I still think this algorithm works better than regular Momentum.
> Bill

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