phc-discussions - Re: [PHC] Compute time hardness

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Message-ID: <551C3508.2050909@dei.uc.pt>
Date: Wed, 01 Apr 2015 19:12:24 +0100
From: Samuel Neves <sneves@....uc.pt>
To: discussions@...sword-hashing.net
Subject: Re: [PHC] Compute time hardness

On 01-04-2015 18:07, Solar Designer wrote:
> On Wed, Apr 01, 2015 at 06:36:32PM +0300, Solar Designer wrote:
>> On Wed, Apr 01, 2015 at 08:20:29AM -0700, Bill Cox wrote:
>>> Alexander's technique of counting sequential operations, and giving
>>> multiply more weight seems reasonable to me, though he's once again
>>> under-cutting himself by only claiming a multiply is worth 3 additions.
>>> Use 8 and you'll be closer to reality, IMO.
>> I agree.  All we know is it's somewhere between 3 and 32.
> OK, it's obviously not 32.  If we count only gate delays, then it's easy
> to see that we need one set of 1024 parallel 1-bit ANDs for the partial
> products, and then 5 sequential groups of (equivalents of) 32-bit ADDs.
> So that's a total of 1 AND + 5 ADD.  But there might be wire delays
> greater than those in a circuit with a more regular structure.  In fact,
> I found a patent (expiring soon) that appears to focus on optimizing the
> structure and it gives "a propagation delay of only 7.5 full adders" for
> a 32x32 multiplier (and 10.5 for 58x58, 11.5 for 61x61).  It isn't
> entirely clear (at least without actually reading and understanding the
> whole thing, which I did not) what exactly they mean by a "full adder"
> here (same bit width as the multiplier's operands? something else?)

You can certainly do the n^2 ANDs and add things up, but that seems like a waste of gates. What is the cost metric being
considered here? Time (circuit depth)? Area? Time * Area?

As a theoretical curiosity, Brent and Kung [1] came up with an optimal area-time n-bit multiplier of depth O(n^(1/2) log
n) and area O(n log n). Melhorn-Preparata's [2] circuit would be O(log n), O(n^2 / log n). The above would be
(asymptotically) depth O(log n) and area O(n^2).

[1] http://maths-people.anu.edu.au/~brent/pub/pub055.html
[2] http://www.sciencedirect.com/science/article/pii/S0019995883800618