phc-discussions - Re: [PHC] multiply latency reduction via table lookups

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Date: Mon, 10 Mar 2014 08:40:50 +0000
From: Samuel Neves <sneves@....uc.pt>
To: discussions@...sword-hashing.net
Subject: Re: [PHC] multiply latency reduction via table lookups

On 10-03-2014 08:02, Solar Designer wrote:
> Now, a 16x16->32 lookup table is still pretty large.  A naive
> implementation of it would take 16 GiB.  A smarter implementation would
> probably halve that, due to multiplication being commutative.  (Can we
> do better yet?)

You can use the quarter-squares identity [1] to keep that table at
around 2^16 entries, in exchange for 2 memory accesses instead of 1.
This seems to be a better tradeoff than 4 accesses (or 3 using
Karatsuba) with an 8x8 table, under the assumption that additions and 
subtractions are 'free'.

> Do e.g. CPUs use table lookups like this for multiplication already?

Not for multiplication as far as I know, but complex floating-point
functions (e.g. inverse square root, trigonometric) usually start with a
table lookup for initial approximations.

[1]
https://en.wikipedia.org/wiki/Multiplication_algorithm#Quarter_square_multiplication