What is the tradeoff between multiplication and addition?
Just a few years ago, mults were slower, but I think now (Intel i9) mult, add and fma are the same.
Show HN: Compute polynomials twice as fast
11–20 of 39 posts
Re: Show HN: Compute polynomials twice as fast
#12Would this be applicable to fast hashes like WyHash and xxh3 or are those not using polynomials? Is this mainly for faster cryptographic hashes?
It is applicable to fast universal hashes like Poly1305 and Polymur (the latter of which I'm the author). However it's not clear to me whether this work improves over the state of the art for that purpose, see some questions here: https://www.reddit.com/r/programming/comments/1wbgcke/comput....
This purpose is however much easier/flexible than actual polynomial equivalence since the requirement here is only that the polynomial is injective, not identical.
WyHash and xxh3 do not have polynomial structures.
Re: Show HN: Compute polynomials twice as fast
#13Would this be applicable to fast hashes like WyHash and xxh3 or are those not using polynomials? Is this mainly for faster cryptographic hashes?
WyHash and xxh3 are not polynomial, in fact this is one of the issues we try to solve in the paper.
Many "practical" hashes use heuristics instead of real field multiplications to be faster. But it means they are vulnerable to adversarial inputs. That means, it's possible to design a set of keys that have much higher probability (under random hash seeds/keys) to collide than you'd expect under a correct hash function.
We actually analyze both WyHash and xxh3 in this setting in section "Adversarial inputs for heuristic hashes" - https://arxiv.org/pdf/2609.06022#page=165
Re: Show HN: Compute polynomials twice as fast
#14Re: Show HN: Compute polynomials twice as fast
#15From the abstract, a name that many on HN would recognize: > We also give an injective polynomial construction for universal hashing that uses N multiplications to hash 2N values with a single random key. This improves the best previous construction by Daniel J. Bernstein (this http URL).
I don't know what happened to the URL, but it's supposed to link to this paper: https://www.gwizfl.org/email/cr.yp.to/antiforgery/pema-20071...
It's a very nice construction (based on Rabin & Winograd's polynomial multiplication method) for building universal hashes with n/2+O(logn) multiplications.
The annoying part is that it's a tree structure, which is not usually what you want in a fast hash that you're folding over a data stream. Some papers like https://eprint.iacr.org/2017/328.pdf try to fix this, but there are a lot of annoying trade-offs.
A famous fast hash is NH, which is just:
H(x) = sum_i (x_{2i} + a_{2i}) * (x_{2i+1} + a_{2i+1})
Our construction (section 5.9 Injective Polynomial Hashing) shows that you can do something a bit similar with polynomials:
P_0 = z
P_i = x_{2i} + (x_{2i+1} + z^3)(x_{2i} + z^2)
Re: Show HN: Compute polynomials twice as fast
#16Would this be applicable to fast hashes like WyHash and xxh3 or are those not using polynomials? Is this mainly for faster cryptographic hashes?
It is applicable to fast universal hashes like Poly1305 and Polymur (the latter of which I'm the author). However it's not clear to me whether this work improves over the state of the art for that purpose, see some questions here: https://www.reddit.com/r/programming/comments/1wbgcke/comput... . This purpose is however much easier/flexible than actual polynomial equivalence since the requirement here is only that the…
It is applicable, but it is not useful.
Universal hashes use the input text as the set of coefficients.
This method requires additional preprocessing of the coefficients, before starting to evaluate the polynomial. That preprocessing would slow the hashing algorithm more than what is gained during evaluation.
This method is useful only when with a given polynomial, i.e. set of polynomial coefficients, you want to evaluate that polynomial many times, so the cost of the preprocessing is amortized.
However, this application is very important because most functions are approximated either with polynomials or with rational functions, so this method can accelerate the evaluation of all such approximated functions.
Re: Show HN: Compute polynomials twice as fast
#17What is the tradeoff between multiplication and addition?
If you are working over floating point, you probably with to use Estrin's method (see https://en.wikipedia.org/wiki/Estrin%27s_scheme - also tab 3 on the website.)
It takes advantage of FMA (fused multiply add), has good numeric stability and uses pipelining optimally.
A while ago I suggested using Estrin's method in Boost, for functions like std::exp. There's some interesting discussions here: https://github.com/boostorg/math/issues/924 if you are interested in all the practical details.
However, for finite fields (e.g. used for hashing and cryptography) multiplication is much more expensive than addition, which is the main use of this algorithm.
Re: Show HN: Compute polynomials twice as fast
#18If you're going to preprocess the polynomial, maybe you want to evaluate it at many different points. But then why not use the FFT?
Re: Show HN: Compute polynomials twice as fast
#19It keeps flipping back to 'monic' from e.g. 'ln(1+x)' when switching between algorithms, and then seems to lock to 'monic'? (Am I missing something?) Also I am curious, in your version vs. horner , how do both algorithms map onto number of fmadd operations?
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Re: Show HN: Compute polynomials twice as fast
#20What is the tradeoff between multiplication and addition?
If you are working over floating point, you probably with to use Estrin's method (see https://en.wikipedia.org/wiki/Estrin%27s_scheme - also tab 3 on the website.) It takes advantage of FMA (fused multiply add), has good numeric stability and uses pipelining optimally. A while ago I suggested using Estrin's method in Boost, for functions like std::exp. There's some interesting discussions here: https://github.com/boo…
Yeah, I just recently learned about Estrin's method when fooling around with some polynomial approximations. I'd been scaling the output by a sqrt term to get better accuracy at small degrees, but it turned out that polynomials of very large degrees can be calculated in the same time as a single correctly-rounded sqrt, especially when fma is available. Seemingly, the only real cost is the added register pressure.
The length of an expression when written out can definitely be deceiving when pipelining is added to the mix.