
Idle Thoughts About Pollard Rho - ColinWright
https://www.solipsys.co.uk/new/IdleThoughtsAboutPollardRho.html?HN_ri11
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vmilner
The asymptotic means for tails and cycles of random functions are in Theorem 3
of
[https://link.springer.com/content/pdf/10.1007/3-540-46885-4_...](https://link.springer.com/content/pdf/10.1007/3-540-46885-4_34.pdf)

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ColinWright
Great catch - thank you! Now to find some time to read it properly.

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vmilner
Antoine Joux's book:

[https://books.google.co.uk/books?id=buQajqt-
_iUC](https://books.google.co.uk/books?id=buQajqt-_iUC)

precis-es this paper in its sections 7.2 and 7.3 and highlights the fact that
looking at "Pollard rho style" random functions with in-degree 0 or 2, merely
changes the multiplicative constants of the tail and cycle statistics without
changing the order.

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ColinWright
That's pretty much what I was expecting, and it's great to see that someone's
done the work. I never expected my idle musings to be in any way new or
insightful, so it's nice to be able to see that at least I'm not completely
off-track.

I'm now wondering about constructing fast-to-compute functions with in-degree
(0,0,3) or (0,0,0,4), which should change the characteristics, but I need to
read a bit first.

Again, thanks.

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tempodox
Do I have to speak TeX to understand this? What's it with all the dollar signs
and periods?

I recognize the graphics, though. Iteration paths of a polynomial modulo N
graphviz'ed. These seemingly simple constructs don't cease to fascinate me.

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ColinWright
Edited ...

I thought something might have broken, but it's working for me. I don't know
how the MathJax works, it might be that you need Javascript turned on.
Certainly usually when I'm using Javascript there is no sensible "fall back"
for times when it's not available, so it doesn't try.

I'm not a front-end person, so I can't do anything more about it now. If
someone else can check it out and let me know the best course of action I'd be
delighted to hear from them - contact details in my profile.

 _Edited again ..._

OK, now looking at it from another browser and it _is_ broken. The other
browser must have something cached. I'll see if I can get it fixed soon.

We apologise for the inconveniece (and incoherence).

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elpres
The problem is that your site is served through HTTPS, but the MathJax code is
loaded via HTTP, and Chrome/Firefox refuse to load external JavaScript due to
mixed content policy. The embedded images also produce a mixed content
warning, but are still loaded because they are not a security risk.

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ColinWright
Thanks ... So the link without the https should work, I'll get to fixing the
proper site when I can back to my desk.

Cheers.

 _Edit: temporary fix in place - working on the rest of the site now 8-(_

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ahh
This is malformatted on my Android chrome--math is overlaying subsequent text
:(

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ColinWright
In the past I've tried repeatedly to get things to layout decently on a wide
selection of web browsers, but I have simply given up completely. I try to get
it to work on a reasonably broad selection, and there are days when I get time
to work on it, but I don't have the front-end skills to cope with the utter
crap-fest that the web has become with regards layout. To be honest, I'm on
the verge of converting things like this to a PDF so I can get proper control
of the formatting and stop pissing away time on something that really, really
should be better.

I'm sorry it doesn't work well on Android/Chrome, but I've tried, and I have
neither the time nor the skills. 8-(

And as a side-comment - kudos to those who work hard to get stuff to render
well on the web. One day there'll be a way for numpties like me to produce
something that looks good and doesn't require endless fiddling.

