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Idle Thoughts About Pollard Rho

solipsys.co.uk

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Re: Idle Thoughts About Pollard Rho

#4

Post by 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.

Re: Idle Thoughts About Pollard Rho

#5

Post by ColinWright »

tempodox wrote: post #4

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.

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).

Re: Idle Thoughts About Pollard Rho

#6

Post by elpres »

ColinWright wrote: post #5
tempodox wrote: post #4

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.

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 co…

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.

Re: Idle Thoughts About Pollard Rho

#7

Post by ColinWright »

elpres wrote: post #6
ColinWright wrote: post #5

Earlier quoted context omitted.

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 co…

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.

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-(

Re: Idle Thoughts About Pollard Rho

#8

Post by vmilner »

ColinWright wrote: post #3
vmilner wrote: post #2

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_...

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

Antoine Joux's book:

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.

Re: Idle Thoughts About Pollard Rho

#9

Post by ColinWright »

vmilner wrote: post #8
ColinWright wrote: post #3

Earlier quoted context omitted.

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

Antoine Joux's book: 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.

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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