Quote from: RetiredCoder on Today at 02:56:03 PM

Quote from: kTimesG on Today at 01:20:28 PM

If we have some optimal average jump distance = m * sqrt(b) / 4 where m = number of kangaroos

Why do you think so? From my experience, optimal average jump distance does not depend on the number of kangs (at least if you have many of them), it's always about sqrt(range).

If you have some generic group, and don't exploit symmetry / negation, and always jump forward, and only have two types of kangaroos, and they all run in parallel, and the jumps are truly random and uniform, and the range is big enough so that you can compare the results better, then "m" is a cofactor of the average jump distance, since it's literally derived (mathematically) from the analysis of the distributed algorithm. This can be verified by experiments, there is a sweet spot for that alpha value, not too small, not too big. But you need to run tens of thousands of simulations on many different interval sizes (and different intervals of the same size), to eliminate noise and biases of specific intervals and the way their elements jump from one to another, and average on those. Else, you'd think you hit the "correct" average, because it seems to work better, but is it?

There's different alpha for different strategies (such as 0.375 for 3-kangaroo, still under a generic group constraint without any symmetry use). It's probably even lower if you add more and more characteristics which you can take advantage of (like, same X for P and -P = how can one even interpret this? 2 DPs in one? how it affects collision? who really knows...). I haven't found any paper that even attempts to analyze this property, but you already know that the damn C of "ops count until solve" is way way lower than 1.72, as that's the value in a generic group. not for secp256k1 ECC

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"m" grows exponentially while the average jump size grows linearly, so whatever cofactor for sqrt(b), even 1.0, will work, but is it really optimal or is it "somewhere about it"?

The cofactor is simplified anyway, in my previous post, no matter of its real value, since it's a constant on both sides. So the comparison yields identical results.