Generate better start values for nth-root

[czurnieden]

This is the implementation I talked about in DCIT/perl-CryptX#123 with @danaj
The runtime is up to standard now.
Needs some work (there is a TODO list at the top of mp_root_n.c, but mostly chores).

[danaj]

I'm getting a failure with a = 2147483645^23 b = 23. The start value t2 = 2147483634 is too low.
[MacOS 26, M1, 64-bit, gcc-15]

[czurnieden]

I had the angst-allowance at the wrong place. Should be added to the log(a).
Worst thing: had it right before ;-)

The number 2147483645^23 is well chosen: log2(2147483645^23) ~ 712.99999995364 so we got 712.99999995364/23 ~ 30.99999999798, which gets got caught in all of the rounding errors. Adding 1 (one) to o the log gives 713.99999995364/23 ~ 31.043478258852, resulting in a roughly 10% higher start-value in theory and is also larger than the actual root. 1/2should do it, too

I also found an nasty little bug if root^base == input, see #599

Thanks for taking a look!

[danaj]

9999999999900000000000000 ^ 1/53 is coming into the loop with t2 = 2. We used to have the comment "Start value must be larger than root" and this isn't true since 2 is the result we're looking for. It doesn't seem to be working right, but with t2=3 it's fine. Sorry for not triaging past that.

[czurnieden]

@danaj

Sorry for not triaging past that

Nah, no reason, I wrote most of it, so it's all on me ;-)

That seems to have been fixed by #599. Will put it in this PR now.

[czurnieden]

Should work now. (famous last words)
Which leads to the question: worth it?

[czurnieden]

Could it be that I was a little bit behind? ;-)