Comments on: Prime numbers, Miller-Rabin

By: Bio-Informatics — Shriphani ‘PSP’ Palakodety

Bio-Informatics — Shriphani ‘PSP’ Palakodety — Fri, 04 Sep 2009 17:20:30 +0000

[...] (which helped me a lot, thanks). And then I mixed it up with my miller rabin implementation and had a bit of fun [...]

By: Shriphani

Shriphani — Thu, 11 Jun 2009 23:43:55 +0000

Matt,

The algorithm uses a check if a^p is congruent to 1 % n. a^p congruent to 1 % n means a^p -1 is divisible by n.

Nice to know SICP has expmod. It uses the idea that a^b % n = ((a % n)^b) % n. I did write it some time back and it is on this blog. Thanks for pointing that out, I will try to integrate modular exponentiation into this script.

By: Matt

Matt — Thu, 11 Jun 2009 15:52:46 +0000

Maybe you can help me, I’m trying to implement Miller-Rabin as described in SICP, and I’m getting a little lost. The algorithm uses ’1 % n’, but I thought this would always be equal to 1? At least that’s what my calculator says. My math is a little rusty and I know I’m being dense here, but Google isn’t helping me.

BTW, if you’re looking for the fastest speed possible you may want to look at SICP’s expmod function (in section 1.2.6), they show a method for calculating a ^ b % n without actually calculating a ^ b. Don’t know if it’s any faster in practice, but worth a shot.

By: Shriphani

Shriphani — Tue, 02 Jun 2009 16:20:15 +0000

Thanks for pointing out that this is wrong. rand_int’s bounds are correct but I also felt that the core of the test could also do with some refactoring since I believe that was also not implemented correctly. Anyway, please let me know if the new snippet is still failing. I will be happy to fix it.

By: Justin

Justin — Tue, 02 Jun 2009 01:39:44 +0000

Just a note for anyone else who stumbles on to this code snippet… this isn’t a good implementation. The rand_int generated is less than n (or s, according to 2^s*d), whereas it needs to be less than the original number.

This implementation is not a good implementation, as 221 for example, is reported as prime more often than composite. Accuracy would be around 50% or less, whereas a good rabin-miller implementation would have 99.999% accuracy.