Primality Checking
May 1, 2009
In a previous exercise you wrote a function that returned a list of prime numbers, and in another exercise you used that function to find a particular prime number. This exercise looks at prime numbers from a different perspective by considering a function that takes a number and determines if it is prime.
The algorithm that we will consider was developed initially by Gary Miller and refined by Michael Rabin, and is probabilistic in nature. It works like this: Express the odd number n to be factored as n = 2^{r} s + 1 with s odd. Then choose a random integer a with 1 ≤ a ≤ n1 and check if a^{s} ≡ 1 (mod n) or a^{2j s} ≡ 1 (mod n) for some 0 ≤ j ≤ r1. (Some browsers render that last equation poorly; it’s a raised to the power 2 to the j times s.) A prime number will pass the check for all a. A composite number will pass the check for about 1/4 of the possible as and fail the check for the remaining 3/4 of the possible as. Thus, to determine if a number n is prime, check multiple as; if k as are checked, this algorithm will err less than one time in 4^{k}. Most primality checkers set k to somewhere between 25 and 50, making the chance of error very small.
Your task is to write a function that determines if an input number n is prime, then to determine if 2^{89}1 is prime. When you are finished, you are welcome to read or run a suggested solution, or post your solution or discuss the exercise in the comments below.
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[…] Praxis – Primality Checking By Remco Niemeijer Today’s Programming Praxis problem is about checking whether or not a number is prime. We’re supposed […]
My Haskell solution (see http://bonsaicode.wordpress.com/2009/05/01/programmingpraxisprimalitychecking/ for a version with comments):
import Control.Arrow
import Data.Bits
import Data.List
import System.Random
isPrime :: Integer > StdGen > Bool
isPrime n g =
let (s, d) = (length *** head) . span even $ iterate (flip div 2) (n – 1)
xs = map (expm n d) . take 50 $ randomRs (2, n – 2) g
in all (\x > elem x [1, n – 1] 
any (== n – 1) (take s $ iterate (expm n 2) x)) xs
expm :: Integer > Integer > Integer > Integer
expm m e b = foldl’ (\r (b’, _) > mod (r * b’) m) 1 .
filter (flip testBit 0 . snd) .
zip (iterate (flip mod m . (^ 2)) b) $
takeWhile (> 0) $ iterate (flip shiftR 1) e
main :: IO ()
main = print . isPrime (2 ^ 89 – 1) =<< getStdGen [/sourcecode]
I’m slowly working my way through old exercises now that I have some free time. Here’s
my attempt in Common Lisp; I’m trying to learn the language, even though I already had Pythoned MillerRabin previously.
Wow, so much code needed in OCaml to solve this exercise in a selfcontained fashion! First of all, a generalpurpose function to return a random
Big_int
between 0 and a specifiedmax
(exclusive):(You can’t go much lower level than this.) Next, some utility functions on
Big_int
s:Finally, an imperativestyle RabinMiller test:
Two optimizations were applied: first, the random a should be greater than 1; second, the successive powers of a in the inner loop are calculated inductively by (modular) squaring.
I made a mistake in
random_big_int
. Lines 10 and 11 are wrong, they should read:This is a Haskell program I wrote (ironically enough around May 2009, though I hadn’t heard about this website back then.) This was my first relatively nontrivial Haskell program.
To test 2^89 1 use,
.\millerrabin.exe t 618970019642690137449562111
Rewrote the Haskell MillerRabin in Clojure as my first Clojure program. It is linebyline translation, pretty much. Main difference is that one can’t generate random numbers > 2^63, so that is taken into account when generating tests.
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