CODEWITHSUNDEEP
algorithmhaskellprimesprimality-test
Slugger
Slugger

Reputation: 675

Why is my Miller Rabin algorithm not working (Haskell)?

I implemented the Miller Rabin test in Haskell. I tried to strictly follow the pseudo code as given for instance in the wikipedia entry for the Miller Rabin test. Now I found online that for certain choices of witnesses, the test is deterministic upto certain given bounds. I am interested in primes under 2^64 and so I found sufficient bounds in this post What witnesses do i need for Rabin-Miller test for numbers up to 10¹⁸?. However, the code seems to work for most small primes I've tested but fails for some larger ones. For instance I tried the ten digit prime 5915587277 and the test returns false. I think my implementation is correct but hopefully someone can spot where I made a mistake and misunderstood something about the MR test. Thanks in advance for any help. Also, sorry for the messy looking code.

isPrime :: Int -> Bool
isPrime n = millerRabinTest n (factorizeN (n-1))

{- factorizeN finds a number s and odd number d such that n -1 = (2^s)d by 
succesively dividing n by two if it is even. -}
factorizeN :: Int -> (Int, Int)
factorizeN n = fN n 0
  where
    fN n s | even n    = fN (n `div` 2) (s + 1)
           | otherwise = (n,s)

{- this is the main function. it takes w values from a set of witnesses
and checks if n passes the test. If it doesn't, n is not prime, if it does 
for all w, it is probably prime. -}
millerRabinTest :: Int -> (Int,Int) -> Bool
millerRabinTest n (d,s) = and [test n (expmod w d n) s | w <- onesToCheck]

{- this is the test that is used in the millerRabinTest function. it sees if
w^d = 1 mod n  or n-1 mod n, if not it multiplies by two
and checks again for a total of s-1 times. If it is never true then the number 
is not prime -}
test :: Int -> Int -> Int -> Bool
test n w s | w `elem` [1,n-1] = True
           | otherwise        = or [ (expmod w (2^k) n) `elem` [1,n-1]| k <- [1..s]]   

{- set of witnesses that should make the Miller Rabin test deterministic if
n < 2^64. -}
onesToCheck :: [Int]
onesToCheck = [2,325,9375,28178,450775,9780504,1795265022]

{- function that calculates a^e mod n. -}
expmod :: Int -> Int -> Int -> Int
expmod a e n  | e == 1           = a `mod` n
              | (e `mod` 2) == 0 = (expmod ((a*a) `mod` n) (e `div` 2) n)
              | otherwise        = (a*(expmod ((a*a) `mod` n) (e `div` 2) n)) `mod` n

Upvotes: 2

Views: 609

Answers (1)

Reid Barton
Reid Barton

Reputation: 15019

Probably your Int is overflowing in expmod when you calculate a*a. Int is a machine-sized integer, no more than 64 bits. You should replace some of the occurrences of Int in your program with Integer, the arbitrary-precision integer type.

Upvotes: 9

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