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Testing primality

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1 Testing Primality

(for a context to this see Prime numbers).

1.1 Primality Test and Integer Factorization

Given an infinite list of prime numbers, we can implement primality test and integer factorization by trial division:

-- isPrime n = head (primeFactors n) == n
isPrime n = n > 1 &&
              foldr (\p r -> p*p > n || ((n `rem` p) /= 0 && r))
                True primes
 
primeFactors n | n > 1 = go n primes   -- or go n (2:[3,5..])
   where                               -- for one-off invocation
     go n ps@(p:t)
        | p*p > n    = [n]
        | r == 0     =  p : go q ps
        | otherwise  =      go n t
                where
                  (q,r) = quotRem n p

When trying to factorize only one number or two, it will be faster to just use (2:[3,5..]) as a source of possible divisors instead of first finding prime numbers and then using them. For more than a few factorizations, when no other primes source is available, just use

primes = 2 : filter isPrime [3,5..]

More at Prime numbers#Optimal trial division.

1.2 Miller-Rabin Primality Test

-- (eq. to) find2km (2^k * n) = (k,n)
find2km :: Integral a => a -> (a,a)
find2km n = f 0 n
    where 
        f k m
            | r == 1 = (k,m)
            | otherwise = f (k+1) q
            where (q,r) = quotRem m 2        
 
-- n is the number to test; a is the (presumably randomly chosen) witness
millerRabinPrimality :: Integer -> Integer -> Bool
millerRabinPrimality n a
    | a <= 1 || a >= n-1 = 
        error $ "millerRabinPrimality: a out of range (" 
              ++ show a ++ " for "++ show n ++ ")" 
    | n < 2 = False
    | even n = False
    | b0 == 1 || b0 == n' = True
    | otherwise = iter (tail b)
    where
        n' = n-1
        (k,m) = find2km n'
        b0 = powMod n a m
        b = take (fromIntegral k) $ iterate (squareMod n) b0
        iter [] = False
        iter (x:xs)
            | x == 1 = False
            | x == n' = True
            | otherwise = iter xs
 
-- (eq. to) pow' (*) (^2) n k = n^k
pow' :: (Num a, Integral b) => (a->a->a) -> (a->a) -> a -> b -> a
pow' _ _ _ 0 = 1
pow' mul sq x' n' = f x' n' 1
    where 
        f x n y
            | n == 1 = x `mul` y
            | r == 0 = f x2 q y
            | otherwise = f x2 q (x `mul` y)
            where
                (q,r) = quotRem n 2
                x2 = sq x
 
mulMod :: Integral a => a -> a -> a -> a
mulMod a b c = (b * c) `mod` a
squareMod :: Integral a => a -> a -> a
squareMod a b = (b * b) `rem` a
 
-- (eq. to) powMod m n k = n^k `mod` m
powMod :: Integral a => a -> a -> a -> a
powMod m = pow' (mulMod m) (squareMod m)

Example:

-- check if '1212121' is prime with several witnesses
> map (millerRabinPrimality 1212121) [5432,1265,87532,8765,26]
[True,True,True,True,True]