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Testing primality

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(Primality Test and Integer Factorization: extra parentheses added for clarity)
(documentation of Miller-Rabin test)
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== Miller-Rabin Primality Test ==
 
== Miller-Rabin Primality Test ==
 
<haskell>
 
<haskell>
  +
-- (eq. to) find2km (2^k * n) = (k,n)
 
find2km :: Integral a => a -> (a,a)
 
find2km :: Integral a => a -> (a,a)
 
find2km n = f 0 n
 
find2km n = f 0 n
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| otherwise = f (k+1) q
 
| otherwise = f (k+1) q
 
where (q,r) = quotRem m 2
 
where (q,r) = quotRem m 2
+
  +
-- n is the number to test; a is the (presumably randomly chosen) witness
 
millerRabinPrimality :: Integer -> Integer -> Bool
 
millerRabinPrimality :: Integer -> Integer -> Bool
 
millerRabinPrimality n a
 
millerRabinPrimality n a
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| x == n' = True
 
| x == n' = True
 
| otherwise = iter xs
 
| 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' :: (Num a, Integral b) => (a->a->a) -> (a->a) -> a -> b -> a
 
pow' _ _ _ 0 = 1
 
pow' _ _ _ 0 = 1
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squareMod :: Integral a => a -> a -> a
 
squareMod :: Integral a => a -> a -> a
 
squareMod a b = (b * b) `rem` 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 :: Integral a => a -> a -> a -> a
 
powMod m = pow' (mulMod m) (squareMod m)
 
powMod m = pow' (mulMod m) (squareMod m)
 
</haskell>
 
</haskell>
   
  +
Example:
  +
  +
<haskell>-- check if '1212121' is prime with several witnesses
  +
> map (millerRabinPrimality 1212121) [5432,1265,87532,8765,26]
  +
[True,True,True,True,True]
  +
</haskell>
   
 
[[Category:Mathematics]]
 
[[Category:Mathematics]]

Revision as of 08:13, 17 August 2011

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 tests and integer factorization:

-- isPrime n = n == head (primeFactors 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
   where
     go n ps@(p:ps')
        | p*p > n        = [n]
        | n `rem` p == 0 =  p : go (n `quot` p) ps
        | otherwise      =      go n ps'

When no other primes source is available, just use

primes = 2 : filter isPrime [3,5..]

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]