Miller Rabin primality test

miller-rabin-prime-test.hs

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+import System.Random (randomRIO)
+import ModularArithmeticUtils (modExp, factorOutTwos)
+
+millerRabinWitness :: Integer -> Integer -> Integer -> Integer -> Bool
+millerRabinWitness n a d s =
+  let x0 = modExp a d n
+  in  x0 == 1 || x0 == n - 1 || loop x0 (s - 1)
+  where
+    loop _ 0 = False
+    loop x r =
+      let x' = (x * x) `mod` n
+      in  x' == n - 1 || loop x' (r - 1)
+
+millerRabin :: Integer -> Integer -> IO Bool
+millerRabin n k
+  | n < 2     = return False
+  | n == 2    = return True
+  | even n    = return False
+  | otherwise = do
+      let (d, s) = factorOutTwos (n - 1)
+      go k d s
+  where
+    go 0 _ _ = return True
+    go i d s = do
+      a <- randomRIO (2, n - 2)
+      if millerRabinWitness n a d s
+        then go (i - 1) d s
+        else return False