September Morn Update

Task/Multiplicative-order/Haskell/multiplicative-order-1.hs

@@ -1 +1,15 @@
-primeFacsExp :: Integer -> [(Integer, Int)]
+powerMod
+  :: (Integral a, Integral b)
+  => a -> a -> b -> a
+powerMod m _ 0 = 1
+powerMod m x n
+  | n > 0 = f x_ (n - 1) x_
+  where
+    x_ = x `rem` m
+    f _ 0 y = y
+    f a d y = g a d
+      where
+        g b i
+          | even i = g (b * b `rem` m) (i `quot` 2)
+          | otherwise = f b (i - 1) (b * y `rem` m)
+powerMod m _ _ = error "powerMod: negative exponent"

Task/Multiplicative-order/Haskell/multiplicative-order-2.hs

@@ -1,9 +1,17 @@
-powerMod :: (Integral a, Integral b) => a -> a -> b -> a
-powerMod m _ 0 =  1
-powerMod m x n | n > 0 = f x' (n-1) x' where
-  x' = x `rem` m
-  f _ 0 y = y
-  f a d y = g a d where
-    g b i | even i    = g (b*b `rem` m) (i `quot` 2)
-          | otherwise = f b (i-1) (b*y `rem` m)
-powerMod m _ _  = error "powerMod: negative exponent"
+import Data.List (foldl1') --'
+
+foldl1_ = foldl1' --'
+
+multOrder a m
+  | gcd a m /= 1 = error "Arguments not coprime"
+  | otherwise = foldl1_ lcm $ map (multOrder_ a) $ primeFacsExp m
+
+multOrder_ a (p, k) = r
+  where
+    pk = p ^ k
+    t = (p - 1) * p ^ (k - 1) -- totient \Phi(p^k)
+    r = product $ map find_qd $ primeFacsExp t
+    find_qd (q, e) = q ^ d
+      where
+        x = powerMod pk a (t `div` (q ^ e))
+        d = length $ takeWhile (/= 1) $ iterate (\y -> powerMod pk y q) x