jacobi symbol

ModularArithmeticUtils.hs

@@ -25,3 +25,36 @@ factorOutTwos n = go n 0
     go x s
       | even x    = go (x `div` 2) (s + 1)
       | otherwise = (x, s)
+
+-- jacobi symbol (a/n)
+-- (a/n) in {-1, 0, 1}
+jacobi :: Integer -> Integer -> Integer
+jacobi a n
+  | n <= 0         = error "jacobi: n must be positive"
+  | even n         = error "jacobi: n must be odd"
+  | a `mod` n == 0 = 0
+  | otherwise      = go (a `mod` n) n 1
+  where
+    go :: Integer -> Integer -> Integer -> Integer
+    go 0 _ _ = 0
+    go 1 _ s = s
+    go x m s =
+      let (xOdd, e) = factorOutTwos x
+          s' = if even e then s else s * jacobi2 m
+      in
+      if xOdd == 1
+        then s'
+        else
+          let s'' = if (xOdd `mod` 4 == 3) && (m `mod` 4 == 3) then -s' else s'
+              xNext = m `mod` xOdd
+          in go xNext xOdd s''
+
+-- Jacobi(2, m), helper function
+jacobi2 :: Integer -> Integer
+jacobi2 m =
+  case m `mod` 8 of
+    1 -> 1
+    7 -> 1
+    3 -> -1
+    5 -> -1
+    _ -> error "jacobi2: unexpected (m mod 8) for odd m"