Jacobi Symbol (Haskell)

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Other implementations: Erlang | Haskell | Python

The Jacobi symbol satisfies the six properties listed below, and these can be used to compute the Jacobi symbol in polynomial time.

1. <<property 1>>=
   jacobi a n | even a         = jacobi 2 n * jacobi (a `div` 2)  n
   

2. For ,.

   <<property 2>>=
              | a >= n         = jacobi (a `mod` n) n
   

3. For odd coprimes a and n, .

   <<property 3>>=
              | a `mod` 4 == 3 &&
                n `mod` 4 == 3 = - jacobi n a
              | otherwise      = jacobi n a
   

4. .

   <<property 4>>=
   jacobi 1 _ = 1
   

5. .

   <<property 5>>=
   jacobi 2 n = case n `mod` 8 of
                  1 -> 1
                  3 -> -1
                  5 -> -1
                  7 -> 1
   

6. .

   <<property 6>>=
   jacobi 0 _ = 0
   

From these properties we can come up with a polynomial time algorithm.

The function always complete as property 1 can be used to remove all even factors, and these will end at property 4. The odd factors will be reduced by property 2 if and if a < n property 3 will swap a and n and property 2 will again reduce a until it matches property 4 or property 6.

<<jacobi symbol>>=
property 4
property 5
property 6
property 1
property 2
property 3

This can be run from the command line with the arguments on the command line

<<jacobi.hs>>=
import System.Environment
jacobi symbol
main = do args <- getArgs
          let a = read (args !! 0)
              b = read (args !! 1)
          print (jacobi a b)

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