module ModularArithmeticUtils (modExp, modMul, legendre, factorOutTwos) where

import Data.Bits (testBit, shiftR)

-- Modular exponentiation: compute a^b mod m
modExp :: Integer -> Integer -> Integer -> Integer
modExp b 0 m = 1
modExp b e m = t * modExp ((b * b) `mod` m) (shiftR e 1) m `mod` m
            where t = if testBit e 0 then b `mod` m else 1

-- Compute (a * b) mod m
modMul :: Integer -> Integer -> Integer -> Integer
modMul a b m = (a * b) `mod` m

-- Compute the Legendre symbol (a/p)
-- (a/p) = a^((p-1)/2) mod p
-- (a/p) in {-1, 0, 1}
legendre :: Integer -> Integer -> Integer
legendre a p = modExp a ((p - 1) `div` 2) p

-- Factor p-1 as q * 2^s, where q is odd
factorOutTwos :: Integer -> (Integer, Integer)
factorOutTwos n = go n 0
  where
    go x s
      | even x    = go (x `div` 2) (s + 1)
      | otherwise = (x, s)