## Shank Tonelli Algorithm

Shanks tonelli algorithm is for finding square roots modulo prime number.

It solves where n is quadratic residue (mod p ) and p is odd prime. Implementation in Haskell.

import Data.List --Calculates x^2=n (mod p) where p is prime n is quadratic residue. legendrE::Integer->Integer->Integer legendrE a p = modpoW a (div (p-1) 2) p modpoW::Integer->Integer->Integer->Integer modpoW a d n= fuN a d where fuN a 1 = mod a n fuN a 2 = mod (a^2) n fuN a d = let p=fuN (mod (a^2) n) (div d 2) q=if odd d then mod (a*p) n else p in mod q n shankS::Integer->Integer->Integer shankS n p | legendrE n p /= 1 = error "non quadratic residude" | mod p 4 == 3 = n^(div (p+1) 4) |otherwise = let q = until (\x->mod x 2 /= 0) (\x->div x 2) $ pred p s = until (\x->q*2^x == pred p) (+1) 0 z = until (\x->legendrE x p == pred p) (+1) 2 c = modpoW z q p r = modpoW n (div (q+1) 2) p t = modpoW n q p m = s in tonellI p c r t m tonellI::Integer->Integer->Integer->Integer->Integer->Integer tonellI p c r t m = if mod t p == 1 then r else let i =repsqR (t^2) p 1 --until (\x->modpoW t (2^x) p == 1) (+1) 1 b = modpoW c (2^(m-i-1)) p r' = mod (r*b) p t' = mod (t*b^2) p c' = mod (b^2) p m' = i in tonellI p c' r' t' m' repsqR t p cnt| mod t p == 1 =cnt | otherwise = repsqR (t^2) p (cnt+1) main::IO() main=do n<-fmap read getLine p<-fmap read getLine; print $ shankS n p

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