Find Square Root under Modulo p | Set 2 (Shanks Tonelli algorithm)

Given a number ‘n’ and a prime ‘p’, find square root of n under modulo p if it exists. 
Examples: 
 

Input: n = 2, p = 113
Output: 62
62^2 = 3844 and 3844 % 113 = 2

Input: n = 2, p = 7
Output: 3 or 4
3 and 4 both are square roots of 2 under modulo
7 because (3*3) % 7 = 2 and (4*4) % 7 = 2

Input: n = 2, p = 5
Output: Square root doesn't exist

We have discussed Euler’s criterion to check if square root exists or not. We have also discussed a solution that works only when p is in form of 4*i + 3 
In this post, Shank Tonelli’s algorithm is discussed that works for all types of inputs.
Algorithm steps to find modular square root using shank Tonelli’s algorithm :
1) Calculate n ^ ((p - 1) / 2) (mod p), it must be 1 or p-1, if it is p-1, then modular square root is not possible.
2) Then after write p-1 as (s * 2^e) for some integer s and e, where s must be an odd number and both s and e should be positive.
3) Then find a number q such that q ^ ((p - 1) / 2) (mod p) = -1 
4) Initialize variable x, b, g and r by following values 

 x = n ^ ((s + 1) / 2 (first guess of square root)
 b = n ^ s 
 g = q ^ s
 r = e (exponent e will decrease after each updation) 

5) Now loop until m > 0 and update value of x, which will be our final answer. 

 Find least integer m such that b^(2^m) = 1(mod p) and 0 <= m <= r – 1 
 If m = 0, then we found correct answer and return x as result
 Else update x, b, g, r as below
 x = x * g ^ (2 ^ (r – m - 1))
 b = b * g ^(2 ^ (r - m))
 g = g ^ (2 ^ (r - m))
 r = m 

so if m becomes 0 or b becomes 1, we terminate and print the result. This loop guarantees to terminate because value of m is decreased each time after updation.
Following is the implementation of above algorithm. 
 

Modular square root of 2 and 113 is 62

For more detail about above algorithm please visit : 
http://cs.indstate.edu/~sgali1/Shanks_Tonelli.pdf
For detail of example (2, 113) see : 
https://math.vt.edu/people/brown/class_homepages/shanks_tonelli.pdf