Tonelli shanks

Tonelli shanks algorithm is used for finding modular square root of number n under mod p in minimal time

def is_quadratic_residue(n,p):
    '''check if the n is qudratic residue in mod p using legendre symbol'''
    if n%p==0:
        return True
    return pow(n,(p-1)//2,p)==1

def tonelli_shanks(n,p):
    '''find the modular square root of n in mod p if possible'''
    if n%p==0:
        # when n = o(mod p) square root of 0 is 0
        return 0
    if not is_quadratic_residue(n,p):
        print("square root does not exits")
        return None
    
    if p%3==4:
        return pow(n,(p+1)//4,p)
    
    # for all p%4==1 we will apply tonelli shanks algorithm
    # step 1
    # p-1 = 2^s.q
    q=p-1
    s=0
    while(q%2==0):
        s+=1
        q=q//2
    # step 2
    n_res = 2
    while is_quadratic_residue(n_res,p):
        n_res+=1
    print(n_res)
    # step 3
    m=s
    c=pow(n_res,q,p)
    t=pow(n,q,p)
    r=pow(n,(q+1)//2,p)
    
    # step 4
    while(t!=1):
        i=0
        temp=t
        while(temp!=1):
            i+=1
            temp = (temp*temp)%p
        pow2 = pow(2,m-i-1)
        b=pow(c,pow2,p)
        m=i
        c = (b**2)%p
        t = (t*b**2)%p
        r = (r*b)%p
    
    return r

x=tonelli_shanks(5,41)
print(x)