Theorem 5: If a prime p ≡ 3 mod 4, then ≡ ∀  either is a or –a is a non-quadratic residue.

Proof: If p ≡ 3 mod 4, then p will be of the form 4l + 3 for some integer l, i.e., p = 2k+1 for some odd  number k where k = 2l+1. We prove the theorem using contradiction. Assume both a and –a are quadratic residues modulo p. We then have x² ≡ a mod p and y² ≡ -a mod p for some x, y ∈Z^*_p . From this we have x^2k ≡ ak mod p and y^2k ≡ (-1)^ka^k mod p. Since k is odd x^2k mod p and y^2k mod p must have opposite signs. But from Fermat’s little theorem, both x^2k and y^2k must be congruent to 1 mod p, which contradicts the assumption. Hence either a or –a is a non-quadratic residue.

If p ≡ 3 mod 4 then either p ≡ 3 mod 8 or p ≡ 7 mod 8. Using the previous theorems we can easily show that the first case 2 is the generator and in the second case p-2 is the generator.  Thus using this characterization we can compute the generator of any odd prime p ≡ 3 mod 4 in O(1) time.

Now we will prove an important theorem for finding out the square root of any quadratic residue.

Theorem 6: If a ∈Z^*_p is a quadratic residue then its square root is mod p.
Proof: Clearly we can see that mod p  =  mod p = mod p = a mod p. This is because Legendre symbol mod p = +1, since a is a quadratic residue.
We can use the above theorem to compute the square root of any quadratic residue  deterministically using modular exponentiation when p ≡ 3 mod 4.        

Reference:

1.   Introduction to Algorithms , Second Edition, T. H. Cormen, C. E. Leiserson, R. Rivest and C. Stein, Prentice Hall India .

2.   Introduction to Analytic Number Theory , T. M. Apostol, Springer International .

3.  Randomized Algorithms , R. Motwani & P. Raghavan, Cambridge University Press .