Let p be an odd prime and let g be a primitive root modulo p. Prove that a has a square root modulo p if and only if its discrete logarithm logg(a) modulo p−1 is even.

For each part, use the data provided to find values of a and b satisfying a2 ≡ b2 (mod N), and then compute gcd(N, a − b) in order to find a nontrivial factor of N, as we did in Examples 3.37 and 3.38. (c) N = 198103 11892 ≡ 27000 (mod 198103) and 27000 = 23 · 33 · 53 16052 ≡ 686 (mod 198103) and 686 = 2 · 73 23782 ≡ 108000 (mod 198103) and 108000 = 25 · 33 · 53 28152 ≡ 105 (mod 198103) and 105 = 3 · 5 · 7   Here is an example of a similar problem solved in the book: We factor N = 914387 using the procedure described in Table 3.4. We first search for integers a with the property that a2 mod N is a product of small primes. For this example, we ask that each a2 mod N be a product of primes in the set {2, 3, 5, 7, 11}. Ignoring for now the question of how to find such a, we observe that 18692 ≡ 750000 (mod 914387) and 750000 = 24 · 3 · 56 19092 ≡ 901120 (mod 914387) and 901120 = 214 · 5 · 11 33872 ≡ 499125 (mod 914387) and 499125 = 3 · 53 · 113 None of the numbers on the right is a…