In this problem we will show that if N = pq is the product of two odd primes, and if x is...

In this problem we will show that if N = pq is the product of two odd primes, and if x is chosen uniformly at random between 0 and N − 1, such that gcd(x, N) = 1, then with probability at least 3/8, the order r of x mod N is even,

and moreover xr/2 is a nontrivial square root of 1 mod N.

(a) Let p be an odd prime and let x be a uniformly random number modulo p. Show that the order of x mod p is even with probability at least 1/2. (Hint: Use Fermat’s little theorem (Section 1.3).)

(b) Use the Chinese remainder theorem (Exercise 1.37) to show that with probability at least 3/4, the order r of x mod N is even.

(c) If r is even, prove that the probability that xr/2 ≡ ±1 is at most 1/2.