Consider a consequence of Euclid's Lemma: a,b,c Z, gcd (a,b) =1 Aa | bca | b and Fermat's Little Theorem: If p is any prime number and a is any integer such that p Xa⇒a=¹= 1 (mod p). Show how these statements are used to show an RSA cypher will work. It is possible in RSA cryptog- raphy to encode a cypher with (pq, e) as public key, by using C=Me (mod pq) and then decode the cypher with d=-e (mod (p-1) (q-1)), using M = Cd (mod pq)

This Intro to Elementary Number Theory Homework problem is very difficult.

Answer the question and gets thumbup

4. Derive each of the following congruences: (b) a² = a (mod 42) for all a.

Both part (a.) and (b.) please.

Fermat's Little Theorem says that for all natural numbers p and integers x, if p is prime then xº = x mod p. (i) Show that the statement is wrong, if we drop the assumption that p is prime. [Remember: All you need to do is to give a counterexample.] (ii) Show, by giving an example, that if p is not prime, then xP = x mod p can still be true for some integers x with x -1, 0, 1.