Alice and Bob secretly agree on a hash function from a $-$independent fam$-$ ily of hash functions. Each maps from a universe of keys to $,$ where is prime. Later, Alice sends a message to Bob over the internet, where $.$ She authenticates this message to Bob by also sending an au$-$ thentication tag $,$ and Bob checks that the pair he receives indeed satis$$es $.$ Suppose that an adversary intercepts en route and tries to fool Bob by replacing the pair with a different pair $.$ Argue that the probability that the adversary succeeds in fooling Bob into ac$-$ cepting is at most $,$ no matter how much computing power the ad$-$ versary has, even if the adversary knows the family of hash functions used.