The encrypt and authenticate scheme is not always secure however, this does not mean that it is never secure. Consider the following:

Where F is a PRF: k= (k1, k2), where k1, k2{0,1}^n. To encrypt, compute encryption by c= (r, Fk1(r)m) and message authentication by t=Fk2(m), and output (c, t).

Prove or refute Is the encrypt and authenticate combination secure when considering only the problem of message authentication?

Prove or refute: Does this encrypt-and-authenticate combination yield a CPA-secure encryption?