4. [Insecure PRF] Let F be a secure PRF defined over (K,X,Y) where K = X = Y = {0,1}", Show that a PRF F(k, (x|lx') ) := F(kx) F(kx') is insecure. Note that F, is defined over the input space {0,1), X.X'E {0,1}", and (x||x') denotes a concatenation of x and x'. Hints and tips: Construct an adversary that distinguishes Fy from a random function. Remember that by the definition of PRF, an adversary is allowed to query PRF on arbitrary inputs. Check the definition of PRF and observe that under a fixed key k, the PRF F(k,-) returns the same values when queried on the same inputs. Think of a PRF as encryption function of a block cipher. E.g., for a fixed key k, when requested to encrypt the same plaintexts, Episk,-) returns the same ciphertexts. Example on Slide 16 of Lecture 3-2 may be helpful. 4. [Insecure PRF] Let F be a secure PRF defined over (K,X,Y) where K = X = Y = {0,1}", Show that a PRF F(k, (x|lx') ) := F(kx) F(kx') is insecure. Note that F, is defined over the input space {0,1), X.X'E {0,1}", and (x||x') denotes a concatenation of x and x'. Hints and tips: Construct an adversary that distinguishes Fy from a random function. Remember that by the definition of PRF, an adversary is allowed to query PRF on arbitrary inputs. Check the definition of PRF and observe that under a fixed key k, the PRF F(k,-) returns the same values when queried on the same inputs. Think of a PRF as encryption function of a block cipher. E.g., for a fixed key k, when requested to encrypt the same plaintexts, Episk,-) returns the same ciphertexts. Example on Slide 16 of Lecture 3-2 may be helpful