4. (40 points) General hash function properties. In this exercise, we discuss the preimage, second preimage, and collision resistance properties of hash functions. (a) Show that collision resistance implies second preimage resistance. (Hint: What does it mean if you can find a second preimage? Now, let h: {0,1} + {0,1}" be a hash function that is both second preimage resistant and collision resistant. We construct another hash function h': {0,1}* {0,1}n+1 by setting Solm h' (m) if m {0,1}"; 1 1 || h(m) else (b) Show that h' is not preimage resistant. () Show that his collision resistant (d) Show that his second preimage resistant. 4. (40 points) General hash function properties. In this exercise, we discuss the preimage, second preimage, and collision resistance properties of hash functions. (a) Show that collision resistance implies second preimage resistance. (Hint: What does it mean if you can find a second preimage? Now, let h: {0,1} + {0,1}" be a hash function that is both second preimage resistant and collision resistant. We construct another hash function h': {0,1}* {0,1}n+1 by setting Solm h' (m) if m {0,1}"; 1 1 || h(m) else (b) Show that h' is not preimage resistant. () Show that his collision resistant (d) Show that his second preimage resistant