Consider the following variants of Merkle–Damgård, where we start with the original construction and make a small change

Problem 3. (20 points) Consider the following variants of Merkle–Damgård, where we start with

the original construction and make a small change. Do the resulting constructions remain secure?

Justify your answers. You may assume that ? and ℓ are at least reasonably large numbers, say at

least 8.

(a) Variant 1: The first bit of the padding block is removed, i.e. ?? := 00…00||⟨?⟩.

(b) Variant 2: Instead of outputting ??, output (??−1,??), i.e. we output the hash of the first (?−1)

blocks along with the last block in plain text.

Problem 4. (20 points) Show that truncating the output of a CRHF even by one bit does not

necessarily result in a CRHF. That is, assume there exists a collision-resistant hash function ? : {0, 1}* → {0, 1}?.

Use ? to construct ?′ : {0, 1}* → {0, 1}^? such that ?′ is a CRHF, but ?′′(?) := ?′(?)[: −1] (in Python notation) is not a CRHF.

Hint: The easiest solution is to artificially modify ? so that it has a collision on the first ? − 1

bits of the output for some messages.

Problem 5. (15 points) Let (?,?) be a CPA-secure cipher defined over (?,ℳ, ?) and let ? : ? → ? be a CRHF.

Define the following cipher (?′,?′) over (?,ℳ, ? × ? ):

?′(?,?) := (?,?(?)) where ? ? ←− ?(?,?);

?′(?, (?1, ?2)) := {︃?(?, ?1), if ?(?1) = ?2; reject, otherwise}.

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