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Let (Gen1,H1) and (Gen2,H2) be two hash functions. Define (Gen,H) so that Gen runs Gen1 and Gen2 to obtain keys s1 and s2, respectively. Then define Hs1,s2(x)=Hs1(x)||Hs2(x).

a) Prove that if at least one of (Gen1,H1) and (Gen2,H2) aris collision resisistant, then (Gen,H) is collision resistant.

b) Determine whether an analogous claim holds for second pre-image resistance and pre-image resistance, respectively. Prove your answer in each case.

Now assume H is any collision reistant hash function. Then is the composition H o H necessarily collision resistant?

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    $\begingroup$ This isn't a question answering service. What have you tried? Where are you stuck? $\endgroup$ – mikeazo Nov 17 '14 at 14:21
  • $\begingroup$ Does the proof-of-work tag even related here? $\endgroup$ – Bush Nov 17 '14 at 23:26
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Since it's your homework, I'll only hive you the direction and not a full formal proof:

a) if at least one of H1 and H2 are collision resistant then H is too. Because if H is not coll-resistant then it's possible to find x and x' s.t. H(x)=H(x') so H(x)=H1(x)||H2(x)=H1(x')||H2(x')=H(x') which means that H1(x)=H1(x') and H2(x)=H2(x') so we found a collision in both H1 and H2.

b) About preimage res, if H is not preimage-res then given y it is possible to find x such that H(x)=y. We know that y=y1||y2=H1(x)||H2(x), thus you can find the preimage of y in both H1 and H2 as before.

c) about the composition, it is clear that a composition is also coll-res. Assume that H(H(•)) is not coll-res then it immediately means that H(•) is not coll-rea, by contradiction to the fact that H is coll-res.

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