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?