Question

5. (Hash, Collision Resistance, Problem 4.9 of D. Stinson, 20 pts) Suppose h1 : {0, 1}2m ? {0, 1}m is a collision resistant hash function. Define h2 : {0, 1}4m ? {0, 1}m as follows: (a) Write x ? {0, 1}4m as x = x1||x2, where x1, x2 ? {0, 1}2m. (b) Define h2(x) = h1(h1(x1)||h1(x2)). Prove that h2 is collision resistant. Hint: Prove by contradiction that h2 is collision resistant: assume that there exists x1, x2 ? {0, 1}4m, such that x1 ? x2, but h2(x1) = h2(x2). Also use the fact that h1 is collision resistant, i.e., there does not exist x1, x2 ? {0, 1}2m such that x1 ? x2 but h1(x1) = h1(x2); in other words, h1(x1) = h1(x2) implies x1 = x2.

          5. (Hash, Collision Resistance, Problem 4.9 of D. Stinson, 20 pts) Suppose h1 : {0, 1}2m ? {0, 1}m is a collision resistant hash function. Define h2 : {0, 1}4m ? {0, 1}m as follows:
(a) Write x ? {0, 1}4m as x = x1||x2, where x1, x2 ? {0, 1}2m.
(b) Define h2(x) = h1(h1(x1)||h1(x2)).

Prove that h2 is collision resistant.
Hint: Prove by contradiction that h2 is collision resistant: assume that there exists x1, x2 ? {0, 1}4m, such that x1 ? x2, but h2(x1) = h2(x2). Also use the fact that h1 is collision resistant, i.e., there does not exist x1, x2 ? {0, 1}2m such that x1 ? x2 but h1(x1) = h1(x2); in other words, h1(x1) = h1(x2) implies x1 = x2.

5. (Hash, Collision Resistance, Problem 4.9 of D. Stinson, 20 pts) Suppose h1 : 0, 12m ? 0, 1m is a collision resistant hash function. Define h2 : 0, 14m ? 0, 1m as follows:
(a) Write x ? 0, 14m as x = x1||x2, where x1, x2 ? 0, 12m.
(b) Define h2(x) = h1(h1(x1)||h1(x2)).

Prove that h2 is collision resistant.
Hint: Prove by contradiction that h2 is collision resistant: assume that there exists x1, x2 ? 0, 14m, such that x1 ? x2, but h2(x1) = h2(x2). Also use the fact that h1 is collision resistant, i.e., there does not exist x1, x2 ? 0, 12m such that x1 ? x2 but h1(x1) = h1(x2); in other words, h1(x1) = h1(x2) implies x1 = x2.

Added by Dennis E.

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