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Unformatted text preview: ion (family) and deﬁne the function H (x) =
H (H (x)). Show that H is not generically collisionresistant (you may assume that H is
known).
(b) Suppose we design a hash function (family) HK : {0, 1}∗ → {0, 1} from the block cipher E :
{0, 1}k × {0, 1} → {0, 1} by ﬁxing a key K ∈ {0, 1}k and setting H (m) = CBCMACE (M ),
K
that is, we divide m into bit blocks m1 , m2 , . . . , mt , set c0 = 0 , ci = EK (mi ⊕ ci−1 , and let
H (m) = ct . Show that this hash function (for known K ) is not collision resistant.
3. MACs and Hashes together. [15 points]
(a) [5 points] Let H : {0, 1}∗ → {0, 1} be a collision resistant hash function (family). Let LBt (x)
return the last t bits of x. Prove that the hash function (family) H (x) = H (x)LBt (x) is
collision resistant.
(b) [10 points] Let H : {0, 1}∗ → {0, 1} be a collisionresistant hash function (family) and
MK : {0, 1}n → {0, 1}t be a (EUFCMA) secure ﬁxedlength MAC. Let FK : {0, 1}∗ →
{0, 1} compute a tag FK (m) by dividing m into nbit blocks m1 , . . . , m...
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This document was uploaded on 04/03/2014.
 Spring '14

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