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(b) [5 points] Prove that the functional graph of any hash function with a ﬁxed output length
must have cycles.
2 cycle length when we use h16 (·tail length is deﬁned print out these 6edges of the path to a cycledeﬁnedain
2.35. To avoid the confusion, ). Your output should as the number numbers. Tail and cycle are from point.
2.35. To avoid the confusion,number of component isthe tail length of node 13 is 3, tailcycle fromnode 12 is 1,
In the following Figure, the tail length is deﬁned as 2, number edges of the path to a length of a point.
In the followingof 6 is 0. Average tailcomponent is 2, tail length 1.75. (i.e. average of tail length starting from
and tail length Figure, the number of length is (3+2+1+1)/4 = of node 13 is 3, tail length of node 12 is 1,
and terminal points is 0. do not have the preimage.) The cycle1.75. (i.e. average of tail 4.
the tail length of 6 that Average tail length is (3+2+1+1)/4 = length of [1, 4, 6, 9]...
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 Spring '14
 Cryptography, hash function, Cryptographic hash function, Block cipher, tail length

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