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I'm reading a proof of the proposition of CollisionResistant Hashing Towards Making UOWHFs Practical

Suppose there exists a compression function $F: \Sigma^k\times\Sigma^{c+m'} \rightarrow \Sigma^c$ with $m'>k$ such that $F$ is $(t',\epsilon')$-resistant to target collisions. Then there exists a compression function $H$ such that $H$ is $(t,\epsilon)$-resistant to target collisions for $t=t'- \Theta (k+m') $ and $\epsilon' = \epsilon + 2^{-k+1}$

To show that, the author constructs the following hash function: $$H(K,x||y||z) = H_K(x||y||z)= \begin{cases} F_K(x||y||z)||K & \text{if }y\neq K \\1^c||1^k & \text{if }y = K \end{cases}$$

Here is the proof:

Blockquote

My questions:

  • I don't understand why a $CF'$ run in time $t+O(k+c+m)$ I know why he add $t$ but I don't undertand Why $O(k+c+m)$?

  • Why a $ProbSuccess(CF',F)\geq 1-2^{-k+1}$? I understand that means $2^{-k}$ but I don't Why $2^{-k}*2$

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  • $\begingroup$ @jauninf- I've suggested a couple of minor edits for clarity (writing the hash function in cases, - into bullets), hope they're helpful $\endgroup$ – figlesquidge Nov 14 '13 at 13:19

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