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I´ve just started with crypto and I am trying to solve an exercise sheet. I am, however, not good at mathematical proofs and got a bit stuck on a certain - probably easy - question:

I´ve got three hash functions:

  • $H_1:= \{0,1\}^* \to \{0,1\}^a$
  • $H_2:= \{0,1\}^* \to \{0,1\}^b$
  • $H_3:=\{0,1\}^* \to \{0,1\}^b$

And I have to proof that: If at least one of $H_1, H_2, H_3$ are collision resistant $M:= \{0,1\}^* \to \{0,1\}^{a+b}$ for $M:= H_1(x)||H_2(x)||H_3(x)$ is collision resistant, too.

Intuitively I would say that $H_2$ and $H_3$ are not collision resistant, since they are shortened to the same length $b$.

I have started with assuming that $M$ is not collision resistant by changing the definition to:

$h(x_1) = h(x_2) = h(x_3) $ for a triplet $(x_1,x_2,x_3)$.

But I am not quite sure, if this is the correct way to do it.

Can I even proof it in one go? Or do I need several steps - like proofing $H_2 = H_3$ first?

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  • $\begingroup$ I'd start with the definition of "collision resistant". Do you have a definition for that from the textbook or class notes? $\endgroup$ – mikeazo Jan 5 '17 at 15:21
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    $\begingroup$ Also, in your notation, you have that the range of $M$ is $\{0,1\}^{a+b}$, should that be $a+2b$? $\endgroup$ – mikeazo Jan 5 '17 at 15:22
  • $\begingroup$ I´ve got one from the lecture notes. It says: " A non-keyed hash function is an efficent function $H: M \to \ T$ And a collision of $H$ is a Tuple $(m_0,m_1)$ with $H(m_0) = H(m_1)$. An it is resistant, if no "efficent" adversary A is known, that finds a collision." This is the one I modified for the triplet. $\endgroup$ – Ajacia Jan 5 '17 at 15:35
  • $\begingroup$ And the range really is the one given. $\endgroup$ – Ajacia Jan 5 '17 at 15:38
  • $\begingroup$ I don't see how the range can be correct. The output of $H_1$ is $a$ bits long, the outputs of $H_2$ and $H_3$ are $b$ bits long each. So if you concatenate them, you should end up with $a+b+b$ (or $a+2b$) bits. $\endgroup$ – mikeazo Jan 5 '17 at 16:15
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Ignoring the issue of output length I mention in the comments, you should be able to prove this without intermediate steps. Here is an outline of how I would do it.

  1. Without loss of generality let one particular hash function be the one that is collision resistant. It doesn't matter which one. Say $H_3$.
  2. Assume that $M$ is not collision resistant. This means, you can find $m_0\neq m_1$ such that $M(m_0)=M(m_1)$.
  3. Show how this would lead to a collision in $H_3$ directly. This leads to a contradiction, so $M$ must be collision resistant.

This is a proof by contradiction.

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  • $\begingroup$ Now I got it. =) Thank you. (And I´ll ask my tutor about the issue you mentioned.) $\endgroup$ – Ajacia Jan 5 '17 at 16:59

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