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I am puzzled with a question that seems to be based on theory. If there is a collision resistant hash function (Since it is not possible for a hash function to be collision free this is a theoretical question) Would it be true to say that there are no two unique hash values for all possible messages x and x' where x ≠ x’? I feel like this would be true....can anyone back me up on this? |
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As you correctly observed, for any function $H\colon \{0,1\}^\ast\to\{0,1\}^n$ collisions must exist, simply because $\{0,1\}^\ast$ is an infinite set and $\{0,1\}^n$ is finite. One could define "hash function" to mean something else (e.g., taking only a bounded input length), but this is non-standard (and of questionable value). However, the term "collision resistance" denotes something different from what you seem to think: It just means that collisions are hard to find, in a computational sense. We expect from a good hash function that nobody is able to find two inputs that lead to the same hash value, but we do not care about their abstract existence, which is a necessary evil. Thus, no: Collision resistance does not mean that $H(x)\neq H(x')$ for any $x\neq x'$. It rather means that practically, nobody should ever find $x\neq x'$ with $H(x)=H(x')$. |
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