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Let's say I have a file A, which is any normal file (pdf, jpeg, mp3 etc.)
Now I get the binary dump of file, say another file B{A}.
And the hexdump of file say, file H{A}.
Now I hash all the three files with any 256 bit HASH (SHA256, BLAKE256 etc.)
I want to know that :
1. What is the possiblity of hash collision in this case (Considering if somehow I find a collison in case of file A, I still can generate the hex and binary dump of that file to see if hashes of B{A} and H{A} matches or not).
2. Will it still be 1/256* 256* 256? Or
3. Will there be 0 collision ? (Considering collision between exactly same size of files)

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Approximating a Hash function $h$ as a function that maps each word in $\{0,1\}^*$ to a word in $\{0,1\}^{256}$ chosen uniformly at random, you can find the probability that at least one of the events:

  • $h(A)=h(B\{A\})$,
  • $h(A)=h(H\{A\})$,
  • $h(B\{A\})=h(H\{A\})$

is true by looking at the complementary probability. The probability that $h(A)$, $h(B\{A\})$, and $h(H\{A\})$ are all distinct is $1 \cdot (1 - \frac{1}{2^{256}}) \cdot (1 - \frac{2}{2^{256}})$. Then, the probability of having at least one collision is then: $$ 1- \left(1 - \frac{1}{2^{256}}\right) \cdot \left(1 - \frac{2}{2^{256}}\right) \approx 2.6 \cdot 10^{-77}. $$

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  • $\begingroup$ Thank you for explanation. $\endgroup$
    – Akash
    Sep 6 '21 at 3:59

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