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Please give help! how can I calculate the probability of collision? I need a mathematical equation for my studying.

Assume, I am using SHA256 to hash 100-bits. Thus:

SHA256 {100} = 256-bits (hash values) I would like to know the probability of collision.

This website: https://preshing.com/20110504/hash-collision-probabilities/

Gave an equation: $K^2 / 2N$

But I could not know what is $K$ and $N$? Is K is the output which is (256-bits) and $N$ is the input which is (100-bits)

Please Help and Thanks in advance...

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    $\begingroup$ Umm... did you read the entire web page you linked to? It defines $k$ and $N$ very close to the top. $\endgroup$ – Ilmari Karonen Dec 8 '18 at 15:26
  • $\begingroup$ $n$ is the output size of the hash function, $k$ is the randomly generated unique inputs. it is $k^2/2n$ $\endgroup$ – kelalaka Dec 8 '18 at 15:28
  • $\begingroup$ well, then based on the equation: 100^2 / 2^256 right? $\endgroup$ – Al-Ani Dec 8 '18 at 15:34
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$n$ is the output size of the given hash function. To find a collision you try randomly generated $k$ different inputs. When we say the output size is $n$, it means that the output space has $2^n$ elements.

What you see on that website is the general case of the collision probability. We normally talk about the 50% probability (birthday attack) on the hash collisions as

$$ k = \sqrt{2^n}$$ You can also see the general result from the birthday paradox.

To have birthday attack with 50% percentage you will need $k = 2^{128} \approx 4.0 × 10^{38}$ randomly generated differently input for a hash function with output size $n= 256$

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