Question

Derive an equation for calculating how many hashes it would take to find a hash value under a particular value for SHA-256. So for example, if you have some ceiling value x, f(x) should tell us the probability that a hash of a random message would be below the value x. For example, if x=1, then only a hash value of 0 would result in something under that value, for 1/(2^256). If you cannot derive the equation, then try calculating specific values and graph the result to get an idea

Answer 1

• There are about 2^256 total combination of hashes in SHA 256., let' call this value N
• The given ceiling value is x. So, there are x values to check for collision.
• The probability that none of the x values will have collision is given by:

This value is approximately equal to

From here , we can say that the atleast one of values below x collides with the hash is given by

Substituting N, we get the answer: