What are the chances that two messages have the same MD5 digest and the same SHA1 digest?

Question

Given two different messages, A and B (maybe 20-80 characters of text, if size matters at all), what is the probability that the MD5 digest of A is the same as the MD5 digest of

Answer 1

Generally, when one picks N elements randomly it is easier to compute the expected number of collision than the probability of a collision. Since the expected number of collisions cannot be smaller than the probability of a collision it can frequently be used as a suitable upper bound.

Assume that p is the probability that two randomly picked elements collide. If we pick N random elements then there are N*(N-1)/2 pair of elements and hence the expected number of collisions is

p * N * (N-1)/2.

E.g., if we assume that the probability for a collision for both MD5 and SHA1 is p=2^-288 then even after randomly picking 2¹⁰⁰ elements we still only expect about 2^-89 collisions.

Another example: if we pick 2³⁰ random elements and only compute the MD5. Assuming that a collision between two MD5 hashes is p=2^-128 this gives an expected number of 2^-59 for the number of collisions. Hence even the probability that the MD5 hash collides for two inputs is already very small.