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

an addendum to Welbog's post:

Ratios of large factorials can be computed without using arbitrary-precision arithmetic, by using Stirling's approximation:

n! ≈ sqrt(2πn) * (n/e)ⁿ

So (S!)/(S^N * (S - N)!) ≈ sqrt(2πS)/sqrt(2π(S-N))*(S/e)^S/((S-N)/e)^S-N/S^N

= sqrt(S/(S-N)) * (S/(S-N))^S-N * e^-N

= sqrt(1 + α) * (1 + α)^S-N * e^-N where α = N/(S-N) is small.

The approximation (1+a/n)^nx ≈ e^ax holds as n → ∞ (or at least becomes very large)

** so this means (1+(N/(S-N)))^S-N ≈ e^N for S-N >> N.

So I would expect that

(S!)/(S^N * (S - N)!) ≈ sqrt(1 + N/(S-N)) * e^N * e^-N = sqrt(1 + N/(S-N)) for S-N >> N....

except this is greater than 1... so one of the approximations isn't good enough. :p

(** caveat: N/S has to be small: for N=22,S=365 this is off by a factor of 2)