8.4-8.5, 8.7

1. I take it the approximation for birthday attack probabilities is taken from Stirling's approximation of large factorials. I need a way to obtain an intuition regarding probabilities. Otherwise, I'm always crunching numbers, and I get distracted.

2. In some circumstances, a probabilistic algorithm is sufficient. This is especially intriguing considering a deterministic primality test exists, but probabilistic tests are still in vogue. I wonder if this is because the running time of the deterministic is significantly higher than the the currently employed ones. However, the discrete logarithm birthday attack is not used because it has a higher time complexity than Baby Step, Giant Step. What I'm getting at is that probabilistic algorithms are probabilistic on the input and not a random process. Otherwise, we could just run the algorithm independently many times until we obtained a solution.