# Why should a distribution being pseudorandom strictly be a sequence of distributions being pseudorandom

According to Katz Lindell book,

Let Dist be a distribution on l-bit strings. Informally, Dist is pseudorandom if the experiment in which a string is sampled from Dist is indistinguishable from the experiment in which a uniform string of length l is sampled. (Strictly speaking, since we are in an asymptotic setting we need to speak of the pseudorandomness of a sequence of distributions Dist = {$$Dist_{n}$$}, where distribution $$Dist_{n}$$ is used for security parameter n. We ignore this point in our current discussion.) More precisely, it should be infeasible for any polynomial-time algorithm to determine (better than guessing) whether it is given a string sampled according to Dist, or whether it is given a uniform l-bit string.

What do they mean here by sequence of distributions?

A sequence of distributions is a definition of a distribution for each $$n$$. For example, consider the distribution $$Dist_n$$ to be a string of length $$l(n)$$ generated using some pseudorandom generator with a seed of length $$n$$. Then, for each $$n$$, the distribution is actually different. The reason why technically you need this is because the notion of negligible is asymptotic - it needs to be a function so that for every polynomial, the function is smaller than it for some large enough $$n$$.