about entropy and random number generator I am reading an article about cryptography of information and Pseudo Random Number Generator. The article said using 1$28$-bit entropy PRNG is safe in some cases of cryptography. I am not familiar with this field. But after doing some research, I got some understanding listed as follows (correct me if it is not correct)


entropy is used to measure how random the information is. One typical definition of entropy is the Shannon entropy which is defined to as $$H(F) = -\sum P(x) \log_2(P(x))$$ So if the information is carried by 128-bit data so each bit is occurred at probability $P(x)=1/2^{128}$, thus the entropy is 128.
I read several posts and they comes with a conclusion that PRNG does not add or increase entropy.


So based on above 2 points, can I say that a $n$-bit entropy PRNG is basically a random generator with initial state chosen as $n$-bit data not the algorithm of PRNG per se (assuming the PRNG implementation takes $128$-bit data of course). Is that correct?

If it is true, my question is as follow:

Should I get any $128$-bit data as initial state will make it result from PRNG random enough or that $128$-bit data should be prepared in specific way ?
Yes, this is correct. (up to the typos that "so each bit is occurred at" in 1. should read "so each 128-bit value occurs at", and that the unit "bit" is missing at the end of that sentence; as well as the not-mentioned assumption that every time a 128-bit value is randomly picked, its value is independent of the choices take before). If the state of a cryptographically secure PRG is exactly 128 bit, then it's not allowed to "loose" any entropy during operation to be 128-bit secure, and it has to be seeded by a perfectly uniform random source.

The problem is how to choose the initial state. An attacker should have only a chance of $2^{-128}$ to guess the initial state, as (s)he could otherwise just calculate the output of the PRG (after guessing the initial state) to predict its output. So you have to find a way to pick a uniformly distributed random 128-bit value for the initial value, which is a real problem: you need a random source (that cannot be observed by the attacker) and a way to extract 128 bit entropy.