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3

There are no security advantages to evaluating the polynomial at random places instead of sequential. The information theoretic security proof of Shamir secret sharing does not depend on the evaluation points being chosen in any specific manner.

1

Low entropy for PRNGs is bad. If you set up a VM by automatized software, you won't have entropy for your PRNG which is not good. What would be the risk level? Weak cryptographic keys (pretty bad if this is f.ex. a SSH log-in key or someting like this). Seeding the VM with entropy rich data from the host is the standard solution and will solve the problem. ...

1

Consider what it means for a key $k$ to be chosen according to some distribution over key space $\kappa$: Assume that $\mathsf{Gen}$ picks key $k$ from key space $\kappa$ with probability $p$. Since $\mathsf{Gen}$ is randomized, this means that a $p$-fraction of all the random tapes will lead it to generate $k$ as the key. If we now conceptually redefine ...

5

Sorry for the late answer, I got busy... So, you know that $\mathsf{Gen}$ is a probabilistic algorithm. What's a probabilistic algorithm? It's an algorithm which, during its execution, can make some random choices, which can be modeled as coin tosses. In programming terms, the algorithm can use a special coin-tossing function, which returns $0$ or $1$ each ...

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It's probably a bit late, but it's relatively easy to get an estimate of entropy. It doesn't matter if some bits are repetitive or not as it's the bits that aren't that contain entropy. You don't need any weird transformations of the data. You have what is called a bit fixing entropy source. Stick all your integers into a single file (binary or ascii - ...

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