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The full $m-$sequence (periodically repeated to avoid modular addition in the subscript} with $$C_{xx}(\tau):=\sum_{k=0}^{2^n-2} (-1)^{x_k+x_{k+\tau}}$$ satisfies $C_{xx}(\tau)=-1+\delta(\tau)(2^n),$ where $\delta$ is the dirac delta function. Now, one might define an $m-$symbol partial correlation function, whose average is proportional to what you want, if ...


It can be consctructed in many different ways. I will just give an example using HMAC-$\mathcal{H}$ where $\mathcal{H}$ is a hash function which returns a $x$-bit hash. So if you want to use a PRNG which takes $n$-bit inputs and returns $2n$-bit output, you could act this way: Divide the input into $\frac{x}{2}$-bit blocks $B_i$ For each block $B_i$ ...


No, A is not true. Suppose that $G_1$ is a secure PRG and $G_2(s) = G_1(s) \oplus 1$, obviously $G_2 \neq G_1$ and $G_2$ is a secure PRG. You can see that $G(s) = G_1(s) \oplus G_2(s) = G_1(s) \oplus G_1(s) \oplus 1 = 1$ which is obviously not a secure PRG. Now you have a hint. You should think the rest of the problems.

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