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What does this mean, exactly? The purpose of the environment is to model "everything else happening in the universe" besides the protocol execution. In the UC model, the adversary is allowed to talk to the environment during the execution of the protocol. So UC security means "security no matter what else is going on in the world, even if other things ...


This is doable by (1) opening all commitments $B_k$ with standard responses $Q_{k}(z)$ that are linear in challenge, (2) proving that all $x_k \in \{0,1\}$ (with at most second-degree polynomials in challenge), and (3) proving $m$ polynomial identities for (at most) third-degree polynomials in challenge $$U_i(z) = T_{i,1}(z) T_{i,2}(z) T_{i,3}(z) $$ where ...


Yes it proves that $y \ne 1 \pmod{n}$ is not an $e$-th root. That is, no $x$ exists such that $x^e = y \pmod{n}$. In particular, if $e | \phi(n)$ then for any $x$ it holds that $\left( x^{ \frac{\phi(n)} {e}} \right)^e = 1 \ne y$. This probably means one last final conclusion designed as a homework. Regarding GQ protocol: prover will be unable to pick an ...


There are no relation we are currently aware of. The reason is as follows. The map $$k \mapsto (k G).x$$ is assumed to be a good pseudo random number generator. (The NSA infiltration of the Dual EC drgb has nothing to do with that fact). This basically says that k and r can be seen as independant random variables.

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