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This is trivially true via the pigeonhole principle. SHA-2/512 has $2^{512}$ possible outputs, but $2^{2^{128}} - 1$ possible inputs. Trying $2^{512}+1$ unique inputs is sufficient to produce at least one collision. That said, SHA-2/512 is designed to be collision resistant, which implies that it should be hard to find two inputs that hash to the same ...


7

Question: Given $n$ values $v_1=\alpha \cdot r_1 \bmod p,..., v_n=\alpha \cdot r_n \bmod p$ for a large $n$ can the adversary learn the value $\alpha$? Answer: assuming that the $r_i$ values are random (that is, equidistributed and uncorrelated), then the attacker gets absolutely no information about $\alpha$ (other than whether or not it's 0). We can ...


1

A fundamental theorem of cryptography (Goldreich, Micali and Wigderson 1986) states that any NP statement can be proven in zero knowledge. So, the answer is yes. For any polynomial-time combination of $X$ and $Y$ into $Z$, it is possible to prove that $Q$ contains $X$ in zero knowledge. Note that the general zero knowledge will not be practically efficient ...



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