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A protocol might rely on using some string $x$ as an index into a set of uniform random bit strings $H(x)$. The security of the protocol may rely on the $H(x)$ being uniform random and independent for different $x$. A complex protocol may rely on having multiple independent uniform random bit strings for any particular string $x$, used for different ...


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The authentication system you are asking about does not mandate different hashing algorithms. $H_0$, $H_1$... $H_n$ can, for example, be implemented with SHA3 in the following way : $H_n(x) = SHA3(n \mathbin\| x)$, the operator $\mathbin\|$ being a concatenation. $H_0$, $H_1$... $H_n$ are said to be different hashing functions from the same family of ...


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It is not hard to construct a secure MAC where $f(k) = \textsf{MAC}(k,0)$ is not one-way. Take any secure $\textsf{MAC}$ and define a new one as $$\textsf{MAC}^*\Bigl( (k_1,k_2), m \Bigr) = \textsf{MAC}(k_1,m) \oplus k_2.$$ It should be relatively clear that $\textsf{MAC}^*$ is also a secure MAC. Now given arbitrary $m$ and $t$, it is easy to find a key ...


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