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Thomas Johnson 11 months ago
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presentation.tex

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 @@ -42,7 +42,7 @@ Given $n = pq$, where $p$ and $q$ are prime: 42 42 43 43 \begin{itemize} 44 44 \item<2-> Pick a random element $a \in \mathbb{Z}/n\mathbb{Z}$. 45 -\item<3-> Compute the order $r$ of $a$ using the oracle, so that $a^r \equiv 0\ (\mathrm{mod}\ n)$. 45 +\item<3-> Compute the order $r$ of $a$ using the oracle, so that $a^r \equiv 1\ (\mathrm{mod}\ n)$. 46 46 \item<4-> If $r$ is odd or $a^{\frac{r}{2}} \equiv -1\ (\mathrm{mod}\ n)$, restart the procedure. 47 47 \item<5-> Let $s \equiv a^{\frac{r}{2}}\ (\mathrm{mod}\ n)$. Compute $s + 1$ and $s - 1$. One of these will be a factor of $n$. 48 48 \end{itemize} @@ -121,7 +121,7 @@ which is true when the string $b$ is lexicographically after $w$. This can be im 121 121 \frametitle{Proof (continued...)} 122 122 At last, consider, given some pivot string $w$ in the $2^k$ space of strings, the formula: 123 123 $$124 -EncryptsToC(X) && LexicographicallyAfterW(X) 124 +EncryptsToC(X) \&\& LexicographicallyAfterW(X) 125 125$$ 126 126 Applying the SAT oracle to this formula will tell us if there is a bitstring in the upper half of the search space that is the desired plaintext string. We can run binary search using this, and acquire the desired plaintext in linear time. 127 127 \end{frame}