43 lines
1.5 KiB
TeX
Executable File
43 lines
1.5 KiB
TeX
Executable File
\section{The Discrete Log Problem}
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\definition{}
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Let $g$ be a generator in $(\mathbb{Z}_p^\times, \ast)$ \par
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Let $n$ be a positive integer.
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\vspace{1mm}
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We now want a function \say{log} from $\mathbb{Z}_p^\times$ to $\mathbb{Z}^+$ so that $\log_g(g^n) = n$. \par
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In other words, we want an inverse of the \say{exponent} function.
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\vspace{1mm}
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This is the \textit{discrete logarithm problem}, often abbreviated \textit{DLP}.
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\problem{}
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Does the discrete log function even exist? \par
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Show that $\exp$ is a bijection, which will guarantee the existence of $\log$. \par
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\note[Note]{Why does this guarantee the existence of log? Recall our lesson on funtions.}
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\vfill
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\problem{}
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What's the simplest (but not the most efficient) way to calculate $\log_g(a)$?
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\vfill
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\problem{}
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Find an efficient way to solve the discrete log problem. \par
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Then learn \LaTeX, write a paper, and enjoy free admission to the graduate program at any university. \par
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\vfill
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The discrete logarithm can be quickly computed in a few special cases, but there is no known way to efficiently compute it in general. Interestingly enough, we haven't been able to prove that an efficient solution \textit{doesn't} exist. The best we can offer is a \say{proof by effort:} many smart people have been trying for long time and haven't solved it yet. It probably doesn't exist.
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\vspace{2mm}
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In the next few pages, we'll see how the assumption \say{DLP is hard} can be used to construct various tools used to secure communications.
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\pagebreak
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