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\section { The Discrete Log Problem}
\definition { }
Let $ g $ be a generator in $ ( \mathbb { Z } _ p ^ \times , \ast ) $ \par
Let $ n $ be a positive integer.
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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
In other words, we want an inverse of the \say { exponent} function.
\vspace { 1mm}
This is the \textit { discrete logarithm problem} , often abbreviated \textit { DLP} .
\problem { }
Does the discrete log function even exist? \par
Show that $ \exp $ is a bijection, which will guarantee the existence of $ \log $ . \par
\note [Note] { Why does this guarantee the existence of log? Recall our lesson on funtions.}
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\problem { }
What's the simplest (but not the most efficient) way to calculate $ \log _ g ( a ) $ ?
\vfill
\problem { }
Find an efficient way to solve the discrete log problem. \par
Then learn \LaTeX , write a paper, and enjoy free admission to the graduate program at any university. \par
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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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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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