# NP

## Refuting random 3SAT-instances on $O(n)$ clauses (weak form) ★★★

Author(s): Feige

\begin{conjecture} For every rational $\epsilon > 0$ and every rational $\Delta$, there is no polynomial-time algorithm for the following problem.

Given is a 3SAT (3CNF) formula $I$ on $n$ variables, for some $n$, and $m = \floor{\Delta n}$ clauses drawn uniformly at random from the set of formulas on $n$ variables. Return with probability at least 0.5 (over the instances) that $I$ is \textit{typical} without returning \textit{typical} for \textit{any} instance with at least $(1 - \epsilon)m$ simultaneously satisfiable clauses. \end{conjecture}

Keywords: NP; randomness in TCS; satisfiability

## Discrete Logarithm Problem ★★★

Author(s):

If $p$ is prime and $g,h \in {\mathbb Z}_p^*$, we write $\log_g(h) = n$ if $n \in {\mathbb Z}$ satisfies $g^n = h$. The problem of finding such an integer $n$ for a given $g,h \in {\mathbb Z}^*_p$ (with $g \neq 1$) is the \emph{Discrete Log Problem}.

\begin{conjecture} There does not exist a polynomial time algorithm to solve the Discrete Log Problem. \end{conjecture}

Keywords: discrete log; NP

## P vs. NP ★★★★

Author(s): Cook; Levin

\begin{problem} Is P = NP? \end{problem}