Polynomial-time many-one reduction

A polynomial-time many-one reduction from a decision problem $A$ to a decision problem $B$ is a polynomial-time computable function $r$ satisfying $x\in A$ exactly when $r(x)\in B$.

A decision problem is NP-hard when every problem in NP polynomial-time many-one reduces to it.

A decision problem is NP-complete when it is both in NP and NP-hard.

The Boolean satisfiability problem asks whether a Boolean formula has an assignment making it true.

Quadratic-equation satisfiability over ${\mathbb{F}}_2$ asks whether a finite system of polynomial equations of degree at most two over the finite field ${\mathbb{F}}_2$ has a common solution.

If $\mathbf{P}\ne \mathbf{NP}$, then NP contains a decision problem that is neither in P nor NP-complete.