  The first purpose of this paper is to provide new finite field extension
theorems for paraboloids and spheres. By using the unusual good Fourier
transform of the zero sphere in some specific dimensions, which has been
discovered recently in the work of Iosevich, Lee, Shen, and the first and
second listed authors (2018), we provide a new $L^2\to L^r$ extension estimate
for paraboloids in dimensions $d=4k+3$ and $q\equiv 3\mod 4$, which improves
significantly the recent exponent obtained by the first listed author. In the
case of spheres, we introduce a way of using \textit{the first association
scheme graph} to analyze energy sets, and as a consequence, we obtain new
$L^p\to L^4$ extension theorems for spheres of primitive radii in odd
dimensions, which break the Stein-Tomas result toward $L^p\to L^4$ which has
stood for more than ten years. Most significantly, it follows from the results
for spheres that there exists a different extension phenomenon between spheres
and paraboloids in odd dimensions, namely, the $L^p\to L^4$ estimates for
spheres with primitive radii are much stronger than those for paraboloids.
Based on new estimates, we will also clarify conjectures on finite field
extension problem for spheres. This results in a reasonably complete
description of finite field extension theorems for spheres. The second purpose
is to show that there is a connection between the restriction conjecture
associated to paraboloids and the Erd\H{o}s-Falconer distance conjecture over
finite fields. The last is to prove that the Erd\H{o}s-Falconer distance
conjecture holds in odd-dimensional spaces when we study distances between two
sets: one set lies on a variety (paraboloids or spheres), and the other set is
arbitrary in $\mathbb{F}_q^d$.
