  In this paper, using the method of compression, we recover the lower bound
for the Erd\H{o}s unit distance problem and provide an alternative proof to the
distinct distance conjecture. In particular, in $\mathbb{R}^k$ for all $k\geq
2$, we have \begin{align}
\#\bigg\{||\vec{x_j}-\vec{x_t}||:~||\vec{x_j}-\vec{x_t}||=1,~1\leq t,j \leq
n,~\vec{x_j},~\vec{x}_t \in \mathbb{R}^k\bigg\}\gg_k
\frac{\sqrt{k}}{2}n^{1+o(1)}.\nonumber \end{align}We also show that
\begin{align} \# \bigg\{d_j:d_j=||\vec{x_s}-\vec{y_t}||,~d_j\neq d_i,~1\leq
s,t\leq n\bigg\}\gg_k \frac{\sqrt{k}}{2}n^{\frac{2}{k}-o(1)}.\nonumber
\end{align}These lower bounds generalizes the lower bounds of the Erd\H{o}s
unit distance and the distinct distance problem to higher dimensions.
