  In this article, we prove that the Chow group of algebraic cycles of a smooth
cubic hypersurface $X$ over an arbitrary field $k$ is generated, via the
natural cylinder homomorphism, by the algebraic cycles of its Fano variety of
lines $F(X)$, under an assumption on the $1$-cycles of $X/k$. As an
application, if $X/\mathbb{C}$ is a smooth complex cubic fourfold, using the
result, we provide a proof of the integral Hodge conjecture for curve classes
on the polarized hyper-K\"ahler variety $F(X)$. In addition, when $X/k$ is a
smooth cubic fourfold over a finitely generated field $k$ with
$\text{char}(k)\neq 2, 3$, our result enable us to prove the integral analog of
the Tate conjecture for $1$-cycles on $F(X)$.
