The cycle space of a graph $G$, denoted $C(G)$, is a vector space over ${\mathbb F}_2$, spanned by all incidence vectors of edge-sets of cycles of $G$. If $G$ has $n$ vertices, then $C_n(G)$ denotes the subspace of $C(G)$, spanned by the incidence vectors of Hamilton cycles of $G$. A classical result in the theory of random graphs asserts that for $G \sim \mathbb{G}(n,p)$, asymptotically almost surely the necessary condition $δ(G) \geq 2$ is also sufficient to ensure Hamiltonicity. Resolving a problem of Christoph, Nenadov, and Petrova, we augment this result by proving that for $G \sim \mathbb{G}(n,p)$, with $n$ being odd, asymptotically almost surely the condition $δ(G) \geq 3$ (observed to be necessary by Heinig) is also sufficient for ensuring $C_n(G) = C(G)$. That is, not only does $G$ typically have a Hamilton cycle, but its Hamilton cycles are typically rich enough to span its cycle space.