We give a randomized, rank-preserving reduction from exact $\ell_p$-CVP to exact Euclidean SVP for every integer $p\ge2$. For rational inputs of bit length $\ell$ on full-rank lattices $L\subset\mathbb R^N$, the reduction uses $(Cp)^N\poly(p,\ell)$ oracle calls and additional bit operations, where $C$ is an absolute constant. It uses $\poly(p,\ell)$ working space outside the oracle, and all oracle queries have rank $N$ and $\poly(p,\ell)$ bit complexity. For $p=2$, the reduction transfers polynomial-space exact SVP algorithms to exact CVP with $2^{O(N)}\poly(\ell)$ time overhead. Instantiating it with the SVP algorithms of Kannan and of Feng, Gao and Liu yields randomized polynomial-space CVP algorithms running in time $N^{N/(2e)+o(N)}$ and $N^{N/(4e)+o(N)}$, respectively. For every fixed integer $p\ge3$, the same reduction yields exact $\ell_p$-CVP algorithms for arbitrary rational targets either in randomized time and space $2^{O(N)}$, or in randomized time $N^{N/(4e)+o(N)}$ and polynomial space. These algorithmic bounds are up to polynomial factors in $\ell$. The reduction performs local search sequentially over the cosets of \(2L\), where \(L\) is the input lattice, while tracking the excess of the current objective value over the optimum. Within a coset containing a closest vector, midpoint inequalities bound the unknown correction in terms of this excess, which in turn allows us to construct many short corrections that reduce the excess bound by a factor of four. A counting argument shows that these improving corrections form a sufficiently large fraction of the lattice points in a known search region, and a sparsification-based sampler assigns sufficiently large probability to every nonzero vector in that region, using at most one SVP call per sample. For \(p\ge3\), the same framework is extended using a diagonal weighted norm computed from the current point.

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