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In this work, we give provable sieving algorithms for the Shortest VectorProblem (SVP) and the Closest Vector Problem (CVP) on lattices in $\ell_p$ norm($1\leq p\leq\infty$). The running time we obtain is better than existingprovable sieving algorithms. We give a new linear sieving procedure that worksfor all $\ell_p$ norm ($1\leq p\leq\infty$). The main idea is to divide thespace into hypercubes such that each vector can be mapped efficiently to asub-region. We achieve a time complexity of $2^{2.751n+o(n)}$, which is muchless than the $2^{3.849n+o(n)}$ complexity of the previous best algorithm.  We also introduce a mixed sieving procedure, where a point is mapped to ahypercube within a ball and then a quadratic sieve is performed within eachhypercube. This improves the running time, especially in the $\ell_2$ norm,where we achieve a time complexity of $2^{2.25n+o(n)}$, while the List SieveBirthday algorithm has a running time of $2^{2.465n+o(n)}$.  We adopt our sieving techniques to approximation algorithms for SVP and CVPin $\ell_p$ norm ($1\leq p\leq\infty$) and show that our algorithm has arunning time of $2^{2.001n+o(n)}$, while previous algorithms have a timecomplexity of $2^{3.169n+o(n)}$.

Faster provable sieving algorithms for the Shortest Vector Problem and the Closest Vector Problem on lattices in $\ell_p$ norm