Paper 2025/2227

Time Memory Trade-off For Enumeration

Abstract

We incorporate a meet-in-the-middle strategy into the enumeration algorithm, enabling a tunable time-memory trade-off. The algorithm can achieve a minimum asymptotic time complexity of \(n^{n/(4e) + o(n)}\), which, in return, demands memory of the same order. This represents a square-root improvement over the state-of-the-art enumeration algorithms. More generally, our approach attains a time complexity of~$n^{cn/(2e) + o(n)}$ with memory usage \(n^{(1-c)n/(2e) + o(n)}\), where $c$ is any constant satisfying \(\frac{1}{2} \leq c < 1\). Our approach decouples the head and tail blocks of the lattice basis. For a properly selected parameter, each enumeration space becomes asymptotically the square root of the original search space. Each tail vector is then extended to the head block space to find its closest vectors using an efficient neighboring search algorithm. Among all pairs of neighboring vectors that we iterate through, the shortest difference vector is then the solution to the Shortest Vector Problem (SVP). Apart from the exact version of the algorithm which is of theoretical interest, we also propose heuristic strategies to improve the practical efficiency. First, we show the adaptation of our algorithm to pruned enumeration. Then we show that with a particularly chosen backbone lattice (rescaled~\(\mathbb{Z}^n\)), we are able to accelerate the neighboring search process to an extremely efficient degree. Finally, we optimize parameters and give a practical cost estimation to show how much acceleration we could bring using this new algorithm.