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In this paper, we present a new methodology to adapt any kind of lattice reduction algorithms to deal with the modular knapsack problem. In general, the modular knapsack problem can be solved using a lattice reduction algorithm, when its density is low. The complexity of lattice reduction algorithms to solve those problems is upper-bounded in the function of the lattice dimension and the maximum norm of the input basis. In the case of a low density modular knapsack-type basis, the weight of maximum norm is mainly from its first column. Therefore, by distributing the weight into multiple columns, we are able to reduce the maximum norm of the input basis. Consequently, the upper bound of the time complexity is reduced.

Lattice Reduction for Modular Knapsack