The aim of this paper is a reduction algorithm for a basis b1, b2, b3 of a 3-dimensional lattice in R n for fixed n ≥ 3. We give a definition of the reduced basis which is equivalent to that of the Minkowski reduced basis of a 3-dimensional lattice. We prove that for b1, b2, b3 ∈ Z, n ≥ 3 and |b1|, |b2|, |b3| ≤ M , our algorithm takes O(log M) binary operations, without using fast integer arith...

Among all lattice reduction algorithms, the LLL algorithm is the first and perhaps the most famous polynomial time algorithm, and it is widely used in many applications. In 2012, S. Qiao [24] introduced another algorithm, the Jacobi method, for lattice basis reduction. S. Qiao and Z. Tian [25] improved the Jacobi method further to be polynomial time but only produces a Quasi-Reduced basis. In t...

In this paper we introduce several new heuristics as to speed up known lattice basis reduction methods and improve the quality of the computed reduced lattice basis in practice. We analyze substantial experimental data and to our knowledge, we are the first to present a general heuristic for determining which variant of the reduction algorithm, for varied parameter choices, yields the most effi...

Euclidean lattices are a rich algebraic object that occurs in a wide variety of contextsin mathematics and in computer science. The present thesis considers several algorithmicaspects of lattices. The concept of lattice basis reduction is thoroughly investigated: in par-ticular, we cover the full range of time-quality trade-offs of reduction algorithms. On thefirst hand, we desc...

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 maximu...

Knapsack public-key encryption schemes are based on the knapsack problem, which is NP-complete. Merkle-Hellman knapsack encryption scheme was the first concrete realization of a public-key encryption scheme. As its secure basis is superincreasing knapsack problem, it has been demonstrated to be insecure. Many variations have subsequently been proposed, whose knapsack vector density are less tha...

Lattice reduction has a wide range of applications. In this paper, we first present a polynomial time Jacobi method for lattice basis reduction by modifying the condition for the Lagrange reduction and integrating the size reduction into the algorithm. We show that the complexity of the modified Jacobi algorithm is O(n5 logB), where n is the dimension of the lattice and B is the maximum length ...

We nd the shortest non-zero vector in the lattice of all integer multiples of the vector (a; b) modulo m, for given integers 0 < a; b < m. We reduce the problem to the computation of a Minkowski-reduced basis for a planar lattice and thereby show that the problem can be solved in O(log m(log logm) 2) bit operations.