We prove that for n b 30, every n-th Lucas and Lehmer number has a primitive divisor. This allows us to list all Lucas and Lehmer numbers without a primitive divisor.

Abstract. The security of lattice-based cryptosystems such as NTRU, GGH and Ajtai-Dwork essentially relies upon the intractability of computing a shortest non-zero lattice vector and a closest lattice vector to a given target vector in high dimensions. The best algorithms for these tasks are due to Kannan, and, though remarkably simple, their complexity estimates have not been improved since over twenty years. Kannan's algorithm for solving the shortest vector problem (SVP) is in particular crucial in Schnorr's celebrated block reduction algorithm, on which rely the best known generic attacks against the lattice-based encryption schemes mentioned above. In this paper we improve the complexity upper-bounds of Kannan's algorithms. The analysis provides new insight on the practical cost of solving SVP, and helps progressing towards providing meaningful key-sizes.

This article presents a multiple-precision binary floating-point library, written in the ISO C language, and based on the GNU MP library. Its particularity is to extend to arbitrary-precision, ideas from the IEEE 754 standard, by providing correct rounding and exceptions . We demonstrate how these strong semantics are achieved---with no significant slowdown with respect to other arbitrary-precision tools---and discuss a few applications where such a library can be useful.

We propose a general method for numerical solution of Thue equations, which allows one to solve in reasonable time Thue equations of high degree (provided necessary algebraic number theory data is available). We illustrate our method, solving completely concrete Thue equations of degrees 19 and 33.1996 Academic Press, Inc.

Abstract. We present the state of the art solvers of the Shortest and Closest Lattice Vector Problems in the Euclidean norm. We recall the three main families of algorithms for these problems, namely the algorithm by Micciancio and Voulgaris based on the Voronoi cell [STOC'10], the Monte-Carlo algorithms derived from the Ajtai, Kumar and Sivakumar algorithm [STOC'01] and the enumeration algorithms originally elaborated by Kannan [STOC'83] and Fincke and Pohst [EUROCAL'83]. We concentrate on the theoretical worst-case complexity bounds, but also consider some practical facets of these algorithms.

Abstract. Strong lattice reduction is the key element for most attacks against lattice-based cryptosystems. Between the strongest but impractical HKZ reduction and the weak but fast LLL reduction, there have been several attempts to find efficient trade-offs. Among them, the BKZ algorithm introduced by Schnorr and Euchner [FCT'91] seems to achieve the best time/quality compromise in practice. However, no reasonable complexity upper bound is known for BKZ, and Gama and Nguyen [Eurocrypt'08] observed experimentally that its practical runtime seems to grow exponentially with the lattice dimension. In this work, we show that BKZ can be terminated long before its completion, while still providing bases of excellent quality. More precisely, we show that if given as inputs a basis (bi) i≤n ∈ Q n×n of a lattice L and a block-size β, and if terminated afterβ 2 (log n + log log maxi bi ) " calls to a β-dimensional HKZ-reduction (or SVP) subroutine, then BKZ returns a basis whose first vector has norm ≤ 2νn , where ν β ≤ β is the maximum of Hermite's constants in dimensions ≤ β. To obtain this result, we develop a completely new elementary technique based on discrete-time affine dynamical systems, which could lead to the design of improved lattice reduction algorithms.

Abstract. The main problem when solving a Thue equation is the computation of the unit group of a certain number field. In this paper we show that the knowledge of a subgroup of finite index is actually sufficient. Two examples linked with the primitive divisor problem for Lucas and Lehmer sequences are given.

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