Abstract
Quadratic form reduction and lattice reduction are fundamental tools in computational number theory and in computer science, especially in cryptography. The celebrated Lenstra–Lenstra–Lovasz reduction algorithm (so-called LLL) has been improved in many ways through the past decades and remains one of the central methods used for reducing integral lattice basis. In particular, its floating-point variants---where the rational arithmetic required by Gram–Schmidt orthogonalization is replaced by floating-point arithmetic---are now the fastest known. However, the systematic study of the reduction theory of real quadratic forms or, more generally, of real lattices is not widely represented in the literature. When the problem arises, the lattice is usually replaced by an integral approximation of (a multiple of) the original lattice, which is then reduced. While practically useful and proven in some special cases, this method doesn't offer any guarantee of success in general. In this work, we present an adaptive-precision version of a generalized LLL algorithm that covers this case in all generality. In particular, we replace floating-point arithmetic by Interval Arithmetic to certify the behavior of the algorithm. We conclude by giving a typical application of the result in algebraic number theory for the reduction of ideal lattices in number fields.
Highlights
A lattice Λ is a free Z-module of finite rank, endowed with a positive-definite bilinear form on its ambient space Λ⊗Z R, as presented for instance in [16]
This definition implies that Λ is discrete in its ambient space for the topology induced by the scalar product
Suppose that the Gram matrix Gγ =(i,j)∈[1 ··· d]2 representing the inner product of the ambient space Γ ⊗Z R in the basis γ is given indirectly by an algorithm or an oracle Oγ that can compute each entry at any desired accuracy
Summary
A lattice Λ is a free Z-module of finite rank, endowed with a positive-definite bilinear form on its ambient space Λ⊗Z R, as presented for instance in [16]. A century later, in 1982, Lenstra, Lenstra and Lovász designed the lll algorithm [14], with the polynomial factorization problem as an application, following the work of Lenstra on integer programming [15] This algorithm constitutes a breakthrough in the history of lattice reduction algorithm, since it is the first to have a runtime polynomial in terms of the dimension. Its main asset—calculating directly on sets—is nowadays used to deterministically determine the global extrema of a continuous function [24] or localizing the zeroes of a function and (dis)proving their existence [11] Another application of Interval Arithmetic is to be able to detect lack of precision at run-time of numerical algorithms, thanks to the guarantees it provides on computations. It is the cost of the multiplication of two floating-point numbers at precision k, since the cost of arithmetic over the exponents is negligible with regards to the cost of arithmetic over the mantissae
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