Section: Scientific Foundations

Euclidean Lattice Reduction and Applications

A Euclidean lattice is the set of integer linear combinations of a finite set of real vectors. Typically, lattices occur when linear algebra questions are asked with discreteness constraints. In the last decade, they have become a classical ingredient in the computer arithmetic toolbox, along with other number-theoretic techniques (continued fractions, diophantine approximation, etc.). Indeed, integers (scaled by powers of the radix) are the essence of the fixed-point and floating-point representations of the real numbers. If the macroscopic properties of floating-point numbers are close to those of the real numbers, the finer properties are definitely related to questions over the integers. Thus, lattices have been successfully used in computer arithmetic to find constrained polynomial approximations to functions, and to attack the Table Maker's Dilemma. They have a potential for further arithmetic applications, for instance the design of digital filters.

Besides, the algorithms on Euclidean lattices are a rich experimentation laboratory for different types of arithmetics. The basis vectors are often represented exactly with long integer arithmetic. Furthermore, the fastest algorithms find the operations to be performed on the basis vectors via approximate computations, typically an approximate Gram-Schmidt orthogonalisation. These approximate computations may be performed with fixed-precision or arbitrary precision floating-point arithmetics. In some time-consuming applications of lattice algorithms, such as cryptanalyses of variants of RSA or lattice-based cryptosystems, integer linear programming, or even for solving the Table Maker's Dilemma, the practical run-time is of utmost importance. This motivates strong optimizations for the underlying arithmetics.

Further, aside from this strong relationship between lattices and arithmetics, the understanding of lattice-based cryptology is developing at a quick pace; making it efficient while remaining secure will require a thorough study, which must involve experts in both arithmetics and cryptology.