## Section: New Results

### Foundations

Participants : Nicolas Gama, Phong Quang Nguyen.

An LLL Algorithm with Quadratic Complexity (SIAM J. Computing, 2009)

LLL is a celebrated algorithm published by Lenstra, Lenstra and Lovász in 1982. This algorithm was the first polynomial-time algorithm that provably finds short vectors in a lattice, which has many applications: given as input a finite number of integral vectors, LLL finds a reasonably short integral linear combination of the input vectors.

This can be viewed as a geometric generalization of the problem of computing greatest common divisors: given as input two integers a and b , Euclid's algorithm computes in quadratic time the gcd of a and b , that is, the nonzero integral linear combination of a and b that is both positive and smallest.

However, the natural analogy between LLL and Euclid's algorithm was not fully satisfactory until now, at least from a computational point of view. Indeed, while the running time of Euclid's algorithm is quadratic without fast integer arithmetic, all LLL-type algorithms known had a cubic running time without fast integer arithmetic, in the sense that their polynomial-time complexity was at least cubic in the bit-size of the largest norm of the input vectors: here, we ignore the additional polynomial term depending on the lattice dimension.

This article [17] presents the first LLL algorithm whose running time is provably quadratic in the bit-size of the largest norm of the input vectors, which makes it similar to Euclid's algorithm. This result was inspired by [18] , which studied low-dimensional lattice reduction.

The LLL Algorithm: Surveys and Applications (Springer book, 2009)

This book [57] , published by Springer, is a follow-up to the 2007 conference that took place in Caen, celebrating the 25th anniversary of the publication of the LLL algorithm. It surveys the foundations and the main applications of LLL and lattice algorithms in computer science and mathematics: for instance, [54] introduces lattices and presents the main provable lattice algorithms. The surveys are written by the invited speakers of the conference.

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