Certified lattice reduction

Quadratic form reduction and lattice reduction are fundamental tools in computational number theory and in computer science, especially in cryptography. The celebrated Lenstra-Lenstra-Lov\'asz 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.Comment: 23 page

Computational linear algebra over finite fields

We present here algorithms for efficient computation of linear algebra problems over finite fields

Real-Time Character Animation for Computer Games

The importance of real-time character animation in computer games has increased considerably over the past decade. Due to advances in computer hardware and the achievement of great increases in computational speed, the demand for more realism in computer games is continuously growing. This paper will present and discuss various methods of 3D character animation and prospects of their real-time application, ranging from the animation of simple articulated objects to real-time deformable object meshes

Exploiting Data Representation for Fault Tolerance

We explore the link between data representation and soft errors in dot products. We present an analytic model for the absolute error introduced should a soft error corrupt a bit in an IEEE-754 floating-point number. We show how this finding relates to the fundamental linear algebra concepts of normalization and matrix equilibration. We present a case study illustrating that the probability of experiencing a large error in a dot product is minimized when both vectors are normalized. Furthermore, when data is normalized we show that the absolute error is less than one or very large, which allows us to detect large errors. We demonstrate how this finding can be used by instrumenting the GMRES iterative solver. We count all possible errors that can be introduced through faults in arithmetic in the computationally intensive orthogonalization phase, and show that when scaling is used the absolute error can be bounded above by one