Abstract:  Our world is not linear. Many phenomena, however, are often "linearized" because only then a reasonably well-working mathematical machinery can describe the phenomena and produce meaningful forecasts. Lattices are geometric objects that have been used to solve many problems in mathematics and computer science. Lattice reduction strategies or the so called LLL-techniques seem inherently linear. The general idea of this technique is to translate our non linear problem to finding a short vector in a lattice built from the nonlinear equation. Then, the so-called Shortest Vector Problem and Closest Vector Problem in lattices play a major role. In recent years, these techniques have been used repeatedly in algorithmic coding theory and cryptography.

In this talk I will investigate lattice reduction technique on some cryptography problems, namely 1) finding roots of multivariate integer polynomials and attacking cryptosystems, 2) Integer factoring and RSA, and 3) predicting pseudorandom number congruential generators over Elliptic Curves.