Ω-Lattices

November 19, 1992 Lecture 19 Lecturer: Michel X. Goemans Scribe: David B. Wilson1 1 Lattices Starting with today's lecture, we will look at problems involving lattices and algorithms for basis reduction of lattices. Applications of this topic include factoring polynomials, breaking cryptosystems, rounding an interior point to an optimal vertex in linear programming, and solving integer programs. We start with de nitions: De nition 1 Given a set of vectors b1; : : : ; bm 2 Qn, we de ne the lattice L = L(b1; : : : ; bm) = fPmi=1 ibi : i 2 Zg. Thus, L is the set of integral combinations of the vectors bi. Example: b1 = (1; 2); b2 = (2; 1); n = m = 2.