This algorithm has revolutionized the computing aspects of geometry of numbers since its introduction in 1982, led to breakthroughs in fields as diverse as computer algebra, cryptology and algorithmic number theory. This book consists of 15 chapters on aspects of the survey grid computing Euclidean and their main applications. Topics covered include polynomial factorization, lattice reduction algorithms, applications in number theory, integer programs, provable security, lattice-based cryptography and complexity. The authors include deep motivations, explanations and examples, and contributed largely independent.