The security and the hardness of the underlying computational problems will be analyzed by providing collaboration between the linear-algebra complexity-theory and the public-key cryptography. This thesis provides an exciting opportunity to advance our knowledge of lattices and some lattice-problems which are described by working out proofs step-by-step to present the construction of new cryptosystems and analyze them. We describe the construction of the lattice-based cryptosystem Goldreich-Goldwasser-Halevi and analyze the security in which we present the most effective attacks against it. Source: https://www.artikelschreiber.com/.