Petros Mol TITLE: Lattices and Cryptography:An Overview of Recent Results with Emphasis on RSA and NTRU Cryptosystems. ABSTRACT: Traditionally, the study of lattices was closely connected to areas such as theory and geometry of numbers. The invention of LLL in 1982 and the intractability of some computational problems related to lattices led to lattice applications in various theoretical areas such as factorization of integer polynomials, integer programming and Public-Key Cryptography. In PKC, lattice theory has played a crucial role in the definition of new cryptosystems, in the study of cryptographic primitives and in cryptanalysis. In this survey, we present some of the most representative applications of lattices in Cryptography with emphasis on RSA and NTRU Cryptosystems.We present how Coppersmith's technique for finding small solutions to polynomial equations can be used to attack certain instances of RSA Cryptosystem. We also highlight the importance of finding short vectors in certain lattices and their corresponding applications in breaking NTRU instances of moderate level of security.Finally, we review some recent advancements which give hope that lattice based cryptography provides a possible alternative to number-theoretic based encryption schemes.