Lattices are geometric objects that can be pictorially described as the set of intersection points of an infinite, regular n-dimensional grid. De spite their apparent simplicity, lattices hide a rich combinatorial struc ture, which has attracted the attention of great mathematicians over the last two centuries. Not surprisingly, lattices have found numerous ap plications in mathematics and computer science, ranging from number theory and Diophantine approximation, to combinatorial optimization and cryptography. The study of lattices, specifically from a computational point of view, was marked by two major breakthroughs: the development of the LLL lattice reduction algorithm by Lenstra, Lenstra and Lovasz in the early 80's, and Ajtai's discovery of a connection between the worst-case and average-case hardness of certain lattice problems in the late 90's. The LLL algorithm, despite the relatively poor quality of the solution it gives in the worst case, allowed to devise polynomial time solutions to many classical problems in computer science. These include, solving integer programs in a fixed number of variables, factoring polynomials over the rationals, breaking knapsack based cryptosystems, and finding solutions to many other Diophantine and cryptanalysis problems."

The papers in this voluriic were presented at the CHYP'I'O 'SS conf- ence on theory and applications of cryptography, Iicld August 21-2, j. 19SS in Sarita Uarbara, ('alifornia. The conference was sponsored hy the Int- national AssociatioIi for C'ryptologic Research (IAC'R) and hosted by the computer science depart incnt at the llniversity of California at Sarita D- ha ra . 'rile 4-1 papers presented hcrc coniprise: 35 papers selected from 61 - tcwded abstracts subniittctl in response to the call for papcrs, 1 invitcd prv sentations, and 6 papers sclccted from a large niiiii1, cr of informal UIIIJ) sewion present at ionc. The papers wcrc chosen by the program committee on the lja\is of tlic perceived originality, quality and relevance to the field of cryptography of the cxtcndcd allst ract5 suhriiitted. 'I'hc su1, missioris wv riot otlierwise rc.fcrcc(l. a id ofteri rcprescnt prcliininary reports on continuing rcscarc.11. It is a pleasure to tharik many colleagues. Ilarold Iredrickscri sing- made CRJ'PTO '88 a successful realit, y. Eric Dacli, Pad Ijnrret. haridedly Tom Bersori, Gilles Brassard, Ocled Goldreich, Andrew Odlyzko. C'liarles Rackoff arid Ron Rivest did excellerit work on the program comrriittcc in piittirig the technical program together, assisted by kind outsick reviekvers.

Lattices are geometric objects that can be pictorially described as the set of intersection points of an infinite, regular n-dimensional grid. De spite their apparent simplicity, lattices hide a rich combinatorial struc ture, which has attracted the attention of great mathematicians over the last two centuries. Not surprisingly, lattices have found numerous ap plications in mathematics and computer science, ranging from number theory and Diophantine approximation, to combinatorial optimization and cryptography. The study of lattices, specifically from a computational point of view, was marked by two major breakthroughs: the development of the LLL lattice reduction algorithm by Lenstra, Lenstra and Lovasz in the early 80's, and Ajtai's discovery of a connection between the worst-case and average-case hardness of certain lattice problems in the late 90's. The LLL algorithm, despite the relatively poor quality of the solution it gives in the worst case, allowed to devise polynomial time solutions to many classical problems in computer science. These include, solving integer programs in a fixed number of variables, factoring polynomials over the rationals, breaking knapsack based cryptosystems, and finding solutions to many other Diophantine and cryptanalysis problems."