NTRU is a fast public-key cryptosystem that is constructed using polynomial rings with coefficients in Z. We present ETRU, an NTRU-like cryptosystem based on the Eisenstein integers Z[ω] where ω is a primitive cube root of unity. We discuss parameter selection and develop a model for the probabilty of decryption failure. We also provide an implementation of ETRU. We use theoretical and experimental data to compare the security and efficiency of ETRU to NTRU with comparable parameter sets and show that ETRU is an improvement over NTRU in terms of security. Contents List of Figures viii List of Tables ix 1 Introduction 1 2 Lattices and Background 4

We relate the partition-type parametrization of rational (arithmetic) nilpotent adjoint orbits of the classical groups SL n and Sp 2n over local non-Archimedean fields with a parametrization, introduced by DeBacker in 2002, which uses the associated Bruhat-Tits building to relate the question to one over the residue field.

Abstract. We explicitly describe the decomposition into irreducibles of the restriction of the principal series representations of SL(2, k), for k a p-adic field, to each of its two maximal compact subgroups (up to conjugacy). We identify these irreducible subrepresentations in the Kirillov-type classification of Shalika. We go on to explicitly describe the decomposition of the reducible principal series of SL (2, k) in terms of the restrictions of its irreducible constituents to a maximal compact subgroup.

The orbit method conjectures a close relationship between the set of irreducible unitary representations of a Lie group G, and admissible coadjoint orbits in the dual of the Lie algebra. We define admissibility for nilpotent coadjoint orbits of p-adic reductive Lie groups, and compute the set of admissible orbits for a range of examples. We find that for unitary, symplectic, orthogonal, general linear and special linear groups over p-adic fields, the admissible nilpotent orbits coincide with the so-called special orbits defined by Lusztig and Spaltenstein in connection with the Springer correspondence.

No abstract