Results on the Secrecy Function Conjecture on the Theta Function of Lattices

This thesis concerns lattice-based communication in a wiretap channel. The Belfiore and Sol ́e secrecy function conjecture, which states that the inverse normalized theta series of a unimodular lattice, when evaluated on the positive y-axis, attains its max- imum at y = 1, is an open problem in this area. We investigate this conjecture for unimodular lattices that arise from self-dual codes over F2 meeting a certain bound on their minimal distance, verifying it numerically for all such unimodular lattices of dimension up to 72. We also investigate the l-modular secrecy function conjecture of Ernvall-Hyt ̈onen and Sethuraman, and prove that the 8-modular lattice C(8) verifies the conjecture. We then make a key conjecture regarding a certain ratio of theta series that if true, will show that all lattices C(l) where l is a positive integer satisfy the secrecy function conjecture. Finally we verify numerically that the l-modular lattice C(l) satisfies the conjecture for many values of l.