The Belfiore--Sol??e Conjecture for unimodular lattices

In this thesis we study the Belfiore-Sol??e conjecture on the secrecy function of unimodular lattices. This conjecture states that for a lattice ??? in Rn, the quotient of the theta series of Zn by the theta series of ???, when restricted to the purely imaginary values z = ???y, y > 0, attains its maximum at y = 1. This conjecture is vitally connected to the confusion at the eavesdropper???s end in wiretap codes for Gaussian channels. We show that if ???1 and ???2 are lattices that satisfy the Ernvall- Hyt??onen criterion on derivatives [1], then so does the direct sum ???1 ??? ???2. It follows immediately that infinitely many lattices satisfy the Belfiore-Sol??e conjecture. Further, we show that all lattices obtained by Construction A from doubly even self-dual codes of lengths up to 40 satisfy the Belfiore-Sol??e conjecture