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.