On the Secrecy Gain of Formally Unimodular Construction A₄ Lattices

Abstract

The theta series of a lattice has been extensively studied in the literature and is closely related to a critical quantity widely used in the fields of physical layer security and cryptography, known as the flatness factor, or equivalently, the smoothing parameter of a lattice. Both fields raise the fundamental question of determining the (globally) maximum theta series over a particular set of volume-one lattices, namely, the stable lattices. In this work, we present a property of unimodular lattices, a subfamily of stable lattices, to verify that the integer lattice Z n achieves the largest possible value of theta series over the set of unimodular lattices. Such a result moves towards proving the conjecture recently stated by Regev and Stephens-Davidowitz: any unimodular lattice, except for those lattices isomorphic to Z n , has a strictly smaller theta series than that of Z n . Our techniques are mainly based on studying the ratio of the theta series of a unimodular lattice to the theta series of Z n , called the secrecy ratio. We relate the Regev and Stephens-Davidowitz conjecture with another conjecture for unimodular lattices, known in the literature as the Belfiore-Solé conjecture. The latter assumes that the secrecy ratio of any unimodular lattice has a symmetry point, which is exactly where the global minimum of the secrecy ratio is achieved.Our technical contributions are three-fold.

Secrecy Gain of Formally Unimodular Lattices from Codes over the Integers Modulo 4

“…by using G C oext as in (20). Thus, this gives that I + a T a + BB T = O η and 2a + 2cB T ≡ 0 (mod 4), which leads to conditions i) and ii) stated in the proposition.…”

“…A lattice Λ is called isodual if it can be obtained from its dual Λ ⋆ by (possibly) a rotation or reflection. In [20], a new and broader family was presented, namely, the formally unimodular lattices, that consists of lattices having the same theta series as their duals, i.e., Θ Λ (z) = Θ Λ ⋆ (z).…”

“…Proposition 18: Consider an odd extension code C oext generated by G C oext as in (20). Then, C oext is self-dual if and only if the following conditions hold i) a T a + BB T ≡ 3I η (mod 4),…”

“…Example 19: Consider a [13, 2 13 ] formally self-dual code C oext over Z 4 generated as in (20), where…”

“…Under the design criterion of the secrecy function, we summarize the following three important observations for the formally unimodular lattices [20].…”