On the Eavesdroppers Correct Decision in Gaussian and Fading Wiretap Channels Using Lattice Codes

In this paper, the probability of Eve the Eavesdropper’s correct decision is considered both in the Gaussian and Rayleigh fading wiretap channels when using lattice codes for the transmission. First, it is proved that the secrecy function determining Eve’s performance attains its maximum at y=1 on all known extremal even unimodular lattices. This is a special case of a conjecture by Belfiore and Sol'e. Further, a very simple method to verify or disprove the conjecture on any given unimodular lattice is given. Second, preliminary analysis on the behavior of Eve’s probability of correct decision in the fast fading wiretap channel is provided. More specifically, we compute the truncated inverse norm power sum factors in Eve’s probability expression. The analysis reveals a performance-secrecy-complexity tradeoff: relaxing on the legitimate user’s performance can significantly increase the security of transmission. The confusion experienced by the eavesdropper may be further increased by using skewed lattices, but at the cost of increased complexity.

💡 Research Summary

This paper investigates the probability that an eavesdropper (Eve) correctly decodes transmitted messages when lattice codes are employed over both Gaussian and Rayleigh fading wiretap channels. The authors first address a conjecture by Belfiore and Solé which states that the secrecy function—defined as the ratio of the theta series of the integer lattice ℤⁿ to that of the employed lattice Λ—attains its maximum at the scaling parameter y = 1. By expressing the theta series of even unimodular lattices in terms of Eisenstein series E₄ and the discriminant Δ, the secrecy function reduces to a simple rational function of the basic theta constants ϑ₂, ϑ₃, and ϑ₄. Lemma 3.2 proves that the product ϑ₄²(y i)·ϑ₄⁴(y i)·ϑ₈³(y i) reaches its maximal value ¼ precisely at y = 1, using symmetry f(y)=f(1/y) and a careful sign analysis of the derivative. Consequently, Theorem 3.3 confirms the conjecture for all known extremal even unimodular lattices (E₈, Λ₂₄, Λ₃₂, etc.). A practical verification method is then presented: for any unimodular lattice, the secrecy function can be written as a polynomial P(z) with z = ϑ₄²·ϑ₄⁴·ϑ₈³, and checking that P(z) attains its minimum at the endpoint z = ¼ on the interval