Diffie-Hellman key exchange on the Heisenberg group

This paper introduces Hadamard-type t-Fibonacci-Lehmer (HTFL) sequences, a new hybrid construction combining Lehmer and Fibonacci recurrences. We establish their fundamental properties, including simple periodicity, and extend the definition to finite groups, with a detailed study of the Heisenberg group. Building on these results, we propose two Diffie–Hellman-style key exchange protocols based on upper-triangular unipotent matrices parameterized by HTFL sequence terms. Our work thus connects sequence theory, group theory, and cryptography in a novel way. While the algebraic framework and periodicity analysis are rigorous, we present the cryptographic constructions primarily as a conceptual foundation. We also discuss potential security considerations and outline directions for strengthening these schemes under formal hardness assumptions. This study demonstrates that HTFL sequences provide a fertile ground for both combinatorial investigations and future cryptographic applications.