This paper is concerned with the question whether Verifiable Delay Functions (VDFs), as introduced by Boneh et al. [CRYPTO 2018], can be constructed in the plain Random Oracle Model (ROM) without any computational assumptions. A first partial answer to this question is due to Mahmoody, Smith, and Wu [ICALP 2020], and rules out the existence of perfectly unique VDFs in the ROM. Building on this result, Guan, Riazanov, and Yuan [CRYPTO 2025] very recently demonstrated that even VDFs with computational uniqueness are impossible under a public-coin setup. However, the case of computationally unique VDFs with private-coin setup remained open. We close this gap by showing that even computationally expensive private-coin setup will not allow to construct VDFs in the ROM.

Let \( E \) be a non-Archimedean local field with residue characteristic \( p \), and let \( \ell \neq p \) be another prime number. Non-Abelian Lubin–Tate theory refers to the appearence of the local Langlands correspondence for \( \operatorname{GL}_n(E) \) within the \( \ell \)-adic cohomology of the Lubin–Tate perfectoid space. Building on a sketch provided in a paper by Yoichi Mieda (itself drawing on work by Drinfeld, Yoshida, and Weinstein), we investigate the geometry of this perfectoid space and define a rational subset admitting a certain formal model whose special fiber is isomorphic to the perfection of a Deligne–Lusztig variety. This connection elucidates links between aspects of the local Langlands correspondence for \( \operatorname{GL}_n(E) \) and Deligne–Lusztig theory, and enables an explicit description of certain components of the local Langlands correspondence for \( \operatorname{GL}_n(E) \). The necessary tools for this explicit description were developed by Yoichi Mieda.