Paper 2026/1278

Barriers for Transparent Algebraic Generation of Hard Supersingular Curves

Abstract

We study transparent public generation of hard supersingular curves: a public, seeded, rerunnable algorithm outputs a supersingular curve while exposing the seed, verification transcript, and all algebraic information reconstructible from the implementation. This setting is distinct from trusted or distributed ceremonies, where a witness may be hidden, erased, or zero-knowledge protected. We define a transcript-security model for this setting and develop barriers for several modeled algebraic generation routes. For modular-conjugacy samplers, we analyze loci of supersingular invariants admitting an isogeny to their Frobenius conjugate; small public parameters give efficiently recognizable sparse support, while extractable high-parameter witnesses yield non-scalar endomorphisms by Frobenius composition. We extend this leakage principle to bounded correspondence witnesses, including kernel, rational-map, Hecke--Brandt, and elliptic-return detour traces. For direct samplers, we separate predicate-only search, local-neighbor exploration, bounded-relation witnesses, low-formal-degree ambient algebraic maps, and degree-one indexed maps from \(\F_p\) to \(\F_{p^2}\). The resulting theorems are deliberately model-bounded: they do not rule out hidden-witness ceremonies, higher-degree indexed maps, compact high-degree root indexing, endpoint-only mechanisms, or nonlocal aggregate sampling. Full proofs are deferred to the appendices.