Steganographic Point Obfuscation via Explicit Inverse Isogenies for $j \in \{0, 1728\}$

Abstract

The mitigation of algebraic invariants in cryptographic protocols is a fundamental requirement for achieving censorship-resistant, information-theoretic indistinguishability uniform communication against Deep Packet Inspection (DPI). While recent literature and standard paradigms (RFC 9380) heavily optimize the forward Hash-to-Curve operation, the exact mathematical inverse---Point-to-Uniform obfuscation---remains a critical research blind spot. For structurally significant curves with $j \in \{0, 1728\}$, such as BLS12-381, direct application of the Simplified Shallue--van de Woestijne--Ulas (SSWU) algorithm is impossible, and attempting to invert recent direct encodings in strict constant time introduces a prohibitive multi-branch computational penalty. In this paper, we resolve this algorithmic bottleneck by deriving explicit analytical expressions for the inverse isogeny mappings required to perform highly efficient, constant-time point obfuscation. Specifically, we present the exact inverse functions for the 11-isogeny ($\phi^{-1}$) and 3-isogeny ($\varphi^{-1}$) over $\mathbb{F}_{p^2}$ for the BLS12-381 curve. Furthermore, we extend this methodology to the family of curves with $j=1728$ ($y^2=x^3+Ax$), constructing the dual 2-isogenies and demonstrating that the target curve uniquely possesses $j=287496$, guaranteeing an unconditionally reversible SSWU mapping. By integrating these deterministic, extraction-free inverse isogenies with the Elligator Squared framework, we provide a rigorous mechanism for the steganographic obfuscation of both public keys ($\mathbb{G}_1$) and digital signatures ($\mathbb{G}_2$). Finally, we introduce a novel steganographic filter for curve generation and present the proof-of-concept BLS12-479+ curve. Empirical benchmarks confirm that its native 2-isogeny bridge yields a $19.7\times$ speedup in deobfuscation and a $2.4\times$ speedup in obfuscation compared to the standard BLS12-381 baseline, establishing a computationally trivial blueprint for unconditionally DPI-resistant consensus layers.