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

The mitigation of algebraic invariants in cryptographic protocols is a fundamental requirement for achieving information-theoretic indistinguishability and zero-metadata communication. While the standard hash-to-curve paradigm (RFC 9380) effectively employs the Simplified Shallue–van de Woestijne–Ulas (SSWU) algorithm alongside low-degree isogenies to map uniform strings to elliptic curves, the inverse problem — obfuscating valid curve points into uniform pseudorandom noise — remains mathematically non-trivial for curves where direct SSWU application is impossible.In this paper, we derive explicit analytical expressions for the inverse isogeny mappings required to perform constant-time point-to-uniform obfuscation on cryptographically significant curves with $j \in \{0, 1728\}$. 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 $j=287496$, which guarantees an unconditionally reversible SSWU mapping.By integrating these explicit inverse isogenies with the Elligator Squared framework, we provide a rigorous, constant-time mechanism for the steganographic obfuscation of both public keys ($\mathbb{G}_1$) and digital signatures ($\mathbb{G}_2$). This ensures that all transmitted cryptographic elements can be deterministically disguised as uniformly distributed random data, rendering heuristic channel analysis ineffective.