Optimizing Elligator Squared for BLS Curves: A Jacobi Quartic Approach

The BLS signature scheme is heavily utilized in modern decentralized protocols due to its signature aggregation capabilities. However, the raw storage and transmission of BLS public keys expose validators to subgroup membership leakage and linkability attacks. While Elligator Squared provides a cryptographic mechanism to obfuscate elliptic curve points into indistinguishable uniform random strings, its application to BLS12-family curves (e.g., BLS12-381, BLS12-479+, BLS12-539+) requires costly inverse isogeny maps and cofactor blinding/clearing. In this paper, we propose a highly optimized obfuscation and deobfuscation pipeline. By exploiting the existence of order-2 points on the isogenous curves, we establish a birational equivalence to the extended Jacobi Quartic. We demonstrate that performing heavy scalar multiplications—specifically cofactor blinding and Fast Cofactor Clearing (FCC)—inside the Jacobi Quartic coordinate space dramatically reduces computational latency. Utilizing wNAF scalar multiplication with optimized dedicated doubling (3M+4S+1C+3add), our implementation achieves unprecedented performance. Our benchmarks confirm that integrating these algebraic optimizations with mathematically trivial 2-isogenies completely shatters the traditional field-scaling penalty, securing the pipeline with an absolute minimum of computational overhead for verifying nodes.