J-Invariant-Based Classification of Elliptic Curves for Homomorphic Digital Signature Schemes

Elliptic curve selection plays a critical role in the efficiency and correctness of elliptic curve-based cryptographic constructions. In this paper, we present a rigorous number-theoretic framework for the classification and construction of elliptic curves optimized for homomorphic digital signature schemes, with a particular focus on the Elliptic Curve Homomorphic Digital Signature Algorithm (EHDSA). Our approach departs from conventional empirically driven curve selection by formally characterizing the algebraic conditions required for the efficient realization of the EHDSA homomorphic mapping ϕ. Using arithmetic invariants of elliptic curves, including j-invariants, complex multiplication structures, and endomorphism ring properties, we identify classes of curves that provably support the required homomorphic and algebraic compatibility conditions. We provide complete mathematical proofs establishing the correctness and necessity of these conditions, along with explicit curve construction algorithms. Under standard assumptions in elliptic curve cryptography, we analyze the impact of curve structure on homomorphic signature efficiency and demonstrate that appropriately classified curves yield measurable improvements over generic curve choices. Our results provide a principled foundation for elliptic curve selection in homomorphic digital signature systems and contribute a mathematically grounded methodology applicable to a broader class of elliptic curve-based cryptographic protocols.