This paper introduces a scalable Additively Homomorphic (A-Homomorphic) encryption framework that bridges the gap between rigorous cryptographic primitives and the flexible, threshold-based requirements of multi-party cryptographic protocols. At the core of our construction is a novel Knapsack-inspired Base-3 packing technique, which provides a mathematically optimal method for emulating bitwise XOR operations within additively homomorphic scalar rings. We prove that a ternary radix (b=3) represents the unique integer optimum for carry-free homomorphic containers, yielding a 1. 58 improvement in information density over traditional binary encodings. To address practical scalability, we adapt the Gaudry--Schost algorithm with equivalence class optimizations, enabling efficient resolution of the short-interval Elliptic Curve Discrete Logarithm Problem (ECDLP). Furthermore, we integrate a steganographic uniform string mapping, which completely eliminates algebraic invariants from the public ciphertexts. This approach achieves perfect information-theoretic indistinguishability while strictly maintaining optimal ciphertext expansion rates without relying on structural parity bits. Finally, by preserving the algebraic structure of underlying error-correcting codes, our framework introduces the paradigm of ``homomorphic blind decoding. '' Grounded in the NP-complete Syndrome Decoding Problem, this approach ensures robust security against advanced algebraic cryptanalysis while paving the way for privacy-preserving, error-tolerant cryptographic architectures.
Andrey Chmora (2026) studied this question.