Scalable Attribute-Based Encryption in Hamming Metric via Packed Additive Homomorphism

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 $1.58\times$ 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.