Optimized Hybrid Homomorphic Signed-Integers Encryption: Addressing Challenges and Enhancing Capabilities

Abstract

Homomorphic cryptography produces encrypted data that supports computation, but current exponential techniques face critical limitations: they leak plaintext information for negative values and fail on trivial values (0, ±1). Additional challenges include private key disclosure, encryption failure under specific conditions, and poor performance and storage efficiency. This paper addresses these limitations through comprehensive analyses, identifying the mathematical conditions that cause encryption failure, and examining their relationship to the modulus’s randomness. A hybrid encryption approach is proposed in which the linear technique complements the exponential technique, particularly for negative and trivial values. The scheme uses the Carmichael function λ(n) to reduce computational costs and provides a unified decryption algorithm that supports signed-integer operations. The unified decryption formula with rounding operations successfully handles positive and negative values, enabling true homomorphic subtraction alongside existing addition and multiplication properties. Decryption is optimized using a single prime key, enhancing security and reducing complexity. Experimental verification shows that the proposed technique passes all 15 National Institute of Standards and Technology tests, with probability values ranging from 0.122325 to 0.991468. Improvements include a 71.8% reduction in exponent size, 89.5% faster encryption, 67.2% faster decryption, and a 33.4% reduction in storage requirements, making the proposed hybrid technique suitable for resource-constrained secure computation applications.