A Comparative Framework for Post-Quantum Cryptography and Homomorphic Encryption: Integration, Methodologies, and Applications

Abstract

Abstract The rapid evolution of quantum computing fundamentally threatens modern cryptographic infrastructures, necessitating a paradigm shift toward advanced encryption techniques. While post-quantum cryptography (PQC) ensures that data remains secure against quantum algorithmic decryption, it does not inherently facilitate the processing of that data in untrusted environments. Homomorphic encryption (HE) addresses this computational privacy requirement, yet traditional HE schemes rely on classical mathematical problems that are easily compromised by quantum systems. This paper presents a comprehensive comparative framework that integrates post-quantum cryptographic principles with homomorphic capabilities to enable quantum-resistant, privacy-preserving computation. By analyzing lattice-based algorithms, code-based cryptography, and hardware acceleration paradigms, this study maps the existing trade-offs between security, computational overhead, and practical deployment feasibility in the impending quantum era. Keywords: PQC, HE. 1.Introduction The rapid evolution of quantum computing technology presents a profound threat to modern cryptographic infrastructures. As quantum algorithms, notably Shor's algorithm, mature, they acquire the capability to solve nondeterministic polynomial time problems in polynomial time, thereby breaking traditional asymmetric encryption standards such as RSA and Elliptic Curve Cryptography (ECC) (Chen, 2024). Consequently, the development of post-quantum cryptography (PQC) has become a global imperative to secure sensitive communications and data against future quantum adversaries (Pranjal & Chaturvedi, 2024). Cryptographers are currently focusing on transitioning secure systems to these newly established PQC standards before large-scale quantum computers become commercially viable. While PQC addresses the secure transmission and storage of data, it does not inherently solve the problem of secure data processing in untrusted cloud environments. Homomorphic encryption (HE) provides a sophisticated mechanism that allows computations to be performed directly on encrypted data without requiring prior decryption (Jain & Cherukuri, 2023). The core problem addressed in this paper is the conceptual and practical integration of PQC with HE shown in Fig1.1 which is essential for ensuring quantum-safe, privacy-preserving computation in distributed architectures. Achieving this synthesis is highly complex due to the massive computational overhead and noise accumulation inherent to both cryptographic domains. However, existing approaches to privacy-preserving computation remain broadly insufficient for the post-quantum era. First, conventional homomorphic encryption schemes rely heavily on mathematical hardness assumptions, such as integer factorization or discrete logarithms, which are demonstrably vulnerable to quantum attacks (Chen, 2024). Second, while some high-performance fully homomorphic implementations exist, they suffer from prohibitive computational overhead and severe ciphertext expansion, making widespread deployment impractical without specialized mitigation strategies (Tseng et al., 2025). Third, hardware-accelerated privacy solutions often rely on trusted execution environments, like Intel SGX, which inherently introduce side-channel vulnerabilities and force reliance on centralized hardware trust rather than pure mathematical security (Sadat et al., 2017). To overcome these critical limitations, this paper proposes a structured comparative methodology for evaluating quantum-resistant homomorphic encryption systems. Specifically, our paper makes the following main contributions: We provide a comprehensive comparative framework that categorizes and evaluates lattice-based and code-based homomorphic encryption schemes against classical performance constraints. We propose a structured, hypothetical evaluation pipeline designed to benchmark the computational overhead and latency of post-quantum homomorphic algorithms in federated networking environments. 2.Privacy-preserving computation The literature surrounding privacy-preserving computation can be broadly classified into three primary categories, the first being classical and fully homomorphic encryption (FHE) systems. The core idea behind these schemes is the preservation of algebraic structures during encryption, allowing operations on ciphertexts to reflect directly on the underlying plaintexts without exposing the raw data (Jain & Cherukuri, 2023). A major strength of these classical FHE systems, such as those utilized in encrypted relational databases like FHE-SQL, is their ability to provide end-to-end cryptographic protection for queries without relying on trusted hardware modules (Tseng et al., 2025). However, a fundamental weakness is that early FHE models, and their underlying privacy homomorphisms, remain fundamentally susceptible to quantum cryptanalysis due to their reliance on classical mathematical problems (Sen, 2013). Compared to these classical paradigms, our work specifically focuses on migrating these operational homomorphic properties into genuinely quantum-safe mathematical frameworks. The second category of related work focuses on hardware acceleration and hybrid computational environments for homomorphic encryption. The core idea here involves leveraging specialized hardware, such as ASIC AI chips or coarse-grained compute architectures like Google's TPUs, to mitigate the enormous computational overhead associated with HE (Tong et al., 2025). The strength of this approach lies in its ability to bridge the energy-efficiency gap, often utilizing intelligent compiler frameworks to convert high-precision modular arithmetic into dense, low-precision matrix multiplications. Conversely, the weakness of hardware-centric methods is that architectural mismatches frequently lead to severe resource under-utilization, and hybrid solutions utilizing components like Intel SGX introduce hardware trust assumptions that can compromise pure cryptographic integrity (Sadat et al., 2017). Unlike these approaches, our framework prioritizes algorithmic resilience and quantum resistance over purely hardware-driven and enclave-based optimizations. The third category encompasses emerging post-quantum and quantum homomorphic paradigms, which seek to unify quantum resistance with computational privacy. Researchers have proposed lattice-based HE schemes to replace vulnerable legacy systems like Paillier and ElGamal, successfully enabling secure applications such as federated learning (Chen, 2024). Similarly, code-based cryptography utilizing McEliece and non-negative matrix factorization has shown promise in providing faster encryption and decryption times while remaining mathematically secure against quantum computers (Chen, 2024). Furthermore, advanced quantum homomorphic encryption protocols have explored information-theoretic security combined with homomorphic quantum error-correction to ensure future-proof data protection (Ouyang & Rohde, 2022). While these systems represent the state-of-the-art, their primary weakness remains the massive memory and communication overhead introduced during distributed training protocols, which negatively impacts classification performance. Our approach builds upon this category by proposing a standardized methodology to systematically compare these disparate post-quantum homomorphic techniques. 3.Multi-stage Method To systematically evaluate and deploy post-quantum homomorphic encryption, we propose a structured, multi-stage methodological framework. The first step involves cryptographic primitive selection, where system architects must evaluate and choose between lattice-based constructs, such as ML-KEM adaptations, and code-based systems like McEliece, depending on their domain constraints (Chen, 2024). The second step requires the implementation of an algorithmic parameter optimization module, which dynamically adjusts polynomial degrees and error distributions to balance security levels with the operational multiplicative depth required by the application. Finally, the third step comprises the integration of a specialized abstraction layer designed to map complex high-precision cryptographic operations into hardware-friendly matrix operations, thereby reducing the inherent bottleneck of ciphertext evaluation. The key design choices within this framework are driven by the absolute necessity to balance quantum resistance with practical algorithmic utility. We prioritize lattice-based cryptography for our primary evaluation pathway because of its established resilience against quantum threats and its recent standardization by the National Institute of Standards and Technology (NIST) (Chen, 2024). Additionally, we explicitly design our framework to avoid reliance on trusted hardware environments, opting instead for pure cryptographic guarantees to prevent the side-channel data leaks that often plague hardware-enclave solutions. By focusing on algorithmic alignment and basis-aligned mathematical transformations, the framework ensures that homomorphic workloads can be optimized effectively without compromising the underlying post-quantum security proofs. To validate the efficacy of the proposed framework, we outline a comprehensive evaluation plan utilizing a hypothetical, high-dimensional healthcare benchmarking dataset. The experimental setup will simulate a secure genome-wide association study (GWAS) executing within a fully decentralized federated learning environment (Sadat et al., 2017). The primary metrics for this hypothetical evaluation will include key generation latency, ciphertext expansion ratio, and the amortized computational time required for homomorphic multiplication operations across distributed nodes. By systematically varying the cryptographic noise parameters and observing the resulting degradation in analytical performance, we aim to empirically map the trad