Post-Quantum Homomorphic Encryption: A Case for Code-Based Alternatives

TL;DR

This review assesses post-quantum homomorphic encryption with a focus on code-based alternatives to the dominant lattice-based approaches. It surveys definitions, generations, and security foundations of HE, then details code-based cryptography (NP-hard decoding and code-equivalence problems, McEliece/Niederreiter/Alekhnovich frameworks) and several CBHE schemes (e.g., Bogdanov–Lee, Armknecht, rank-metric variants). The authors compare code-based HE to lattice-based schemes, highlighting quantum resistance and simpler arithmetic as key advantages, while noting practical barriers such as key/ciphertext sizes and nascent FHE tooling. The piece concludes with five proposed research directions to advance post-quantum code-based HE, aiming to provide robust, scalable alternatives that complement lattice-based PQHE in secure computation under quantum threat.

Abstract

Homomorphic Encryption (HE) allows secure and privacy-protected computation on encrypted data without the need to decrypt it. Since Shor's algorithm rendered prime factorisation and discrete logarithm-based ciphers insecure with quantum computations, researchers have been working on building post-quantum homomorphic encryption (PQHE) algorithms. Most of the current PQHE algorithms are secured by Lattice-based problems and there have been limited attempts to build ciphers based on error-correcting code-based problems. This review presents an overview of the current approaches to building PQHE schemes and justifies code-based encryption as a novel way to diversify post-quantum algorithms. We present the mathematical underpinnings of existing code-based cryptographic frameworks and their security and efficiency guarantees. We compare lattice-based and code-based homomorphic encryption solutions identifying challenges that have inhibited the progress of code-based schemes. We finally propose five new research directions to advance post-quantum code-based homomorphic encryption.

Figures (3)

• Figure 1: Distribution of review papers on Homomomorphic Encryption published in last 5 years by year of publication.
• Figure 2: Scimagosci journal quartile distribution of review papers on Homomorphic Encryption in 5 years
• Figure 3: Illustration of the general Homomorphic Encryption process. Here $X, Y$ are plaintexts, $A, B$ are encryptions of $X, Y$ respectively, $C$ is the homomorphic evaluation of $A, B$, while $Z$ is the decrypted value of $C$.

Theorems & Definitions (1)

• Definition 1