[Paper Review] A Survey on Homomorphic Encryption Schemes: Theory and Implementation

Abbas Acar, Hidayet Aksu+2 more|arXiv (Cornell University)|Apr 12, 2017

Cryptography and Data Security133 references184 citations

TL;DR

This paper surveys homomorphic encryption (HE) including partially, somewhat, and fully homomorphic schemes, their theory, implementations, and future directions, with emphasis on Gentry-type FHE progress post-2009.

ABSTRACT

Legacy encryption systems depend on sharing a key (public or private) among the peers involved in exchanging an encrypted message. However, this approach poses privacy concerns. Especially with popular cloud services, the control over the privacy of the sensitive data is lost. Even when the keys are not shared, the encrypted material is shared with a third party that does not necessarily need to access the content. Moreover, untrusted servers, providers, and cloud operators can keep identifying elements of users long after users end the relationship with the services. Indeed, Homomorphic Encryption (HE), a special kind of encryption scheme, can address these concerns as it allows any third party to operate on the encrypted data without decrypting it in advance. Although this extremely useful feature of the HE scheme has been known for over 30 years, the first plausible and achievable Fully Homomorphic Encryption (FHE) scheme, which allows any computable function to perform on the encrypted data, was introduced by Craig Gentry in 2009. Even though this was a major achievement, different implementations so far demonstrated that FHE still needs to be improved significantly to be practical on every platform. First, we present the basics of HE and the details of the well-known Partially Homomorphic Encryption (PHE) and Somewhat Homomorphic Encryption (SWHE), which are important pillars of achieving FHE. Then, the main FHE families, which have become the base for the other follow-up FHE schemes are presented. Furthermore, the implementations and recent improvements in Gentry-type FHE schemes are also surveyed. Finally, further research directions are discussed. This survey is intended to give a clear knowledge and foundation to researchers and practitioners interested in knowing, applying, as well as extending the state of the art HE, PHE, SWHE, and FHE systems.

Motivation & Objective

• Explain the basic concepts and taxonomy of homomorphic encryption (HE, PHE, SWHE, FHE).
• Review major HE families and how they enable different algebraic operations on encrypted data.
• Summarize practical implementations, performance improvements, and hardware/software considerations.
• Discuss open research challenges and future directions in HE research and applications.

Proposed method

• Introduce the HE landscape and definitions of Enc, Dec, Eval, and KeyGen.
• Classify HE schemes into PHE, SWHE, and FHE and describe their operation limits.
• Present well-known HE schemes and their homomorphic properties (addition, multiplication, or both).
• Survey Gentry-type FHE developments and subsequent improvements and implementations.
• Discuss hardware and software implementation aspects and practical performance concerns.
• Outline future research directions and knowledge gaps in HE research and practice.

Experimental results

Research questions

• RQ1 What are the core differences between PHE, SWHE, and FHE in terms of supported operations and practical limitations?
• RQ2 Which well-known HE schemes support additive and/or multiplicative homomorphism, and what are their concrete limitations?
• RQ3 What progress and implementations have emerged following Gentry’s 2009 FHE breakthrough?
• RQ4 What are the key research directions and open challenges in making HE practical across platforms?

Key findings

• HE schemes are categorized into PHE (unlimited number of operations), SWHE (limited number of operations), and FHE (universal computability).
• Gentry’s 2009 FHE breakthrough remains foundational; subsequent work focuses on practical performances, reductions, and hardware/architecture optimizations.
• A broad survey of HE families (RSA, El-Gamal, Benaloh, Goldwasser–Micali, GS, KT1/KTx, Boyen–Goh–Nis sinh, CPS, and more) shows diverse tradeoffs in additively and/or multiplicatively homomorphic capabilities and security assumptions.
• Many real-world HE implementations emphasize optimized evaluation strategies, encoding schemes, and circuit-friendly approaches, aiming to reduce noise growth and expansion to improve practicality.
• Post-Gentry work often targets efficient HE in practice, including hardware acceleration, optimized schemes, and hybrid approaches to balance security and performance.
• The survey provides a consolidated reference for researchers and practitioners to understand theoretical foundations, implementation options, and future research directions in HE.