The Investigation of Fully Homomorphic Encryption: Techniques,

Applications, and Future Directions

Yiling Xu

Data Science, University of Washington, 1320 NE Campus Pkwy, Seattle, U.S.A.

Keywords: Fully Homomorphic Encryption, CKKS Scheme, Gentry's First Scheme, BGV Scheme, FHE Application.

Abstract: Fully Homomorphic Encryption (FHE) is an encryption method that enables direct computation on encrypted

data without revealing its contents. This review offers a concise yet comprehensive overview of FHE, a pivotal

cryptographic technology enabling calculations on encrypted data without necessitating decryption. The paper

delves into the core concepts and principles underpinning FHE, elucidating the specific methodologies

integral to its application. A thorough exploration of various FHE schemes forms a significant part of the

study, where each scheme's functionality, advancements, and their broader implications on the field are

meticulously examined. The discourse extends to a critical analysis of FHE's practical applications across

diverse sectors, assessing its effectiveness and influence within the contemporary technological milieu. By

providing an in-depth look at both the theoretical and practical aspects of FHE, this paper aims to highlight

its potential and challenges, offering insights into its evolving role in the landscape of secure data processing

and privacy-preserving computations.

1 INTRODUCTION

The journey of cryptography through the ages has

been marked by continuous evolution, with each

advancement bringing with it a new realm of

possibilities. Among these, the concept of Fully

Homomorphic Encryption (FHE) stands as a

monumental achievement, a beacon of potential in the

vast sea of data security. This revolutionary idea, first

conceived by Rivest et al. in 1978, lay dormant for

decades, more a theoretical curiosity than a practical

tool (Wikipedia Contributors 2019). It wasn't until

2009, with Craig Gentry's pioneering work, that FHE

transitioned from an elusive cryptographic dream to a

tangible, albeit complex, reality. This marked a

seminal moment in data security, opening new

horizons in how sensitive data is handled and

processed (Wikipedia Contributors 2019).

Fully Homomorphic Encryption is a

cryptographic paradigm unlike any other. In essence,

it allows for computations to be performed on data

while it remains in an encrypted state, negating the

need for decryption during processing. This

groundbreaking capability ensures that data is not just

protected at rest or in transit, but also while in use, a

feat that traditional cryptographic methods have long

struggled to achieve (Van Dijk et al. 2010,

Armknecht et al. 2015). The implications of this are

vast and profound, particularly in an era where data is

not merely an asset but the very lifeblood of the

technological existence. In the digital world inhabit,

where data breaches and cyber threats loom large,

FHE offers a beacon of hope - a means to protect the

most sensitive of information in ways previously

deemed unattainable.

The inception of FHE traces back to a theoretical

framework, but it was Gentry's work that illuminated

its practical potential. This exploration commences

with an in-depth explication of the fundamental tenets

of FHE, progressing into a comprehensive

examination of the complex mathematical algorithms

that constitute the backbone of this intriguing

technological advance. It represents an intellectual

voyage across a terrain of polynomial algebra and

lattice-based cryptographic techniques, a domain

where theoretical constructs converge with practical

applications, and the esoteric realms of cryptography

are rendered comprehensible. This exploration

traverses a landscape of polynomial algebra and

lattice-based encryption, marrying theoretical

concepts with practical applications (Joye 2021).

Furthermore, this exploration is not confined to

the theoretical realm. The practical applications of

FHE are as diverse as they are impactful. In the realm

Xu, Y.

The Investigation of Fully Homomorphic Encryption: Techniques, Applications, and Future Directions.

DOI: 10.5220/0012819200004547

Paper published under CC license (CC BY-NC-ND 4.0)

In Proceedings of the 1st International Conference on Data Science and Engineering (ICDSE 2024), pages 253-259

ISBN: 978-989-758-690-3

253

of cloud computing, FHE offers a transformative

approach to data processing, enabling secure

computations on encrypted data within cloud

environments, thus ensuring privacy and

confidentiality. In sectors such as healthcare and

finance, where the sensitivity of data is paramount,

FHE provides a means to process and analyze

personal and financial information without

compromising security. Governmental agencies,

grappling with the dual challenges of data utility and

privacy, can leverage FHE to secure national interests

while safeguarding individual rights.

The Defense Advanced Research Projects Agency

(DARPA) has initiated a program called Data

Protection in Virtual Environments (DPRIVE)

(Uppal 2024). This program aims to develop an

accelerator for FHE. A key aspect of this initiative is

a collaboration between Intel and Microsoft. Intel has

signed an agreement with DARPA to participate in

the DPRIVE program, focusing on designing an

application-specific integrated circuit (ASIC)

accelerator to enhance the performance of FHE

computations. Microsoft is a key partner in this

initiative, responsible for leading the commercial

adoption of the technology. The goal is to enable

computations on fully encrypted data without needing

decryption, significantly reducing potential cyber

threats. This program is expected to have a broad

impact, including applications in healthcare,

insurance, and finance, as it allows organizations to

utilize and extract value from sensitive data without

risking exposure (Uppal 2024, Darpa.mil 2024, Intel

2024).

The DARPA DPRIVE program is a multiyear

effort that will involve several phases, starting with

the design, development, and verification of

foundational IP blocks, leading to the integration into

a system-on-chip and a full software stack (Uppal

2024). The collaboration between Intel and Microsoft

is aimed at bringing this technology into commercial

use, potentially revolutionizing the way sensitive data

is handled across various industries.

Inpher, a technology company, has developed

XOR Secret Computing®, which utilizes

cryptographic techniques including FHE to enable

secure, privacy-preserving analytics and machine

learning on encrypted data (Anon 2024). This

technology allows organizations to compute and

derive insights from encrypted datasets across

different jurisdictions and environments without

exposing the underlying data. It's particularly

beneficial for multi-party collaborations where data

privacy is crucial.

In this scholarly examination, it provides a

concise overview of the fundamental concepts and

underlying principles of FHE. It delves into the

specific methodologies employed in the application

of FHE, as well as an analysis of the diverse sectors

where FHE finds utility. The study encompasses a

comprehensive exploration of various FHE schemes,

detailing their functionalities, advancements, and the

consequent implications they have on the field.

Furthermore, it engages in a critical discourse on the

contemporary applications of FHE, assessing its

practicality and impact in the current technological

landscape.

2 METHOD

2.1 The Structure of FHE

Fully Homomorphic Encryption is a type of

encryption that allows computation on ciphertexts,

generating an encrypted result that, when decrypted,

matches the result of operations performed on the

plaintext. This means that data can be encrypted and

processed without ever exposing it in its unencrypted

form, thereby maintaining confidentiality and

security.

The structure and workflow of FHE shown in Fig.

1 can be divided into three stages. The first is the

encryption stage, which starts with the encryption of

plaintext data. Each piece of data is converted into

ciphertext using a public encryption key. This

encrypted data is now secure and can be stored or

transmitted without being read by unauthorized

parties (Marget 2022). The second is the calculation

stage. The difference between FHE and other

encryption methods is that it can directly perform

calculations on these encrypted data (ciphertext).

Operations such as addition, multiplication, and more

complicated functions can be performed without

decrypting the data. These calculations are performed

using algorithms specifically designed for encrypted

data. The third is the decryption stage. Once the

necessary calculations are completed, the resulting

ciphertext can be converted back into readable

plaintext. This is implemented using a private

decryption key. The result of decryption is the same

as when the same calculation is performed on the

unencrypted data.

ICDSE 2024 - International Conference on Data Science and Engineering

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Figure 1: FHE basic structure (Picture credit: Original).

2.2 Key Terminologies

In FHE, several key terminologies are crucial for

understanding its framework:

Plaintext: This refers to the original, unencrypted

data that is clear and readable.

Ciphertext: Once plaintext is encrypted using

FHE, it becomes ciphertext, which is unreadable

without the appropriate decryption key.

Encryption Key (Public Key): This is the key

used to encrypt plaintext. It's public, meaning it can

be shared without compromising security.

Decryption Key (Private Key): This key

decrypts the ciphertext back into plaintext and is kept

confidential by its owner.

Homomorphic Operations: These are

mathematical operations performed on ciphertexts

that yield results equivalent to operations done on

plaintext.

2.3 Four Properties of FHE

Fully Homomorphic Encryption offers four

properties, and here are two pivotal computational

abilities on encrypted data: homomorphic addition

and homomorphic multiplication. Homomorphic

addition, denoted by

𝑬𝒂 ⊕ 𝑬𝒃  𝑬𝒂 

𝒃

allows for the addition of two encrypted values,

E(a) and E(b), producing an encrypted sum that, when

decrypted, equals the sum of the original plaintext

values. This special addition operation works directly

on ciphertexts, enabling addition without decryption.

Figure 2: Translating Algorithms to Handle Fully

Homomorphic Encrypted Data on the Cloud (Marcolla et al.

2022).

Similarly, homomorphic multiplication,

expressed as 𝑬𝒂 ⊕ 𝑬𝒃  𝑬𝒂  𝒃, permits

the multiplication of encrypted data. The

multiplication of two encrypted values E(a) and E(b)

results in an encrypted product that reveals the

product of the plaintext values upon decryption. This

is essential as it allows for any computation reducible

to additions and multiplications to be executed on

encrypted data (Armknecht et al. 2015).

The most significant advantage of FHE lies in its

robust data privacy protection. Computation on

encrypted data ensures that the underlying data

remains confidential throughout the process (Michael

1970). This feature is crucial in scenarios where

sensitive data is processed by third parties, such as in

cloud computing, or where maintaining data privacy

The Investigation of Fully Homomorphic Encryption: Techniques, Applications, and Future Directions

255

is legally mandated. FHE enables these third parties

to perform necessary computations without ever

accessing the actual data, thus preserving privacy.

However, FHE faces a substantial challenge in

terms of computational efficiency. The intricate

mathematical operations involved in encryption,

decryption, and computation on ciphertexts are

resource-intensive (Yousuf et al. 2020). This makes

FHE considerably slower compared to traditional

encryption methods. Despite this, the field is evolving

rapidly, with new algorithms and hardware

advancements aiming to enhance efficiency. While

FHE hasn’t reached the efficiency of conventional

methods yet, its potential for secure data processing

in sensitive applications continues to drive interest

and research in the field (Michael 1970).

2.4 Key Schemes in FHE

In the realm of Fully Homomorphic Encryption,

several key schemes have been developed over the

years, each with unique characteristics and

contributions to the field. These schemes represent

various approaches to solving the challenges inherent

in FHE, such as computational efficiency and noise

management. Some of the notable FHE schemes

include:

Gentry's First Scheme: Craig Gentry's original

FHE scheme, introduced in his groundbreaking 2009

thesis, laid the foundation for FHE. This scheme was

based on lattice cryptography and introduced the

concept of "bootstrapping" to control the growth of

noise in ciphertexts during computations.

BGV Scheme: Named after its creators, the BGV

scheme improved upon Gentry's initial work. It

offered a more efficient approach to handling noise

and enabled more practical implementations of FHE

(Khnlil et al. 2017).

Cheon-Kim-Kim-Song (CKKS) Scheme: The

CKKS scheme, developed by Jung Hee Cheon et al.,

is particularly efficient for arithmetic on encrypted

floating-point numbers. This makes it well-suited for

applications in fields such as data science and

machine learning.

Ring Learning with Errors (RLWE) Schemes:

These schemes, which are variants of the original

Learning with Errors (LWE) problem, use ring-based

structures to achieve more efficient encryption and

computation. RLWE-based schemes are often more

practical for real-world implementations of FHE

(Christian et al. 2021, Translating Algorithms to

Handle Fully Homomorphic Encrypted Data on the

Cloud - Scientific Figure on ResearchGate 2023,

Marcolla et al. 2022) as shown in Fig. 2.

Fan-Vercauteren (FV) Scheme: Developed by

Junfeng Fan and Frederik Vercauteren, this scheme is

notable for its simplicity and efficiency. It has been

widely used in various software libraries

implementing FHE.

Gentry-Sahai-Waters (GSW) Scheme: This

scheme, proposed by Craig Gentry et al. is known for

its simplicity and the conceptual clarity it brings to

FHE. The GSW scheme has also influenced the

development of subsequent FHE constructions.

3 DISCUSSION

3.1 Advancement in FHE

FHE is an encryption technology that allows

calculations to be performed on encrypted data

without first decrypting it. This means data can

remain secure throughout its entire lifecycle –

whether at rest, in transit or, more importantly, during

processing.

One of the key developments in FHE in recent

years has been the introduction of various solutions

and libraries that enhance its feasibility and

performance. These include CONCRETE, TFHE,

FHEW, HEAAN, SEAL, PALISADE, Lattigo and

HELib, each supporting different schemes such as

CGGI, FHEW, CKKS, BGV and BFV. These

libraries and compilers address different aspects of

FHE, such as homomorphic arithmetic operations and

machine learning workloads (Gorantala et al. 2023).

Tools like nGraph-HE, SEALion, CHET, and EVA

are designed specifically for machine learning

applications, while tools like Cingulata and Encrypt-

Everything Everywhere focus on more general

computations.

3.1.1 CKKS Scheme

In the field of FHE, one of the most important and

widely used schemes is the Cheon-Kim-Kim-Song

(CKKS) scheme. The CKKS scheme is particularly

notable for its efficiency in handling floating-point

calculations, which makes it well suited for

applications in machine learning and data analysis.

This relevance to machine learning is crucial given

the growing importance of artificial intelligence and

big data across various industries (Notes on Lattices

2024). The CKKS scheme is able to efficiently

manage complex computations while maintaining

encryption.

The operation of the CKKS scheme involves

several key steps:

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Encoding/Decoding: The CKKS scheme

encodes real or complex vectors into polynomials

and, similarly, decodes these polynomials back to

vectors. This procedure uses the roots of cyclotomic

polynomials. To decode a polynomial into a vector,

the polynomial needs to be evaluated at certain values

(specifically the roots of the cycloidal polynomial)

(Notes on Lattices 2024). The encoding process, on

the other hand, is about finding a polynomial given a

vector such that when the polynomial is evaluated at

the roots of the cyclotoid polynomial, it is equal to the

original vector. This involves solving a system of

linear equations using a Vandermond matrix.

Security Basics: The security of the CKKS

scheme, like many other FHE schemes, is based on

difficult problems in lattice cryptography,

specifically the error learning (LWE) and error loop

learning (RLWE) problems (Notes on Lattices 2024).

These problems are considered difficult for

computers, including quantum computers, to

effectively solve.

Practical Applications: In practical applications,

CKKS can be used to perform complex calculations

on encrypted data. For example, in privacy-

preserving machine learning, CKKS allows training

machine learning models on encrypted data, ensuring

the privacy of the data while still enabling valuable

computations such as training and inference.

3.1.2 Practical Applications

The practical applications of FHE are very broad,

covering various industries such as finance,

healthcare, government, cloud computing, security

analysis and privacy-preserving machine learning.

FHE is also considered quantum-resistant, providing

an additional layer of security against potential

quantum computing threats (IBM 2021, Iapp 2013).

Healthcare Data Analytics: In the healthcare

industry, FHE enables secure and private analysis of

sensitive medical data. Hospitals and research

institutions can use FHE to perform computations on

encrypted patient data for research purposes without

exposing personal patient information. The

application is particularly valuable for collaborative

research involving multiple institutions where data

privacy is critical.

Financial Services: In the financial sector, FHE

can be used for secure data sharing and analysis. For

example, banks can analyze encrypted financial

records for fraud detection, risk analysis or credit

scoring, while ensuring the confidentiality of

individual customer data. This allows for deeper data

analysis without compromising privacy.

Cloud Computing and Data Outsourcing: FHE

implements secure computing on cloud platforms.

Enterprises can store and process encrypted data in

the cloud and perform computations on the encrypted

data itself. This means that sensitive data such as

business analytics, personal information or

proprietary algorithms can be processed in the cloud

without exposing the actual data to the cloud

provider.

3.2 The Challenges of FHE

However, FHE is not without its challenges. The

main issue currently hindering its widespread

adoption is the complexity of developing efficient

FHE applications. Performance, while greatly

improved, is still an issue, especially for advanced

imperative programs. The restrictive nature of FHE

operations and its non-intuitive performance

characteristics often require significant

transformations of the program to achieve efficiency.

To address these issues, recent work has focused on

developing compilers like HECO, which aims to

transform high-level programs into efficient and safe

FHE implementations.

Performance Issues: One of the main limitations

of FHE is its computational performance. Initially,

the FHE scheme was much slower than operating on

unencrypted data, sometimes a trillion times slower.

Although progress has been made in this area, and

recent advances have reduced performance

degradation, FHE operations are still quite slow, up

to a million times slower than unencrypted operations

(Baffle 2017). This significant speed reduction poses

a huge challenge for practical applications, especially

in situations where fast data processing is critical,

such as database queries.

Implementation Complexity: The development

of FHE applications is complex and requires expertise

in cryptography and programming. Developers need

to choose from various FHE schemes and libraries,

such as CONCRETE, TFHE, FHEW, HEAAN,

SEAL, PALISADE, Lattigo, and HELib, each with

their own advantages and limitations. Furthermore,

domain-specific compilers (e.g., ALCHEMY,

Marble, RAMPARTS) focus on a subset of

computations but suffer from inefficient

bootstrapping operations. General-purpose compilers

such as Cingulata and Encrypt-Everything-

Everywhere offer wider applicability, but are often

slower and require specific configuration (Gorantala

et al. 2023). The complexity of selecting the right

tools and implementing them effectively limits the

practical deployment of FHE.

The Investigation of Fully Homomorphic Encryption: Techniques, Applications, and Future Directions

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Lack of Unified APIs and Benchmarks: FHE

lacks standardized application programming

interfaces (APIs) and concrete benchmarks, further

complicating its integration with existing systems

(Gorantala et al. 2023). The lack of standardization

makes it difficult for developers to evaluate the

performance and security of their FHE

implementations and to ensure compatibility across

different systems and applications.

Standardization Efforts: Recognizing these

challenges, researchers are continually working to

develop FHE standards. The International

Organization for Standardization (ISO) and the

International Electrotechnical Commission (IEC)

have launched the FHE standards project, which is

currently in the consultation phase. Future phases of

standardization may involve the development of

standards for application areas including memory

encryption, wireless communications and portable

devices (Iapp 2013). Key management is an

important aspect of FHE and the focus of these

standardization efforts, as the current public key

infrastructure is not well suited for FHE.

3.3 The Development of FHE in the

Future

Regarding future developments, there are ongoing

efforts to standardize and benchmark FHE. The

International Organization for Standardization (ISO)

and the International Electrotechnical Commission

(IEC) have launched the FHE standards project,

which is currently in the consultation phase. This

standardization may extend to applications such as

memory encryption, wireless communications, and

multi-stakeholder software-defined networking (Iapp

2013). Key management is another area of focus,

especially given that the current public key

infrastructure is not well suited for FHE. In the future

they should accelerate FHE performance,

standardization and benchmarking, develop user-

friendly tools and libraries, collaborative research and

development.

4 CONCLUSION

This paper mainly describes the background

knowledge of FHE and some FHE methods. This is a

cutting-edge encryption technology that can calculate

encrypted data without decryption. Great strides have

been made in this area, with various schemes such as

CKKS, CONCRETE, TFHE and FHEW playing a

key role. Despite these advances, FHE still faces

challenges such as computational inefficiency,

implementation complexity, and lack of standardized

APIs and benchmarks. This paper studies in detail the

CKKS scheme, which is known for its efficiency in

handling real or complex numbers and its suitability

for applications such as machine learning. However,

an overarching theme in FHE development is

balancing its groundbreaking potential for secure data

processing with the ongoing challenge of optimizing

its performance and usability for wider practical

applications. It is anticipated that these challenges

and shortcomings can be overcome one by one in the

future. This article may lack some actual data, which

will be filled in in the future.

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