What is Homomorphic Encryption?

Homomorphic Encryption

Definition and basic concept

Homomorphic
encryption is a revolutionary advancement in
cryptography that allows computations to be performed directly on encrypted
data . Unlike traditional methods where data must be decrypted to be used,
homomorphic encryption ensures that information remains protected at all stages
of processing.

This
process offers complete confidentiality: only those with the decryption keys
can access the results. This makes it particularly valuable for sensitive
sectors, such as healthcare , financial services , and cloud
computing , where data protection is paramount.

In
short, this technology combines advanced security with convenience ,
eliminating the need to compromise between privacy and functionality.

History and evolution

The
concept of homomorphic encryption first appeared in 1978 thanks to visionary
researchers such as Ronald Rivest, Leonard Adleman , and Michael Dertouzos .
However, it wasn't until 2009 that Craig Gentry proposed the first
working implementation with a scheme based on cryptographic lattices .

Since
then, significant progress has been made:

• Cryptographic
  noise reduction , which
  improved the accuracy of calculations.
• Optimization
  of algorithms , making processing
  faster.
• Development of open
  source libraries such as HElib (IBM) and SEAL (Microsoft),
  promoting wider adoption.

Comparison with other cryptographic approaches

Homomorphic
encryption is distinguished from other cryptographic methods, such as symmetric
algorithms or Zero-Knowledge
(ZK) proofs , by its unique ability to preserve the confidentiality
of data during processing .

Cryptographic approach	Main feature	Key advantage	Main limitation
Homomorphic encryption	Calculations on numerical data	Permanent confidentiality	High computational complexity
Symmetric encryption	Fast encryption and decryption	Simplicity and speed	Data not protected during calculation
Zero-Knowledge (ZK) Proofs	Verification without revealing information	Scalability on blockchain	Centralization of calculations

Unlike
symmetric encryption , FHE ( Fully Homomorphic Encryption ) guarantees
protection even during processing. Although it shares with ZK a
confidentiality objective, homomorphic encryption does not require a
centralized prover .

Principles and algorithms of homomorphic encryption

General operation

Homomorphic
encryption is revolutionizing the way
sensitive data is used. Unlike traditional methods that require decrypting
data to perform processing, FHE ( Fully Homomorphic Encryption )
allows direct calculations on encrypted data , without ever revealing
their contents.

Basic
principle : The results of calculations on
encrypted data are identical to those obtained on clear data, but remain protected
by permanent encryption .

Example
: A company may encrypt its data
internally before transmitting it to an external service provider, such as a
cloud provider. The latter performs the necessary calculations without ever
accessing the unencrypted data. The results, still encrypted, are then returned
to the company, where they are decrypted locally.

Conditions for effective encryption

To
be fully functional, a homomorphic encryption system must meet several
conditions:

• Correction
  : The results of encrypted operations, once
  decrypted, must be faithful to those obtained with unencrypted data.
• Error
  handling : Calculations on
  encrypted data often generate cryptographic noise . This noise must
  be limited to ensure the accuracy and reliability of the results.

Types of homomorphic encryption

1. Partially
   homomorphic encryption (PHE)
   Allows a single operation (addition or multiplication) to be
   performed on encrypted data. This type of encryption is lightweight and
   fast, ideal for simple use cases .
   
   
• Example : A bank can add up its
  customers' encrypted deposits to calculate the total without ever
  decrypting each deposit .
2. Fully
   homomorphic encryption (FHE)
   offers complete flexibility , supporting an unlimited sequence
   of arithmetic operations . However, this method is more complex and
   resource-intensive.
   
   
• Example : Secure machine learning ,
  where models can be trained on encrypted data without compromising their
  confidentiality.

Mathematical properties

Homomorphic
encryption is based on solid foundations in linear algebra and number
theory . Key features include:

• Homomorphic
  addition : Ability to add two
  encrypted data without decryption.
• Homomorphic
  multiplication : Ability
  to multiply encrypted data.
• Combining
  operations : Performing complex
  calculations by combining several types of operations.

To
ensure robust security , even in the face of threats from quantum
computers , the FHE uses tools such as cryptographic lattices (mathematical
networks).

Examples of popular algorithms

• Gentry's
  FHE : The first fully homomorphic algorithm,
  proposed in 2009.
• BGV : A scheme optimized for complex calculations, reducing processing
  times.
• CKKS : Designed for approximate calculations, often used in applications
  like machine learning .

Practical applications of homomorphic encryption

Privacy in blockchain

Homomorphic
encryption brings a new dimension of privacy
to the traditionally transparent and public blockchain . With encrypted blockchains, even smart contracts can remain confidential, accessible only to authorized
users. Validators , on the other hand, do not have access to sensitive
data, ensuring a high level of protection.

Securing cloud computing

In
cloud computing , homomorphic encryption transforms computing spaces
into truly secure environments . It allows complex computations to be
performed without the data ever being decrypted.

Example
: A company can analyze a fully
encrypted customer database through a cloud service. The provider then has no
access to sensitive information, thus complying with regulations such as GDPR
in Europe.

Protection of medical data

Homomorphic
encryption meets the critical needs of the medical sector by ensuring the
confidentiality of sensitive data while facilitating its exploitation.
It complies with strict regulations such as GDPR ( European Union) and HIPAA
(United States).

Example
: Researchers can analyze encrypted
medical records without compromising patient privacy. This technology promotes secure
collaboration between medical institutions while reducing the risk of
sensitive data breaches.

Secure financial transactions

Banks
and financial institutions are adopting homomorphic encryption to protect
sensitive data such as revenue or transaction history. This technology
allows for analysis without decryption, ensuring complete confidentiality .

Example
: A bank can assess a customer's
creditworthiness or calculate financial risks while maintaining the
confidentiality of the information. This also helps meet regulatory data
standards.

Secure Search and Privacy

Homomorphic
encryption facilitates secure searches by protecting both queries and
database contents. It is particularly important in sensitive areas such as national
defense or advanced technologies .

Example
: Machine learning models
can be trained directly on encrypted data, ensuring complete confidentiality,
even in shared or collaborative environments.

Towards a quantified daily life

In
the future, homomorphic encryption could become as common a technology as HTTPS
connections on the Internet.

Example
: A concept similar to " httpz
" could emerge, ensuring end-to-end encryption not only for data
transmission, but also for data processing and storage. This would provide comprehensive
protection against cyber threats in diverse sectors, ranging from
healthcare to finance, blockchain, and online gaming.

Recent advances in homomorphic encryption

Technological evolution and advances

Since
the 1970s, homomorphic encryption has come a long way, evolving from a
purely theoretical concept to a practical technology . This evolution
was made possible by the development of open source tools such as HElib (IBM)
and SEAL (Microsoft). These tools have enabled its adoption in strategic
sectors such as cloud computing , healthcare and finance .

Significant progress:

• Cryptographic
  noise reduction : Modern
  schemes such as BGV and CKKS have limited the accumulation
  of noise generated by successive calculations. This has significantly
  improved the speed and reliability of operations.
• Algorithmic
  optimizations : Processing times have
  been reduced while increasing the accuracy of calculations , making
  the technology more efficient.
• Dedicated
  hardware support :
  Cryptography-specific processors accelerate computations, making
  homomorphic encryption more accessible to resource-intensive
  applications .

However,
challenges remain , particularly for environments requiring low latency,
such as the Internet of Things (IoT) .

Recent technological innovations

• Homomorphic
  machine learning : This
  innovation enables predictive models to be trained on encrypted
  data, guaranteeing total confidentiality. It revolutionizes the secure
  analysis of sensitive data , particularly in health and finance.
• Secure
  Multi-Party Computation : By
  combining homomorphic encryption with other cryptographic techniques, this
  approach facilitates collaboration between companies while protecting the
  information exchanged.
• Post-quantum
  resilience : Current research is
  moving towards schemes that are resistant to attacks on quantum
  computers , thus ensuring the sustainability of this technology in the
  future.

Tools and Libraries for Adoption

Library	Main features
Microsoft SEAL	Versatile and open source, suitable for research and industries.
IBM HELib	Optimized for complex arithmetic calculations, with great flexibility.
Palisade	Ideal for cloud and IoT, meeting the needs of large-scale applications.
TenSEAL	Specially designed for the machine secure learning , facilitating the training of models on encrypted data.

These
advancements enable wider adoption, particularly in demanding environments such
as cloud, machine learning and interconnected systems.

Advantages of homomorphic encryption

Homomorphic
encryption (FHE) is a leading innovation in the protection of
sensitive data , providing unprecedented guarantees of security and confidentiality
. This technology is essential for companies and institutions handling
large amounts of critical information , particularly in the fields of healthcare
, finance and cloud computing .

Enhanced confidentiality

FHE
ensures that data remains encrypted throughout processing , eliminating
the risk of accidental or malicious exposure. Sensitive data is always
protected, even from systems or third parties performing the calculations.
Service providers and other stakeholders never have access to the information
in plain text, which enhances confidentiality.

Secure outsourcing

Homomorphic
encryption allows complex processing to be outsourced to cloud service
providers or other third parties without compromising data confidentiality.
Service providers can perform the necessary operations without ever seeing the
data content. This mechanism is particularly useful for business
collaborations, where data protection is crucial.

Regulatory compliance

The
FHE helps businesses comply with strict data protection regulations,
such as the GDPR in Europe and the CCPA in the United States.
Businesses reduce the risk of legal sanctions by ensuring the security of
personal information. Users and partners are reassured by rigorous and
transparent data management.

Versatility and flexibility

Homomorphic
encryption offers unparalleled flexibility compared to other encryption
methods, allowing complex computations to be performed directly on encrypted
data .

Balanced traceability

FHE
offers an ideal combination of anonymity and traceability ,
making it suitable for sensitive scenarios such as blind auctions (participants
remain anonymous while ensuring the integrity of the process.) or OTC market transactions (sensitive data is protected while meeting compliance
requirements .)

Disadvantages of homomorphic encryption

Despite
its many advantages, full homomorphic encryption (FHE) has significant
limitations that hinder its widespread adoption. In areas such as IoT or
real -time computing platforms , speed and power limitations reduce
the effectiveness of this technology.

High computational complexity

Computations
performed on encrypted data require significantly more computing resources than
those performed on plaintext data. The direct consequence is increased
slowness, which is particularly problematic for applications requiring low
latency or real-time processing , such as cloud services or IoT devices.

Size of encrypted files

Data
encrypted with homomorphic algorithms is much larger than its plaintext
counterparts. This increase in size complicates its use in environments where storage
space is limited or expensive. As a result , infrastructures must be sized
to accommodate these volumes, increasing operating costs.

Complex implementation

Implementing
homomorphic encryption requires advanced technical expertise , making
its deployment complex for many organizations. Therefore, developers must
redesign their algorithms to work without access to cleartext data ,
which represents a major programming challenge. The tools needed for
large-scale deployment require significant investment , thus limiting
access to FHE to large companies and institutions with sufficient resources.

Table of advantages and disadvantages of homomorphic encryption

Benefits	Disadvantages
Data remains encrypted throughout processing, ensuring its confidentiality.	Calculations require high computing resources, slowing down processing.
Allows complex processing to be outsourced without compromising data confidentiality.	Encrypted files are much larger, increasing storage requirements and costs.
Helps comply with regulations like GDPR and CCPA, reducing legal risks.	Implementation requires advanced expertise and expensive infrastructure.
Allows complex calculations directly on encrypted data, without decryption.	Current performance is insufficient for real-time or low-latency applications.
Offers a balance between anonymity and traceability, suitable for cases such as auctions or regulated transactions.	Limited to resource-rich environments, hindering its widespread adoption.

Conclusion: A promising technology facing many challenges

Homomorphic
encryption (FHE) is emerging as a critical solution to address the
growing cyber threats and the need to protect sensitive data. By
enabling direct computation on encrypted data , it ensures complete
confidentiality while meeting compliance requirements, such as GDPR and
CCPA .

However,
this technology is hampered by major challenges: computational complexity, high
costs, and implementation difficulties. These limitations restrict its adoption
to large enterprises and raise questions about its accessibility.

To
democratize FHE, initiatives such as the development of open source tools and
the establishment of audit standards are necessary. At the same time, it opens
up promising prospects, particularly in confidential auctions , secure
transactions and blockchain applications .

By
combining security , flexibility and confidentiality ,
homomorphic encryption positions itself as a pillar of data protection in an
increasingly digital world.

Recommendations

To
effectively integrate homomorphic encryption , it is essential to adopt
a structured approach:

• Needs
  Assessment : Identify priority use
  cases, such as secure processing of medical or financial data.
• Tool
  selection : Select libraries that
  match the requirements, such as Microsoft SEAL or HElib .
• Resource
  planning : Consider costs related
  to implementation, infrastructure, and team training.
• Phased
  Adoption : Deploy FHE in stages to
  anticipate and resolve technical or operational challenges.