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Homomorphic Encryption: What Is It and How Is It Used

Homomorphic Encryption: What Is It and How Is It Used

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July 15, 2022 | Anastasios Arampatzis

Every day, organizations handle a lot of sensitive information that needs to be encrypted both when it is stored (data at rest) and when it is being transmitted (data in transit). 

The problem with encrypting data is that sooner or later, you have to decrypt it. And decrypting data makes it vulnerable to hackers. You can keep your cloud files cryptographically scrambled using a secret key, but as soon as you want to actually do something with those files—anything from editing a word document or querying a database of financial data—you have to unlock the data and leave it vulnerable. Homomorphic encryption, an advancement in the science of cryptography, changes that.

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What is Homomorphic Encryption?

The purpose of homomorphic encryption is to allow computation on encrypted data. Thus data can remain confidential while it is processed, enabling useful tasks to be accomplished with data residing in untrusted environments. In a world of distributed computation and heterogeneous networking this is extremely valuable.

A homomorphic cryptosystem is like other forms of public encryption in that it uses a public key to encrypt data and allows only the individual with the matching private key to access its unencrypted data. However, what sets it apart from other forms of encryption is that it uses an algebraic system to allow a variety of computations (or operations) on the encrypted data.




In mathematics, homomorphic describes the transformation of one data set into another while preserving relationships between elements in both sets. The term is derived from the Greek words for “same structure.” Because the data in a homomorphic encryption scheme retains the same structure, identical mathematical operations, whether they are performed on encrypted or decrypted data, will result in equivalent results.

In practice, most homomorphic encryption schemes work best with data represented as integers and while using addition and multiplication as the operational functions. This means that the encrypted data can be manipulated and analyzed as though it’s in plaintext format without actually being decrypted. The encrypted data can be computed and processed to get an encrypted answer, but only you can decrypt the ciphertext and understand what it means. 

Types of Homomorphic Encryption:

There are three types of homomorphic encryption. The primary difference between them is related to the types and frequency of mathematical operations that can be performed on the ciphertext. The three types are:

  • Partially Homomorphic Encryption
  • Somewhat Homomorphic Encryption
  • Fully Homomorphic Encryption

Partially homomorphic encryption (PHE) allows only select mathematical functions to be performed on encrypted values. This means that only one operation, either addition or multiplication, can be performed an unlimited number of times on the ciphertext. PHE with multiplicative operations is the foundation for RSA encryption, which is commonly used in establishing secure connections through SSL/TLS.

A somewhat homomorphic encryption (SHE) scheme is one that supports select operation (either addition or multiplication) up to a certain complexity, but these operations can only be performed a set number of times.

Fully Homomorphic Encryption

Fully homomorphic encryption (FHE) has a lot of potential for making functionality consistent with privacy by helping to keep information secure and accessible at the same time. Developed from the SHE scheme, FHE is capable of using both addition and multiplication any number of times and makes secure multi-party computation more efficient. Unlike other forms of homomorphic encryption, it can handle arbitrary computations on your ciphertexts.

The goal behind FHE is to allow anyone to use encrypted data to perform useful operations without access to the encryption key. In particular, this concept has applications for improving cloud computing security. If you want to store encrypted data in the cloud but don’t want to run the risk of a hacker breaking in your cloud account, it provides you with a way to pull, search, and manipulate your data without having to allow the cloud provider access to your data.

Security of Fully Homomorphic Encryption

The security of the homomorphic encryption schemes is based on the Ring-Learning With Errors (RLWE) problem, which is a hard mathematical problem related to high-dimensional lattices. A great number of peer-reviewed research confirming the hardness of the RLWE problem gives us confidence that these schemes are indeed at least as secure as any standardized encryption scheme.

Applications of Fully Homomorphic Encryption

Craig Gentry mentioned in his graduation thesis that “Fully homomorphic encryption has numerous applications. For example, it enables private queries to a search engine—the user submits an encrypted query and the search engine computes a succinct encrypted answer without ever looking at the query in the clear. It also enables searching on encrypted data—a user stores encrypted files on a remote file server and can later have the server retrieve only files that (when decrypted) satisfy some Boolean constraint, even though the server cannot decrypt the files on its own. More broadly, fully homomorphic encryption improves the efficiency of secure multi party computation.”

Researchers have already identified several practical applications of FHE, some of which are discussed herein:

  • Securing Data Stored in the Cloud. Using homomorphic encryption, you can secure the data that you store in the cloud while also retaining the ability to calculate and search ciphered information that you can later decrypt without compromising the integrity of the data as a whole.
  • Enabling Data Analytics in Regulated Industries. Homomorphic encryption allows data to be encrypted and outsourced to commercial cloud environments for research and data-sharing purposes while protecting user or patient data privacy. It can be used for businesses and organizations across a variety of industries including financial services, retail, information technology, and healthcare to allow people to use data without seeing its unencrypted values. Examples include predictive analysis of medical data without putting data privacy at risk, preserving customer privacy in personalized advertising, financial privacy for functions like stock price prediction algorithms, and forensic image recognition.
  • Improving Election Security and Transparency. Researchers are working on how to use homomorphic encryption to make democratic elections more secure and transparent. For example, the Paillier encryption scheme, which uses addition operations, would be best suited for voting-related applications because it allows users to add up various values in an unbiased way while keeping their values private. This technology could not only protect data from manipulation, it could allow it to be independently verified by authorized third parties.
Limitations of Fully Homomorphic Encryption

There are currently two known limitations of FHE. The first limitation is support for multiple users. Suppose there are many users of the same system (which relies on an internal database that is used in computations), and who wish to protect their personal data from the provider. One solution would be for the provider to have a separate database for every user, encrypted under that user’s public key. If this database is very large and there are many users, this would quickly become infeasible.

Next, there are limitations for applications that involve running very large and complex algorithms homomorphically. All fully homomorphic encryption schemes today have a large computational overhead, which describes the ratio of computation time in the encrypted version versus computation time in the clear. Although polynomial in size, this overhead tends to be a rather large polynomial, which increases runtimes substantially and makes homomorphic computation of complex functions impractical.

Implementations of Fully Homomorphic Encryption

Some of the world’s largest technology companies have initiated programs to advance homomorphic encryption to make it more universally available and user-friendly.

Microsoft, for instance, has created SEAL (Simple Encrypted Arithmetic Library), a set of encryption libraries that allow computations to be performed directly on encrypted data. Powered by open-source homomorphic encryption technology, Microsoft’s SEAL team is partnering with companies like IXUP to build end-to-end encrypted data storage and computation services. Companies can use SEAL to create platforms to perform data analytics on information while it’s still encrypted, and the owners of the data never have to share their encryption key with anyone else. The goal, Microsoft says, is to “put our library in the hands of every developer, so we can work together for more secure, private, and trustworthy computing.”

Google also announced its backing for homomorphic encryption by unveiling its open-source cryptographic tool, Private Join and Compute. Google’s tool is focused on analyzing data in its encrypted form, with only the insights derived from the analysis visible, and not the underlying data itself. 

Finally, with the goal of making homomorphic encryption widespread, IBM released its first version of its HElib C++ library in 2016, but it reportedly “ran 100 trillion times slower than plaintext operations.” Since that time, IBM has continued working to combat this issue and have come up with a version that is 75 times faster, but it is still lagging behind plaintext operations.

(This post has been updated. It was originally posted on January 1, 2020.

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About the author

Anastasios Arampatzis
Anastasios Arampatzis

Anastasios Arampatzis is a retired Hellenic Air Force officer with over 20 years of experience in evaluating cybersecurity and managing IT projects. He works as an informatics instructor at AKMI Educational Institute, while his interests include exploring the human side of cybersecurity.

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