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. The users or service providers with the key have exclusive rights on the data. Especially with popular cloud services, 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. Therefore, this survey focuses on HE and FHE schemes. 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 for 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, and extending the state-of-the-art HE, PHE, SWHE, and FHE systems.

https://dl.acm.org/doi/pdf/10.1145/3214303

A Survey on Homomorphic Encryption Schemes: Theory and Implementation

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.

Cloud computing and cloud storage have contributed to a big shift in data processing and its use. Availability and accessibility of resources with the reduction of substantial work is one of the main reasons for the cloud revolution. With this cloud computing revolution, outsourcing applications are in great demand. The client uses the service by uploading their data to the cloud and finally gets the result by processing it. It benefits users greatly, but it also exposes sensitive data to third-party service providers. In the healthcare industry, patient health records are digital records of a patient's medical history kept by hospitals or health care providers. Patient health records are stored in data centers for storage and processing. Before doing computations on data, traditional encryption techniques decrypt the data in their original form. As a result, sensitive medical information is lost. Homomorphic encryption can protect sensitive information by allowing data to be processed in an encrypted form such that only encrypted data is accessible to service providers. In this paper, an attempt is made to present a systematic review of homomorphic cryptosystems with its categorization and evolution over time. In addition, this paper also includes a review of homomorphic cryptosystem contributions in healthcare. 

/pdf/a-systematic-review-of-homomorphic-encryption-and-its-11f16j3p.pdf

A systematic review of homomorphic encryption and its contributions in healthcare industry

Data privacy concerns are increasing significantly in the context of the Internet of Things, cloud services, edge computing, artificial intelligence applications, and other applications enabled by next-generation networks. Homomorphic encryption addresses privacy challenges by enabling multiple operations to be performed on encrypted messages without decryption. This article comprehensively addresses homomorphic encryption from both theoretical and practical perspectives. This article delves into the mathematical foundations required to understand fully homomorphic encryption ( <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$\textsf {FHE}$ </tex-math></inline-formula> ). It consequently covers design fundamentals and security properties of <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$\textsf {FHE}$ </tex-math></inline-formula> and describes the main <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$\textsf {FHE}$ </tex-math></inline-formula> schemes based on various mathematical problems. On a more practical level, this article presents a view on privacy-preserving machine learning using homomorphic encryption and then surveys <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$\textsf {FHE}$ </tex-math></inline-formula> at length from an engineering angle, covering the potential application of <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$\textsf {FHE}$ </tex-math></inline-formula> in fog computing and cloud computing services. It also provides a comprehensive analysis of existing state-of-the-art <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$\textsf {FHE}$ </tex-math></inline-formula> libraries and tools, implemented in software and hardware, and the performance thereof. 

https://zenodo.org/record/7152464/files/FHE%20survey.pdf

Survey on Fully Homomorphic Encryption, Theory, and Applications

Partially homomorphic encryption systems may be transformed into fully homomorphic encryption systems that are scalable, rapid in translation speed, difficult to invert or break, capable of enabling various types of public and/or private key generation protocols and semantically secure. Input plaintext data are transformed into modified plaintext data using a prime number operation and the modified plaintext data is then encrypted using any number of conventional encryption schemes. Desired computations on the encrypted data are transformed into homomorphic operations, based on the nature of the encryption format, and the homomorphic operations are applied to yield manipulated encrypted data. The manipulated encrypted data may be decrypted and the decrypted plaintext data may be modified into final, output plaintext data using a similar prime number operation as applied during encryption. The final, output plaintext is equivalent to plaintext data that would have been generated by just applying the desired computations to the input plaintext data.

Systems and methods for implementing an efficient, scalable homomorphic transformation of encrypted data with minimal data expansion and improved processing efficiency

Capability of operating over encrypted data makes Fully Homomorphic Encryption(FHE)the Holy Grail for secure data processing applications. Though many applications need only secret keys, FHE has not been achieved properly through symmetric cryptography. Major hurdle is the need to refresh noisy ciphertexts which essentially requires public key and bootstrapping. We introduce a refreshing procedure to make a somewhat homomorphic scheme, fully homomorphic without requiring bootstrapping. Our scheme uses symmetric keys and has performance superior to existing public-key schemes.

Fully Homomorphic symmetric scheme without bootstrapping

Today, cloud offers unlimited computing capabilities at affordable price. However, securing data in cloud ensuring usability and availability of data presents a challenge. Benefits promised by cloud computing cannot be availed fully until security issues are properly addressed. Homomorphic encryption ensures data security during computation as well. Most of the currently available homomorphic encryption schemes are based on public key cryptosystems causing large overhead. We propose a symmetric key encryption scheme with fully homomorphic evaluation capabilities, applicable even in multiuser environment. We also present a protocol for using the proposed scheme to private data processing in clouds. The proposed scheme requires a key size linearly proportional to security parameter causing computational overhead proportional to security parameter.

Making data in cloud secure and usable: fully homomorphic encryption with symmetric keys

Addressing Data Aggregation Using Polynomial Regression in WSNs

Introduction of IPv6 with increased address length and different prefix length distribution poses new challenges to the IP lookup and forwarding algorithms. Algorithms traditionally used for IPv4 cannot be directly modified for IPv6. In this paper, we propose a new data structure named ‘Disjoint Set Tries’. The aim of new design emphasis on building routing tables based on prefix length distribution in IPv6. It has lower lookup complexity compared to previous trie-based approaches and stable memory consumption. This makes it a good candidate for IPv6 routing lookup table.

Prefix Length-Based Disjoint Set Tries for IPv6 Lookup

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