Popping Bubbles: Cryptanalysis of Homomorphic Encryption

Imagine an encryption scheme where it is possible to add and multiply numbers without any knowledge of the numbers. Instead one could manipulate encryptions of the numbers and then the decryption of the result would give the result of the arithmetic on the original numbers. Encryption algorithms with this property are called homomorphic and have various applications in cloud computing. Homomorphic encryption schemes exist but are generally so inefficient that they are not practical. This report introduces a toy cryptosystem called Bubbles: a somewhat homomorphic encryption scheme created by Professor Martin and Professor Sunar at Worcester Polytechnic Institute. We will show that the original scheme is insecure and may be efficiently popped . We will then examine two variations of the scheme that introduce noise to increase security and show that Bubbles is still vulnerable except when parameters are carefully chosen. However these safe parameter choices make Bubbles more inefficient than other recent homomorphic schemes

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.Comment: - Updated. (October 6, 2017) - This paper is an early draft of the survey that is being submitted to ACM CSUR and has been uploaded to arXiv for feedback from stakeholder

Conditionals in Homomorphic Encryption and Machine Learning Applications

Homomorphic encryption aims at allowing computations on encrypted data without decryption other than that of the final result. This could provide an elegant solution to the issue of privacy preservation in data-based applications, such as those using machine learning, but several open issues hamper this plan. In this work we assess the possibility for homomorphic encryption to fully implement its program without relying on other techniques, such as multiparty computation (SMPC), which may be impossible in many use cases (for instance due to the high level of communication required). We proceed in two steps: i) on the basis of the structured program theorem (Bohm-Jacopini theorem) we identify the relevant minimal set of operations homomorphic encryption must be able to perform to implement any algorithm; and ii) we analyse the possibility to solve -- and propose an implementation for -- the most fundamentally relevant issue as it emerges from our analysis, that is, the implementation of conditionals (requiring comparison and selection/jump operations). We show how this issue clashes with the fundamental requirements of homomorphic encryption and could represent a drawback for its use as a complete solution for privacy preservation in data-based applications, in particular machine learning ones. Our approach for comparisons is novel and entirely embedded in homomorphic encryption, while previous studies relied on other techniques, such as SMPC, demanding high level of communication among parties, and decryption of intermediate results from data-owners. Our protocol is also provably safe (sharing the same safety as the homomorphic encryption schemes), differently from other techniques such as Order-Preserving/Revealing-Encryption (OPE/ORE).Comment: 14 pages, 1 figure, corrected typos, added introductory pedagogical section on polynomial approximatio

Verifiable Encodings for Secure Homomorphic Analytics

Homomorphic encryption, which enables the execution of arithmetic operations directly on ciphertexts, is a promising solution for protecting privacy of cloud-delegated computations on sensitive data. However, the correctness of the computation result is not ensured. We propose two error detection encodings and build authenticators that enable practical client-verification of cloud-based homomorphic computations under different trade-offs and without compromising on the features of the encryption algorithm. Our authenticators operate on top of trending ring learning with errors based fully homomorphic encryption schemes over the integers. We implement our solution in VERITAS, a ready-to-use system for verification of outsourced computations executed over encrypted data. We show that contrary to prior work VERITAS supports verification of any homomorphic operation and we demonstrate its practicality for various applications, such as ride-hailing, genomic-data analysis, encrypted search, and machine-learning training and inference.Comment: update authors, typos corrected, scheme update

Improving the Efficiency of Homomorphic Encryption Schemes

In this dissertation, we explore different approaches to practical homomorphic encryption schemes. For partial homomorphic encryption schemes, we observe that the versatility is the main bottleneck. To solve this problem, we propose general approaches to improve versatility of them by either extending the range of supported circuits or extending the message space. These general approaches can be applied to a wide range of partial HE schemes and greatly increase the number of applications that they support. For fully homomorphic encryption schemes, the slow running speed and the large ciphertext are the main challenges. Therefore, we propose efficient implementations as well as methods to compress the ciphertext. In detail, the Gentry Halevi FHE scheme and the LTV FHE scheme are implemented and the resulting performance shows significant improvement over previous works. For ciphertext compression, the concept of scheme conversion is proposed. Given a scheme converter, we can convert between schemes with compact ciphertext for communication and homomorphic schemes for computation