Showing papers on "Homomorphic encryption published in 2014"

Leveled) Fully Homomorphic Encryption without Bootstrapping

Abstract: We present a novel approach to fully homomorphic encryption (FHE) that dramatically improves performance and bases security on weaker assumptions. A central conceptual contribution in our work is a new way of constructing leveled, fully homomorphic encryption schemes (capable of evaluating arbitrary polynomial-size circuits of a-priori bounded depth), without Gentry’s bootstrapping procedure. Specifically, we offer a choice of FHE schemes based on the learning with error (LWE) or Ring LWE (RLWE) problems that have 2 λ security against known attacks. We construct the following. (1) A leveled FHE scheme that can evaluate depth-L arithmetic circuits (composed of fan-in 2 gates) using O(λ. L3) per-gate computation, quasilinear in the security parameter. Security is based on RLWE for an approximation factor exponential in L. This construction does not use the bootstrapping procedure. (2) A leveled FHE scheme that can evaluate depth-L arithmetic circuits (composed of fan-in 2 gates) using O(λ2) per-gate computation, which is independent of L. Security is based on RLWE for quasipolynomial factors. This construction uses bootstrapping as an optimization. We obtain similar results for LWE, but with worse performance. All previous (leveled) FHE schemes required a per-gate computation of Ω(λ3.5), and all of them relied on subexponential hardness assumptions. We introduce a number of further optimizations to our scheme based on the Ring LWE assumption. As an example, for circuits of large width (e.g., where a constant fraction of levels have width Ω(λ)), we can reduce the per-gate computation of the bootstrapped version to O(λ), independent of L, by batching the bootstrapping operation. At the core of our construction is a new approach for managing the noise in lattice-based ciphertexts, significantly extending the techniques of Brakerski and Vaikuntanathan [2011b].

Algorithms in HElib

Abstract: HElib is a software library that implements homomorphic encryption (HE), specifically the Brakerski-Gentry-Vaikuntanathan (BGV) scheme, focusing on effective use of the Smart-Vercauteren ciphertext packing techniques and the Gentry-Halevi-Smart optimizations. The underlying cryptosystem serves as the equivalent of a “hardware platform” for HElib, in that it defines a set of operations that can be applied homomorphically, and specifies their cost. This “platform” is a SIMD environment (somewhat similar to Intel SSE and the like), but with unique cost metrics and parameters. In this report we describe some of the algorithms and optimization techniques that are used in HElib for data movement, linear algebra, and other operations over this “platform.”

Fully homomorphic SIMD operations

Abstract: At PKC 2010 Smart and Vercauteren presented a variant of Gentry's fully homomorphic public key encryption scheme and mentioned that the scheme could support SIMD style operations. The slow key generation process of the Smart---Vercauteren system was then addressed in a paper by Gentry and Halevi, but their key generation method appears to exclude the SIMD style operation alluded to by Smart and Vercauteren. In this paper, we show how to select parameters to enable such SIMD operations. As such, we obtain a somewhat homomorphic scheme supporting both SIMD operations and operations on large finite fields of characteristic two. This somewhat homomorphic scheme can be made fully homomorphic in a naive way by recrypting all data elements separately. However, we show that the SIMD operations can be used to perform the recrypt procedure in parallel, resulting in a substantial speed-up. Finally, we demonstrate how such SIMD operations can be used to perform various tasks by studying two use cases: implementing AES homomorphically and encrypted database lookup.

Efficient Fully Homomorphic Encryption from (Standard) $\mathsf{LWE}$

Abstract: A fully homomorphic encryption (FHE) scheme allows anyone to transform an encryption of a message, $m$, into an encryption of any (efficient) function of that message, $f(m)$, without knowing the secret key. We present a leveled FHE scheme that is based solely on the (standard) learning with errors ($\mathsf{LWE}$) assumption. (Leveled FHE schemes are initialized with a bound on the maximal evaluation depth. However, this restriction can be removed by assuming “weak circular security.'') Applying known results on $\mathsf{LWE}$, the security of our scheme is based on the worst-case hardness of “short vector problems” on arbitrary lattices. Our construction improves on previous works in two aspects: 1. We show that “somewhat homomorphic” encryption can be based on $\mathsf{LWE}$, using a new relinearization technique. In contrast, all previous schemes relied on complexity assumptions related to ideals in various rings. 2. We deviate from the “squashing paradigm” used in all previous works. We introduce a n...

Lattice cryptography for the internet

Abstract: In recent years, lattice-based cryptography has been recognized for its many attractive properties, such as strong provable security guarantees and apparent resistance to quantum attacks, flexibility for realizing powerful tools like fully homomorphic encryption, and high asymptotic efficiency. Indeed, several works have demonstrated that for basic tasks like encryption and authentication, lattice-based primitives can have performance competitive with (or even surpassing) those based on classical mechanisms like RSA or Diffie-Hellman. However, there still has been relatively little work on developing lattice cryptography for deployment in real-world cryptosystems and protocols.

Lattice Cryptography for the Internet

Abstract: In recent years, lattice-based cryptography has been recognized for its many attractive properties, such as strong provable security guarantees and apparent resistance to quantum attacks, flexibility for realizing powerful tools like fully homomorphic encryption, and high asymptotic efficiency Indeed, several works have demonstrated that for basic tasks like encryption and authentication, lattice-based primitives can have performance competitive with (or even surpassing) those based on classical mechanisms like RSA or Diffie-Hellman However, there still has been relatively little work on developing lattice cryptography for deployment in real-world cryptosystems and protocols

Privacy Preserving Back-Propagation Neural Network Learning Made Practical with Cloud Computing

Abstract: To improve the accuracy of learning result, in practice multiple parties may collaborate through conducting joint Back-Propagation neural network learning on the union of their respective data sets. During this process no party wants to disclose her/his private data to others. Existing schemes supporting this kind of collaborative learning are either limited in the way of data partition or just consider two parties. There lacks a solution that allows two or more parties, each with an arbitrarily partitioned data set, to collaboratively conduct the learning. This paper solves this open problem by utilizing the power of cloud computing. In our proposed scheme, each party encrypts his/her private data locally and uploads the ciphertexts into the cloud. The cloud then executes most of the operations pertaining to the learning algorithms over ciphertexts without knowing the original private data. By securely offloading the expensive operations to the cloud, we keep the computation and communication costs on each party minimal and independent to the number of participants. To support flexible operations over ciphertexts, we adopt and tailor the BGN "doubly homomorphic" encryption algorithm for the multiparty setting. Numerical analysis and experiments on commodity cloud show that our scheme is secure, efficient, and accurate.

Faster Bootstrapping with Polynomial Error

Abstract: Bootstrapping is a technique, originally due to Gentry (STOC 2009), for “refreshing” ciphertexts of a somewhat homomorphic encryption scheme so that they can support further homomorphic operations. To date, bootstrapping remains the only known way of obtaining fully homomorphic encryption for arbitrary unbounded computations.

Two-Round Secure MPC from Indistinguishability Obfuscation

Abstract: One fundamental complexity measure of an MPC protocol is its round complexity. Asharov et al. recently constructed the first three round protocol for general MPC in the CRS model. Here, we show how to achieve this result with only two rounds. We obtain UC security with abort against static malicious adversaries, and fairness if there is an honest majority. Additionally the communication in our protocol is only proportional to the input and output size of the function being evaluated and independent of its circuit size. Our main tool is indistinguishability obfuscation, for which a candidate construction was recently proposed by Garg et al.

Efficiently Verifiable Computation on Encrypted Data

Abstract: We study the task of verifiable delegation of computation on encrypted data. We improve previous definitions in order to tolerate adversaries that learn whether or not clients accept the result of a delegated computation. In this strong model, we construct a scheme for arbitrary computations and highly efficient schemes for delegation of various classes of functions, such as linear combinations, high-degree univariate polynomials, and multivariate quadratic polynomials. Notably, the latter class includes many useful statistics. Using our solution, a client can store a large encrypted dataset on a server, query statistics over this data, and receive encrypted results that can be efficiently verified and decrypted. As a key contribution for the efficiency of our schemes, we develop a novel homomorphic hashing technique that allows us to efficiently authenticate computations, at the same cost as if the data were in the clear, avoiding a $10^4$ overhead which would occur with a naive approach. We support our theoretical constructions with extensive implementation tests that show the practical feasibility of our schemes.

Bootstrapping for HElib.

Abstract: Gentry’s bootstrapping technique is still the only known method of obtaining fully homomorphic encryption where the system’s parameters do not depend on the complexity of the evaluated functions. Bootstrapping involves a recryption procedure where the scheme’s decryption algorithm is evaluated homomorphically. So far, there have been precious few implementations of recryption, and fewer still that can handle “packed ciphertexts” that encrypt vectors of elements. In the current work, we report on an implementation of recryption of fully-packed ciphertexts using the HElib library for somewhat-homomorphic encryption. This implementation required extending the recryption algorithms from the literature, as well as many aspects of the HElib library. Our implementation supports bootstrapping of packed ciphertexts over many extension fields/rings. One example that we tested involves ciphertexts that encrypt vectors of 1024 elements from GF(2). In that setting, the recryption procedure takes under 5.5 minutes (at security-level ≈ 76) on a single core, and allows a depth-9 computation before the next recryption is needed.

Homomorphic Encryption and Applications

Abstract: This book introduces the fundamental concepts of homomorphic encryption. From these foundations, applications are developed in the fields of private information retrieval, private searching on streaming data, privacy-preserving data mining, electronic voting and cloud computing. The content is presented in an instructional and practical style, with concrete examples to enhance the reader's understanding. This volume achieves a balance between the theoretical and the practical components of modern information security. Readers will learn key principles of homomorphic encryption as well as their application in solving real world problems.

A Comparison of the Homomorphic Encryption Schemes FV and YASHE.

Abstract: We conduct a theoretical and practical comparison of two Ring-LWE-based, scale-invariant, leveled homomorphic encryption schemes – Fan and Vercauteren’s adaptation of BGV and the YASHE scheme proposed by Bos, Lauter, Loftus and Naehrig. In particular, we explain how to choose parameters to ensure correctness and security against lattice attacks. Our parameter selection improves the approach of van de Pol and Smart to choose parameters for schemes based on the Ring-LWE problem by using the BKZ-2.0 simulation algorithm. We implemented both encryption schemes in C++, using the arithmetic library FLINT, and compared them in practice to assess their respective strengths and weaknesses. In particular, we performed a homomorphic evaluation of the lightweight block cipher SIMON. Combining block ciphers with homomorphic encryption allows to solve the gargantuan ciphertext expansion in cloud applications.

A Comparison of the Homomorphic Encryption Schemes FV and YASHE

Abstract: We conduct a theoretical and practical comparison of two Ring-LWE-based, scale-invariant, leveled homomorphic encryption schemes – Fan and Vercauteren’s adaptation of BGV and the YASHE scheme proposed by Bos, Lauter, Loftus and Naehrig. In particular, we explain how to choose parameters to ensure correctness and security against lattice attacks. Our parameter selection improves the approach of van de Pol and Smart to choose parameters for schemes based on the Ring-LWE problem by using the BKZ-2.0 simulation algorithm.

Scale-Invariant Fully Homomorphic Encryption over the Integers

Abstract: At Crypto 2012, Brakerski constructed a scale-invariant fully homomorphic encryption scheme based on the LWE problem, in which the same modulus is used throughout the evaluation process, instead of a ladder of moduli when doing "modulus switching". In this paper we describe a variant of the van Dijk et al. FHE scheme over the integers with the same scale-invariant property. Our scheme has a single secret modulus whose size is linear in the multiplicative depth of the circuit to be homomorphically evaluated, instead of exponential; we therefore construct a leveled fully homomorphic encryption scheme. This scheme can be transformed into a pure fully homomorphic encryption scheme using bootstrapping, and its security is still based on the Approximate-GCD problem. We also describe an implementation of the homomorphic evaluation of the full AES encryption circuit, and obtain significantly improved performance compared to previous implementations: about 23 seconds resp. 3 minutes per AES block at the 72-bit resp. 80-bit security level on a mid-range workstation. Finally, we prove the equivalence between the error-free decisional Approximate-GCD problem introduced by Cheon et al. Eurocrypt 2013 and the classical computational Approximate-GCD problem. This equivalence allows to get rid of the additional noise in all the integer-based FHE schemes described so far, and therefore to simplify their security proof.

Crypto-Nets: Neural Networks over Encrypted Data

Abstract: The problem we address is the following: how can a user employ a predictive model that is held by a third party, without compromising private information. For example, a hospital may wish to use a cloud service to predict the readmission risk of a patient. However, due to regulations, the patient's medical files cannot be revealed. The goal is to make an inference using the model, without jeopardizing the accuracy of the prediction or the privacy of the data. To achieve high accuracy, we use neural networks, which have been shown to outperform other learning models for many tasks. To achieve the privacy requirements, we use homomorphic encryption in the following protocol: the data owner encrypts the data and sends the ciphertexts to the third party to obtain a prediction from a trained model. The model operates on these ciphertexts and sends back the encrypted prediction. In this protocol, not only the data remains private, even the values predicted are available only to the data owner. Using homomorphic encryption and modifications to the activation functions and training algorithms of neural networks, we show that it is protocol is possible and may be feasible. This method paves the way to build a secure cloud-based neural network prediction services without invading users' privacy.

Private Computation on Encrypted Genomic Data

Abstract: A number of databases around the world currently host a wealth of genomic data that is invaluable to researchers conducting a variety of genomic studies. However, patients who volunteer their genomic data run the risk of privacy invasion. In this work, we give a cryptographic solution to this problem: to maintain patient privacy, we propose encrypting all genomic data in the database. To allow meaningful computation on the encrypted data, we propose using a homomorphic encryption scheme.

Homomorphic Signatures with Efficient Verification for Polynomial Functions

Abstract: A homomorphic signature scheme for a class of functions \(\mathcal{C}\) allows a client to sign and upload elements of some data set D on a server At any later point, the server can derive a (publicly verifiable) signature that certifies that some y is the result computing some \(f\in\mathcal{C}\) on the basic data set D This primitive has been formalized by Boneh and Freeman (Eurocrypt 2011) who also proposed the only known construction for the class of multivariate polynomials of fixed degree d ≥ 1 In this paper we construct new homomorphic signature schemes for such functions Our schemes provide the first alternatives to the one of Boneh-Freeman, and improve over their solution in three main aspects First, our schemes do not rely on random oracles Second, we obtain security in a stronger fully-adaptive model: while the solution of Boneh-Freeman requires the adversary to query messages in a given data set all at once, our schemes can tolerate adversaries that query one message at a time, in a fully-adaptive way Third, signature verification is more efficient (in an amortized sense) than computing the function from scratch The latter property opens the way to using homomorphic signatures for publicly-verifiable computation on outsourced data Our schemes rely on a new assumption on leveled graded encodings which we show to hold in a generic model

Towards Efficient Privacy-preserving Image Feature Extraction in Cloud Computing

Abstract: As the image data produced by individuals and enterprises is rapidly increasing, Scalar Invariant Feature Transform (SIFT), as a local feature detection algorithm, has been heavily employed in various areas, including object recognition, robotic mapping, etc. In this context, there is a growing need to outsource such image computation with high complexity to cloud for its economic computing resources and on-demand ubiquitous access. However, how to protect the private image data while enabling image computation becomes a major concern. To address this fundamental challenge, we study the privacy requirements in outsourcing SIFT computation and propose SecSIFT, a high performance privacy-preserving SIFT feature detection system. In previous private image computation works, one common approach is to encrypt the private image in a public key based homomorphic scheme that enables the original processing algorithms designed for plaintext domain to be performed over ciphertext domain. In contrast to these works, our system is not restricted by the efficiency limitations of homomorphic encryption scheme. The proposed system distributes the computation procedures of SIFT to a set of independent, co-operative cloud servers, and keeps the outsourced computation procedures as simple as possible to avoid utilizing homomorphic encryption scheme. Thus, it enables implementation with practical computation and communication complexity. Extensive experimental results demonstrate that SecSIFT performs comparably to original SIFT on image benchmarks while capable of preserving the privacy in an efficient way.

Survey of Various Homomorphic Encryption algorithms and Schemes

Abstract: Homomorphic encryption is the encryption scheme which means the operations on the encrypted data. Homomorphic encryption can be applied in any system by using various public key algorithms. When the data is transferred to the public area, there are many encryption algorithms to secure the operations and the storage of the data. But to process data located on remote server and to preserve privacy, homomorphic encryption is useful that allows the operations on the cipher text, which can provide the same results after calculations as the working directly on the raw data. In this paper, the main focus is on public key cryptographic algorithms based on homomorphic encryption scheme for preserving security. The case study on various principles and properties of homomorphic encryption is given and then various homomorphic algorithms using asymmetric key systems such as RSA, ElGamal, Paillier algorithms as well as various homomorphic encryption schemes such as BrakerskiGentry-Vaikuntanathan (BGV), Enhanced homomorphic Cryptosystem (EHC), Algebra homomorphic encryption scheme based on updated ElGamal (AHEE), Non-interactive exponential homomorphic encryption scheme (NEHE) are investigated.

Confidentiality-Preserving Image Search: A Comparative Study Between Homomorphic Encryption and Distance-Preserving Randomization

Abstract: Recent years have seen increasing popularity of storing and managing personal multimedia data using online services. Preserving confidentiality of online personal data while offering efficient functionalities thus becomes an important and pressing research issue. In this paper, we study the problem of content-based search of image data archived online while preserving content confidentiality. The problem has different settings from those typically considered in the secure computation literature, as it deals with data in rank-ordered search, and has a different security-efficiency requirement. Secure computation techniques, such as homomorphic encryption, can potentially be used in this application, at a cost of high computational and communication complexity. Alternatively, efficient techniques based on randomizing visual feature and search indexes have been proposed recently to enable similarity comparison between encrypted images. This paper focuses on comparing these two major paradigms of techniques, namely, homomorphic encryption-based techniques and feature/index randomization-based techniques, for confidentiality-preserving image search. We develop novel and systematic metrics to quantitatively evaluate security strength in this unique type of data and applications. We compare these two paradigms of techniques in terms of their search performance, security strength, and computational efficiency. The insights obtained through this paper and comparison will help design practical algorithms appropriate for privacy-aware cloud multimedia systems.

Secure query processing with data interoperability in a cloud database environment

Abstract: We address security issues in a cloud database system which employs the DBaaS model. In such a model, a data owner (DO) exports its data to a cloud database service provider (SP). To provide data security, sensitive data is encrypted by the DO before it is uploaded to the SP. Existing encryption schemes, however, are only partially homomorphic in the sense that each of them was designed to allow one specific type of computation to be done on encrypted data. These existing schemes cannot be integrated to answer real practical queries that involve operations of different kinds. We propose and analyze a secure query processing system (SDB) on relational tables and a set of elementary operators on encrypted data that allow data interoperability, which allows a wide range of SQL queries to be processed by the SP on encrypted information. We prove that our encryption scheme is secure against two types of threats and that it is practically efficient.

Non-malleability from Malleability: Simulation-Sound Quasi-Adaptive NIZK Proofs and CCA2-Secure Encryption from Homomorphic Signatures

Abstract: Verifiability is central to building protocols and systems with integrity. Initially, efficient methods employed the Fiat-Shamir heuristics. Since 2008, the Groth-Sahai techniques have been the most efficient in constructing non-interactive witness indistinguishable and zero-knowledge proofs for algebraic relations in the standard model. For the important task of proving membership in linear subspaces, Jutla and Roy (Asiacrypt 2013) gave significantly more efficient proofs in the quasi-adaptive setting (QA-NIZK). For membership of the row space of a t ×n matrix, their QA-NIZK proofs save Ω(t) group elements compared to Groth-Sahai. Here, we give QA-NIZK proofs made of a constant number group elements – regardless of the number of equations or the number of variables – and additionally prove them unbounded simulation-sound. Unlike previous unbounded simulation-sound Groth-Sahai-based proofs, our construction does not involve quadratic pairing product equations and does not rely on a chosen-ciphertext-secure encryption scheme. Instead, we build on structure-preserving signatures with homomorphic properties. We apply our methods to design new and improved CCA2-secure encryption schemes. In particular, we build the first efficient threshold CCA-secure keyed-homomorphic encryption scheme (i.e., where homomorphic operations can only be carried out using a dedicated evaluation key) with publicly verifiable ciphertexts.

Distinguisher-based attacks on public-key cryptosystems using Reed---Solomon codes

Abstract: Because of their interesting algebraic properties, several authors promote the use of generalized Reed---Solomon codes in cryptography. Niederreiter was the first to suggest an instantiation of his cryptosystem with them but Sidelnikov and Shestakov showed that this choice is insecure. Wieschebrink proposed a variant of the McEliece cryptosystem which consists in concatenating a few random columns to a generator matrix of a secretly chosen generalized Reed---Solomon code. More recently, new schemes appeared which are the homomorphic encryption scheme proposed by Bogdanov and Lee, and a variation of the McEliece cryptosystem proposed by Baldi et al. which hides the generalized Reed---Solomon code by means of matrices of very low rank. In this work, we show how to mount key-recovery attacks against these public-key encryption schemes. We use the concept of distinguisher which aims at detecting a behavior different from the one that one would expect from a random code. All the distinguishers we have built are based on the notion of component-wise product of codes. It results in a powerful tool that is able to recover the secret structure of codes when they are derived from generalized Reed---Solomon codes. Lastly, we give an alternative to Sidelnikov and Shestakov attack by building a filtration which enables to completely recover the support and the non-zero scalars defining the secret generalized Reed---Solomon code.

Efficient homomorphic encryption on integer vectors and its applications

Abstract: Homomorphic encryption, aimed at enabling computation in the encrypted domain, is becoming important to a wide and growing range of applications, from cloud computing to distributed sensing. In recent years, a number of approaches to fully (or nearly fully) homomorphic encryption have been proposed, but to date the space and time complexity of the associated schemes has precluded their use in practice. In this work, we demonstrate that more practical homomorphic encryption schemes are possible when we require that not all encrypted computations be supported, but rather only those of interest to the target application. More specifically, we develop a homomorphic encryption scheme operating directly on integer vectors that supports three operations of fundamental interest in signal processing applications: addition, linear transformation, and weighted inner products. Moreover, when used in combination, these primitives allow us to efficiently and securely compute arbitrary polynomials. Some practically relevant examples of the computations supported by this framework are described, including feature extraction, recognition, classification, and data aggregation.

Privacy of outsourced k-means clustering

Abstract: It is attractive for an organization to outsource its data analytics to a service provider who has powerful platforms and advanced analytics skills. However, the organization (data owner) may have concerns about the privacy of its data. In this paper, we present a method that allows the data owner to encrypt its data with a homomorphic encryption scheme and the service provider to perform k-means clustering directly over the encrypted data. However, since the ciphertexts resulting from homomorphic encryption do not preserve the order of distances between data objects and cluster centers, we propose an approach that enables the service provider to compare encrypted distances with the trapdoor information provided by the data owner. The efficiency of our method is validated by extensive experimental evaluation.

Practical homomorphic encryption: A survey

Abstract: Cloud computing technology has rapidly evolved over the last decade, offering an alternative way to store and work with large amounts of data. However data security remains an important issue particularly when using a public cloud service provider. The recent area of homomorphic cryptography allows computation on encrypted data, which would allow users to ensure data privacy on the cloud and increase the potential market for cloud computing. A significant amount of research on homomorphic cryptography appeared in the literature over the last few years; yet the performance of existing implementations of encryption schemes remains unsuitable for real time applications. One way this limitation is being addressed is through the use of graphics processing units (GPUs) and field programmable gate arrays (FPGAs) for implementations of homomorphic encryption schemes. This review presents the current state of the art in this promising new area of research and highlights the interesting remaining open problems.

Accelerating NTRU based homomorphic encryption using GPUs

Abstract: We introduce a large polynomial arithmetic library optimized for Nvidia GPUs to support fully homomorphic encryption schemes. To realize the large polynomial arithmetic library we convert polynomials with large coefficients using the Chinese Remainder Theorem into many polynomials with small coefficients, and then carry out modular multiplications in the residue space using a custom developed discrete Fourier transform library. We further extend the library to support the homomorphic evaluation operations, i.e. addition, multiplication, and relinearization, in an NTRU based somewhat homomorphic encryption library. Finally, we put the library to use to evaluate homomorphic evaluation of two block ciphers: Prince and AES, which show 2.57 times and 7.6 times speedup, respectively, over an Intel Xeon software implementation.

A scalable implementation of fully homomorphic encryption built on NTRU

Abstract: In this paper we report on our work to design, implement and evaluate a Fully Homomorphic Encryption (FHE) scheme. Our FHE scheme is an NTRU-like cryptosystem, with additional support for efficient key switching and modulus reduction operations to reduce the frequency of bootstrapping operations. Ciphertexts in our scheme are represented as matrices of 64-bit integers. The basis of our design is a layered software services stack to provide high-level FHE operations supported by lower-level lattice-based primitive implementations running on a computing substrate. We implement and evaluate our FHE scheme to run on a commodity CPU-based computing environment. We implemented our FHE scheme to run in a compiled C environment and use parallelism to take advantage of multi-core processors. We provide experimental results which show that our FHE implementation provides at least an order of magnitude improvement in runtime as compared to recent publicly known evaluation results of other FHE software implementations.