Fully Homomorphic Encryption Scheme Research Articles

Recent advances of Privacy-Preserving Machine Learning based on (Fully) Homomorphic Encryption

Fully Homomorphic Encryption (FHE), known for its ability to process encrypted data without decryption, is a promising technique in solving the privacy concerns in the machine learning era. However, there are many kinds of available FHE schemes and way more FHE-based solutions in the literature, and they are still fast evolving, making it difficult to get a complete view. This article aims to introduce recent representative results of FHE-based privacy-preserving machine learning, helping users understand the pros and cons of different kind of solutions, and choose an appropriate approach for their needs.

Private pathological assessment via machine learning and homomorphic encryption

PurposeThe objective of this research is to explore the applicability of machine learning and fully homomorphic encryption (FHE) in the private pathological assessment, with a focus on the inference phase of support vector machines (SVM) for the classification of confidential medical data.MethodsA framework is introduced that utilizes the Cheon-Kim-Kim-Song (CKKS) FHE scheme, facilitating the execution of SVM inference on encrypted datasets. This framework ensures the privacy of patient data and negates the necessity of decryption during the analytical process. Additionally, an efficient feature extraction technique is presented for the transformation of medical imagery into vectorial representations.ResultsThe system’s evaluation across various datasets substantiates its practicality and efficacy. The proposed method delivers classification accuracy and performance on par with traditional, non-encrypted SVM inference, while upholding a 128-bit security level against established cryptographic attacks targeting the CKKS scheme. The secure inference process is executed within a temporal span of mere seconds.ConclusionThe findings of this study underscore the viability of FHE in enhancing the security and efficiency of bioinformatics analyses, potentially benefiting fields such as cardiology, oncology, and medical imagery. The implications of this research are significant for the future of privacy-preserving machine learning, promoting progress in diagnostic procedures, tailored medical treatments, and clinical investigations.

Coupling bit and modular arithmetic for efficient general-purpose fully homomorphic encryption

Fully Homomorphic Encryption (FHE) enables computation directly on encrypted data. This property is desirable for outsourced computation of sensitive data as it relies solely on the underlying security of the cryptosystem and not in access control policies. Even though FHE is still significantly slower than unencrypted computation, practical times are possible for applications easily representable as low-order polynomials, since most FHE schemes support modular addition and multiplication over ciphertexts. If, however, an application cannot be expressed with low-order polynomials, then Boolean logic must be emulated. This bit-level arithmetic enables any computation to be performed homomorphically. Nevertheless, as it runs on top of the natively supported modular arithmetic, it has poor performance, which hinders its use in the majority of scenarios. In this work, we propose Bridging, a technique that allows conversion from bit-level to modular arithmetic and vice-versa. This enables the use of the comprehensive computation provided by bit-level arithmetic and the performance of modular arithmetic within the same application. Experimental results show that Bridging can lead to 1-2 orders of magnitude performance improvement for tested benchmarks and two real-world applications: URL denylisting and genotype imputation. Bridging performance comes from two factors: reduced number of operations and smaller multiplicative depth.

Verifiable FHE via Lattice-based SNARKs

Fully Homomorphic Encryption (FHE) is a prevalent cryptographic primitive that allows for computation on encrypted data. In various cryptographic protocols, this enables outsourcing computation to a third party while retaining the privacy of the inputs to the computation. However, these schemes make an honest-but-curious assumption about the adversary. Previous work has tried to remove this assumption by combining FHE with Verifiable Computation (VC). Recent work has increased the flexibility of this approach by introducing integrity checks for homomorphic computations over rings. However, efficient FHE for circuits of large multiplicative depth also requires non-ring computations called maintenance operations, i.e. modswitching and keyswitching, which cannot be efficiently verified by existing constructions. We propose the first efficiently verifiable FHE scheme that allows for arbitrary depth homomorphic circuits by utilizing the double-CRT representation in which FHE schemes are typically computed, and using lattice-based SNARKs to prove components of this computation separately, including the maintenance operations. Therefore, our construction can theoretically handle bootstrapping operations. We also present the first implementation of a verifiable computation on encrypted data for a computation that contains multiple ciphertext-ciphertext multiplications. Concretely, we verify the homomorphic computation of an approximate neural network containing three layers and >100 ciphertexts in less than 1 second while maintaining reasonable prover costs.

Algebraic Attack on FHE-Friendly Cipher HERA Using Multiple Collisions

Fully homomorphic encryption (FHE) is an advanced cryptography technique to allow computations (i.e., addition and multiplication) over encrypted data. After years of effort, the performance of FHE has been significantly improved and it has moved from theory to practice. The transciphering framework is another important technique in FHE to address the issue of ciphertext expansion and reduce the client-side computational overhead. To apply the transciphering framework to the CKKS FHE scheme, a new transciphering framework called the Real-to-Finite-Field (RtF) framework and a corresponding FHE-friendly symmetric-key primitive called HERA were proposed at ASIACRYPT 2021. Although HERA has a very similar structure to AES, it is considerably different in the following aspects: 1) the power map x → x3 is used as the S-box; 2) a randomized key schedule is used; 3) it is over a prime field Fp with p > 216. In this work, we perform the first third-party cryptanalysis of HERA, by showing how to mount new algebraic attacks with multiple collisions in the round keys. Specifically, according to the special way to randomize the round keys in HERA, we find it possible to peel off the last nonlinear layer by using collisions in the last-round key and a simple property of the power map. In this way, we could construct an overdefined system of equations of a much lower degree in the key, and efficiently solve the system via the linearization technique. As a esult, for HERA with 192 and 256 bits of security, respectively, we could break some parameters under the same assumption made by designers that the algebra constant ω for Gaussian elimination is ω = 2, i.e., Gaussian elimination on an n × n matrix takes O(nω) field operations. If using more conservative choices like ω ∈ {2.8, 3}, our attacks can also successfully reduce the security margins of some variants of HERA to only 1 round. However, the security of HERA with 80 and 128 bits of security is not affected by our attacks due to the high cost to find multiple collisions. In any case, our attacks reveal a weakness of HERA caused by the randomized key schedule and its small state size.

A Comparative Study of BFV and CKKs Schemes to Secure IoT Data Using TenSeal and Pyfhel Homomorphic Encryption Libraries

Internet of things (IoT) devices and applications are on the rise, generating large amounts of sensitive and confidential data that need to be processed securely. Due to resource constraints, the data generated is often stored and processed in the cloud. The drawback of data cloud storage and processing is the fact that it can be hacked, leaked, or sold by cloud companies. Fully homomorphic encryption (FHE) allows computation on encrypted data using basic mathematical operations and has recently been successfully implemented using schemes and libraries with better performance. In this paper, the authors propose a mixture of edge-cloud-based security schemes using FHE to secure IoT data. The authors evaluate the performance of two FHE schemes (BFV and CKKS) based on data: encoding speed, encryption speed, arithmetic operations (addition and multiplication) speed, and decryption decoding speed using two Python libraries (TenSEAL and PyFHEl). The encryption and decryption are done at the edge node using a Raspberry Pi 4, while the processing is done at the cloud node using a laptop.

Optimizing Homomorphic Evaluation Circuits by Program Synthesis and Time-bounded Exhaustive Search

We present a new and general method for optimizing homomorphic evaluation circuits. Although fully homomorphic encryption (FHE) holds the promise of enabling safe and secure third party computation, building FHE applications has been challenging due to their high computational costs. Domain-specific optimizations require a great deal of expertise on the underlying FHE schemes and FHE compilers that aim to lower the hurdle, generate outcomes that are typically sub-optimal, as they rely on manually-developed optimization rules. In this article, based on the prior work of FHE compilers, we propose a method for automatically learning and using optimization rules for FHE circuits. Our method focuses on reducing the maximum multiplicative depth, the decisive performance bottleneck, of FHE circuits by combining program synthesis, term rewriting, and equality saturation. It first uses program synthesis to learn equivalences of small circuits as rewrite rules from a set of training circuits. Then, we perform term rewriting on the input circuit to obtain a new circuit that has lower multiplicative depth. Our rewriting method uses the equational matching with generalized version of the learned rules, and its soundness property is formally proven. Our optimizations also try to explore every possible alternative order of applying rewrite rules by time-bounded exhaustive search technique called equality saturation. Experimental results show that our method generates circuits that can be homomorphically evaluated 1.08×–3.17× faster (with the geometric mean of 1.56×) than the state-of-the-art method. Our method is also orthogonal to existing domain-specific optimizations.

Accelerating Finite Field Arithmetic for Homomorphic Encryption on GPUs

Fully Homomorphic Encryption (FHE) is a rapidly developing technology that enables computation directly on encrypted data, making it a compelling solution for security in cloud-based systems. In addition, modern FHE schemes are believed to be resistant to quantum attacks. Although FHE offers unprecedented potential for security, current implementations suffer from prohibitively high latency. Finite field arithmetic operations, particularly the multiplication of high-degree polynomials, are key computational bottlenecks. The parallel processing capabilities provided by modern Graphical Processing Units (GPUs) make them compelling candidates to target these highly parallelizable workloads. In this article, we discuss methods to accelerate polynomial multiplication with GPUs, with the goal of making FHE practical.

SoK: New Insights into Fully Homomorphic Encryption Libraries via Standardized Benchmarks

Fully homomorphic encryption (FHE) enables arbitrary computation on encrypted data, allowing users to upload ciphertexts to cloud servers for computation while mitigating privacy risks. Many cryptographic schemes fall under the umbrella of FHE, and each scheme has several open-source implementations with its own strengths and weaknesses. Nevertheless, developers have no straightforward way to choose which FHE scheme and implementation is best suited for their application needs, especially considering that each scheme offers different security, performance, and usability guarantees. To allow programmers to effectively utilize the power of FHE, we employ a series of benchmarks called the Terminator 2 Benchmark Suite and present new insights gained from running these algorithms with a variety of FHE back - ends. Contrary to generic benchmarks that do not take into consideration the inherent challenges of encrypted computation, our methodology is tailored to the secure computational primitives of each target FHE implementation. To ensure fair comparisons, we developed a versatile compiler(called T2) that converts arbitrary benchmarks written in a domain - specific language into identical encrypted programs running on different popular FHE libraries as a backend.Our analysis exposes for the first time the advantages and disadvantages of each FHE library as well as the types of applications most suited for each computational domain(i.e., binary, integer, and floating - point).

Pipelined Key Switching Accelerator Architecture for CKKS-Based Fully Homomorphic Encryption

The increasing ubiquity of big data and cloud-based computing has led to increased concerns regarding the privacy and security of user data. In response, fully homomorphic encryption (FHE) was developed to address this issue by enabling arbitrary computation on encrypted data without decryption. However, the high computational costs of homomorphic evaluations restrict the practical application of FHE schemes. To tackle these computational and memory challenges, a variety of optimization approaches and acceleration efforts are actively being pursued. This paper introduces the KeySwitch module, a highly efficient and extensively pipelined hardware architecture designed to accelerate the costly key switching operation in homomorphic computations. Built on top of an area-efficient number-theoretic transform design, the KeySwitch module exploited the inherent parallelism of key switching operation and incorporated three main optimizations: fine-grained pipelining, on-chip resource usage, and high-throughput implementation. An evaluation on the Xilinx U250 FPGA platform demonstrated a 1.6× improvement in data throughput compared to previous work with more efficient hardware resource utilization. This work contributes to the development of advanced hardware accelerators for privacy-preserving computations and promoting the adoption of FHE in practical applications with enhanced efficiency.

Energy Efficient and Secure Neural Network–based Disease Detection Framework for Mobile Healthcare Network

Adopting mobile healthcare network (MHN) services such as disease detection is fraught with concerns about the security and privacy of the entities involved and the resource restrictions at the Internet of Things (IoT) nodes. Hence, the essential requirements for disease detection services are to (i) produce accurate and fast disease detection without jeopardizing the privacy of health clouds and medical users and (ii) reduce the computational and transmission overhead (energy consumption) of the IoT devices while maintaining the privacy. For privacy preservation of widely used neural network– (NN) based disease detection, existing literature suggests either computationally heavy public key fully homomorphic encryption (FHE), or secure multiparty computation, with a large number of interactions. Hence, the existing privacy-preserving NN schemes are energy consuming and not suitable for resource-constrained IoT nodes in MHN. This work proposes a lightweight, fully homomorphic, symmetric key FHE scheme (SkFhe) to address the issues involved in implementing privacy-preserving NN. Based on SkFhe, widely used non-linear activation functions ReLU and Leaky ReLU are implemented over the encrypted domain. Furthermore, based on the proposed privacy-preserving linear transformation and non-linear activation functions, an energy-efficient, accurate, and privacy-preserving NN is proposed. The proposed scheme guarantees privacy preservation of the health cloud’s NN model and medical user’s data. The experimental analysis demonstrates that the proposed solution dramatically reduces the overhead in communication and computation at the user side compared to the existing schemes. Moreover, the improved energy efficiency at the user is accomplished with reduced diagnosis time without sacrificing classification accuracy.

HE-Booster: An Efficient Polynomial Arithmetic Acceleration on GPUs for Fully Homomorphic Encryption

Fully Homomorphic Encryption (FHE) enables secure offloading of computations to untrusted cloud servers as it allows computing on encrypted data. However, existing well-known FHE schemes suffer from heavy performance overheads. Thus numerous accelerations based on FPGAs, ASICs, and GPUs have been proposed. Compared to FPGAs and ASICs, GPUs have obvious advantages in productivity and development costs. And also, GPUs have already been widely deployed in commercial cloud or supercomputing centers. Therefore, we present HE-Booster, an efficient GPU-based FHE acceleration design. For single-GPU acceleration, a thorough systematic design is exploited to map five common phases in typical FHE schemes to the GPU parallel architecture. In particular, inspired by the regular architecture of NTT/INTT, a novel inter-thread local synchronization is proposed to exploit thread-level parallelism. For multi-GPU acceleration, we propose a scalable parallelization design that exploits <italic xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">data-level parallelism</i> through fine-grained data partition under different representations. Finally, experiments on 1 NVIDIA GPU demonstrate that our work outperforms 251.7×, 78.5× and 164.9× than three mainstream CPU-based libraries HElib, SEAL, and PALISADE, and up to 170.5× speedup is obtained compared to the GPU-accelerated library cuHE. What's more, performing 8 homomorphic multiplications on 8 GPUs can deliver up to a 7.66× performance boost compared to a single-GPU implementation.

CoFHE: Software and hardware Co-design for FHE-based machine learning as a service

Introduction: Privacy concerns arise whenever sensitive data is outsourced to untrusted Machine Learning as a Service (MLaaS) platforms. Fully Homomorphic Encryption (FHE) emerges one of the most promising solutions to implementing privacy-preserving MLaaS. But prior FHE-based MLaaS faces challenges from both software and hardware perspectives. First, FHE can be implemented by various schemes including BGV, BFV, and CKKS, which are good at different FHE operations, e.g., additions, multiplications, and rotations. Different neural network architectures require different numbers of FHE operations, thereby preferring different FHE schemes. However, state-of-the-art MLaaS just naïvely chooses one FHE scheme to build FHE-based neural networks without considering other FHE schemes. Second, state-of-the-art MLaaS uses power-hungry hardware accelerators to process FHE-based inferences. Typically, prior high-performance FHE accelerators consume &gt;160 Watt, due to their huge capacity (e.g., 512 MB) on-chip SRAM scratchpad memories.Methods: In this paper, we propose a software and hardware co-designed FHE-based MLaaS framework, CoFHE. From the software perspective, we propose an FHE compiler to select the best FHE scheme for a network architecture. We also build a low-power and high-density NAND-SPIN and SRAM hybrid scratchpad memory system for FHE hardware accelerators.Results: On average, under the same security and accuracy constraints, on average, CoFHE accelerates various FHE-based inferences by 18%, and reduces the energy consumption of various FHE-based inferences by 26%.Discussion: CoFHE greatly improves the latency and energy efficiency of FHE-based MLaaS.

Toward Secure Data Computation and Outsource for Multi-User Cloud-Based IoT

Cloud computing has promoted the success of Internet of Things (IoT) with offering abundant storage and computation resources where the data from IoT sensors can be remotely outsourced to the cloud servers, whereas storing, exchanging and processing data collected through IoT sensors via centralised or decentralised cloud servers make cloud-based IoT systems prone to internal or external attacks. To protect IoT data against potential malicious users and adversaries, some cryptographic schemes have been applied to ensure confidentiality and integrity of IoT data. It is however a challenging task to perform any arithmetical computations once data items are encrypted. Fully-homomorphic encryption which is based on lattices can, in principle, provide a solution, but it is unfortunately inefficient in computation and hence cannot be applied to IoT. Fully-homomorphic encryption is feasible when we allow an involvement of semi-trusted server. However, it is challenging to provide such a system in the situation of distributed environments for shared IoT data. We solve this problem and provide a fully-homomorphic encryption scheme for cloud-based IoT applications. We introduce a new method with the aid of semi-trusted server who can help in the computation of the homomorphic multiplications without gaining any useful information of the encrypted data.

HEDA

Recent years have witnessed the rapid development of the encrypted database, due to the increasing number of data privacy breaches and the corresponding laws and regulations that caused millions of dollars in loss. These encrypted databases may rely on different techniques, such as cryptographic primitives and trusted execution environments. In this work, we investigate the feasibility of utilizing fully homomorphic encryption (FHE) to support unbounded database aggregation queries, which typically involve comparisons as filtering predicates and a final aggregation. These operators are theoretically supported by FHE, but need careful algorithm design to maximize the efficiency and have not been explored before. We creatively use two types of FHE schemes, i.e. , one for numerical and one for binary value, to enjoy their advantages respectively. To bridge the encrypted values between these two schemes for seamless query processing without client-server interaction, we propose a novel ciphertext transformation mechanism, which is of independent research interest, to close this gap. We further implement our system and test it over three TPC-H queries and a query over a real social media e-commerce database. Evaluation results show that, to process an aggregation query over 8 k encrypted rows takes about 430 seconds. Although it is slower than plaintext processing in magnitudes and still has much room for improvement, as the very first work in this domain, our system demonstrates the feasibility of using FHE to process OLAP queries.

FDFB: Full Domain Functional Bootstrapping Towards Practical Fully Homomorphic Encryption

Computation on ciphertexts of all known fully homomorphic encryption (FHE) schemes induces some noise, which, if too large, will destroy the plaintext. Therefore, the bootstrapping technique that re-encrypts a ciphertext and reduces the noise level remains the only known way of building FHE schemes for arbitrary unbounded computations. The bootstrapping step is also the major efficiency bottleneck in current FHE schemes. A promising direction towards improving concrete efficiency is to exploit the bootstrapping process to perform useful computation while reducing the noise at the same time. We show a bootstrapping algorithm, which embeds a lookup table and evaluates arbitrary functions of the plaintext while reducing the noise. Depending on the choice of parameters, the resulting homomorphic encryption scheme may be either an exact FHE or homomorphic encryption for approximate arithmetic. Since we can evaluate arbitrary functions over the plaintext space, we can use the natural homomorphism of Regev encryption to compute affine functions without bootstrapping almost for free. Consequently, our algorithms are particularly suitable for arithmetic circuits over a finite field with many additions and scalar multiplication gates. We achieve significant speedups when compared to binary circuit-based FHE. For example, we achieve 280-1200x speedups when computing an affine function of size 784 followed by any univariate function when compared to FHE schemes that compute binary circuits. With our bootstrapping algorithm, we can efficiently convert between arithmetic and boolean plaintexts and extend the plaintext space using the Chinese remainder theorem. Furthermore, we can run the computation in an exact and approximate mode where we trade-off the size of the plaintext space with approximation error. We provide a tight error analysis and show several parameter sets for our bootstrapping. Finally, we implement our algorithm and provide extensive tests. We demonstrate our algorithms by evaluating different neural networks in several parameter and accuracy settings.

Verifiable quaternion fully homomorphic encryption scheme for mitigating false data injection attacks by privacy preservation in fog environment

Fog computing as an extension of cloud computing has attracted maximum attention over the recent years due to its suitability and applicability. The Fog computing is identified to handle the issues of inadequate mobility support and location sensitiveness and maximized latency. In the Internet of Things (IoT), a collection of IoT devices needs to be connected to the Fog nodes for supporting the center of the cloud service to process and store partial data part in prior. This possibility of connecting IoT devices with Fog nodes has the maximum possibility of enhancing the service quality and minimize the pressure involved in data processing in real time. However, Fog nodes used in data processing are considered from a diversified number of challenging issues introduced by the privacy violation of IoT devices, modification attacks and false data injection attack. In this paper, a Verifiable Quaternion Fully Homomorphic Encryption (VQFHE) scheme is proposed for mitigating False Data Injection Attacks by means of privacy preservation and fault tolerance in Fog computing scenario. This proposed VQFHE scheme derives the benefits of Quaternions for ensures that data injection always happens from the legitimate devices, inspite to the honest or curious behavior of the cloud control center (CCC) and Fog nodes. The simulation results conducted using iFogSim simulator confirmed its predominance over the benchmarked schemes with respect to communication overhead and computation overhead.

EDLaaS:Fully Homomorphic Encryption over Neural Network Graphs for Vision and Private Strawberry Yield Forecasting.

We present automatically parameterised Fully Homomorphic Encryption (FHE) for encrypted neural network inference and exemplify our inference over FHE-compatible neural networks with our own open-source framework and reproducible examples. We use the fourth generation Cheon, Kim, Kim, and Song (CKKS) FHE scheme over fixed points provided by the Microsoft Simple Encrypted Arithmetic Library (MS-SEAL). We significantly enhance the usability and applicability of FHE in deep learning contexts, with a focus on the constituent graphs, traversal, and optimisation. We find that FHE is not a panacea for all privacy-preserving machine learning (PPML) problems and that certain limitations still remain, such as model training. However, we also find that in certain contexts FHE is well-suited for computing completely private predictions with neural networks. The ability to privately compute sensitive problems more easily while lowering the barriers to entry can allow otherwise too-sensitive fields to begin advantaging themselves of performant third-party neural networks. Lastly, we show how encrypted deep learning can be applied to a sensitive real-world problem in agri-food, i.e., strawberry yield forecasting, demonstrating competitive performance. We argue that the adoption of encrypted deep learning methods at scale could allow for a greater adoption of deep learning methodologies where privacy concerns exist, hence having a large positive potential impact within the agri-food sector and its journey to net zero.

An ultra-compact and high-speed FFT-based large-integer multiplier for fully homomorphic encryption using a dual spike-based arithmetic circuit over GF(p)

During last years, fully homomorphic encryption (FHE) has attracted great interest since enables computation on encrypted data and thus can be considered as a potential solution in advanced applications, such as privacy-preserving cloud computing, genomics, electronic voting and bio-metric authentication. Nowadays, software simulations of the FHE schemes on general purpose computers are extremely slow. Therefore, the development of specific hardware architectures open up new horizons in efficient simulation of FHE schemes. Until date, the implementation of FHE schemes in embedded devices remain impractical due to very high processing time and area consumption required to process very large-integer numbers. Specifically, the development of efficient very large-integer finite-field adder and multiplier potentially allows the throughput and area consumption to be improved since these circuits are the most used components in the computation of FHE schemes. This work presents, for the first time, the development of a compact and highly dual finite-field circuit based on spiking neural P systems (SN P), dendritic growth, dendritic pruning, communication on request and structural plasticity. Specifically, this circuit performs either the finite-field addition or multiplication of two variable integers by only reconnecting their synapses dynamically. Hence, the proposed circuit performs both operations employing the same neural network. In this way, we achieve a significant improvement in terms of area consumption and throughput. Since the computation of FHE schemes requires the multiplication of very large-integer numbers, we use the proposed dual finite-field circuit as the basic processing unit to create a very large-integer multiplier based on fast Fourier transform (FFT). Specifically, the creation of the proposed very large-integer multiplier has allowed us to accelerate the key generation and encryption processes involved in the computation of FHE algorithm. This multiplier was implemented in scalable compact neuromorphic architecture, which mimic the dynamic dendritic phenomena, such as dendritic growth and dendritic pruning. To achieve this, we propose a dynamic multiplexing mechanism based on simple multiplexers and an optimized control unit. This have allowed us to significantly improve the area consumption compared with previously reported solution.

A Highly Unified Reconfigurable Multicore Architecture to Speed Up NTT/INTT for Homomorphic Polynomial Multiplication

The ring learning with error (RLWE)-based fully homomorphic encryption (FHE) scheme has become one of the most promising FHE schemes. However, its performance is limited by the homomorphic multiplication, especially the polynomial multiplication which occupies major computing resources. Therefore, efficient implementation of polynomial multiplication is crucial for high-performance FHE applications. In this article, we present an area-efficient and highly unified reconfigurable multicore number theoretic transform (NTT)/inverse NTT (INTT) architecture (named MCNA), which employs NTT and INTT for polynomial multiplier with a variable number of reconfigurable processing elements. To reduce latency, MCNA merges the preprocessing and postprocessing into the constant-geometry NTT and INTT, respectively. Also, a reconfigurable modular multiplier based on digital signal processor (DSP) is proposed to speed up the modular multiplication. In order to avoid designing independent memory access pattern for INTT, a unified read/write structure of NTT/INTT is presented. Furthermore, a novel memory access pattern named “cyclic-sharing” is proposed to reduce 25% memory capacity. MCNA is evaluated on a Xilinx Virtex-7 field-programmable gate array (FPGA) platform. Running at 250-MHz clock frequency, the throughput of MCNA for NTT/INTT achieves <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$2.78\times \sim 9.32\times $ </tex-math></inline-formula> improvements in comparison to prior works, while the area efficiency of lookup table (LUT) and flip-flop (FF) is improved by <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$1.25\times \sim 4.79\times $ </tex-math></inline-formula> . For polynomial multiplication, the throughput of MCNA achieves <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$3.73\times \sim 7.69\times $ </tex-math></inline-formula> enhancements, as well as <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$1.13\times \sim 14.8\times $ </tex-math></inline-formula> area efficiency improvements.