Artix-7 Field Programmable Gate Array Research Articles

Voltage Scaled Low Power DNN Accelerator Design on Reconfigurable Platform

The exponential emergence of Field-Programmable Gate Arrays (FPGAs) has accelerated research on hardware implementation of Deep Neural Networks (DNNs). Among all DNN processors, domain-specific architectures such as Google’s Tensor Processor Unit (TPU) have outperformed conventional GPUs (Graphics Processing Units) and CPUs (Central Processing Units). However, implementing low-power TPUs in reconfigurable hardware remains a challenge in this field. Voltage scaling, a popular approach for energy savings, can be challenging in FPGAs, as it may lead to timing failures if not implemented appropriately. This work presents an ultra-low-power FPGA implementation of a TPU for edge applications. We divide the systolic array of a TPU into different FPGA partitions based on the minimum slack value of different design paths of Multiplier Accumulators (MACs). Each partition uses different near-threshold (NTC) biasing voltages to run its FPGA cores. The biasing voltage for each partition is roughly calculated by the proposed static schemes. However, further calibration of biasing voltage is performed by the proposed runtime scheme. To overcome the timing failure caused by NTC, the MACs with higher minimum slack are placed in lower-voltage partitions, while the MACs with lower minimum slack paths are placed in higher-voltage partitions. The proposed architecture is implemented in a commercial platform, namely Vivado with Xilinx Artix-7 FPGA and academic platform VTR with 22 nm, 45 nm and 130 nm FPGAs. Any timing error caused by NTC can be caught by the Razor flipflop used in each MAC. The proposed voltage-scaled, partitioned systolic array can save 3.1% to 11.6% of dynamic power in Vivado and VTR tools, respectively, depending on the FPGA technology, partition size, number of partitions and biasing voltages. The normalized performance and accuracy of benchmark models running on our low-power TPU are very competitive compared to existing literature.

CASA: A Compact and Scalable Accelerator for Approximate Homomorphic Encryption

Approximate arithmetic-based homomorphic encryption (HE) scheme CKKS [CKKS17] is arguably the most suitable one for real-world data-privacy applications due to its wider computation range than other HE schemes such as BGV [BGV14], FV and BFV [Bra12, FV12]. However, the most crucial homomorphic operation of CKKS called key-switching induces a great amount of computational burden in actual deployment situations, and creates scalability challenges for hardware acceleration. In this paper, we present a novel Compact And Scalable Accelerator (CASA) for CKKS on the field-programmable gate array (FPGA) platform. The proposed CASA addresses the aforementioned computational and scalability challenges in homomorphic operations, including key-exchange, homomorphic multiplication, homomorphic addition, and rescaling.On the architecture layer, we propose a new design methodology for efficient acceleration of CKKS. We design this novel hardware architecture by carefully studying the homomorphic operation patterns and data dependency amongst the primitive oracles. The homomorphic operations are efficiently mapped into an accelerator with simple control and smooth operation, which brings benefits for scalable implementation and enhanced pipeline and parallel processing (even with the potential for further improvement).On the component layer, we carry out a detailed and extensive study and present novel micro-architectures for primitive function modules, including memory bank, number theoretic transform (NTT) module, modulus switching bank, and dyadic multiplication and accumulation.On the arithmetic layer, we develop a new partially reduction-free modular arithmetic technique to eliminate part of the reduction cost over different prime moduli within the moduli chain of the Residue Number System (RNS). The proposed structure can support arbitrary numbers of security primes of CKKS during key exchange, which offers better security options for adopting the scalable design methodology.As a proof-of-concept, we implement CASA on the FPGA platform and compare it with state-of-the-art designs. The implementation results showcase the superior performance of the proposed CASA in many aspects such as compact area, scalable architecture, and overall better area-time complexities.In particular, we successfully implement CASA on a mainstream resource-constrained Artix-7 FPGA. To the authors’ best knowledge, this is the first compact CKKS accelerator implemented on an Artix-7 device, e.g., CASA achieves a 10.8x speedup compared with the state-of-the-art CPU implementations (with power consumption of only 5.8%). Considering the power-delay product metric, CASA also achieves 138x and 105x improvement compared with the recent GPU implementation.

An efficient Grain-80 stream cipher with unrolling features to enhance the throughput on hardware platform

<span>The stream cipher is a fundamental component of symmetric cryptography and offers unique implementation speed and scalability advantages. Additionally, the complexity of the cipher algorithm deployment environment forces new, appropriate designs and challenges on the already-existing cipher algorithms. To increase throughput, an efficient Grain-80 stream cipher with unrolling features is designed in this manuscript. The Grain-80 cipher uses an 80-bit key, and a 64-bit initialization vector (IV) and contains two feedback shift registers (linear and non-linear) and an output function. The register balancing and unrolling features of the proposed Grian-80 cipher combine to increase throughput while requiring little additional hardware. Low latency, fast throughput, excellent efficiency, and reduced attack susceptibility are all features of the unrolling architecture. The proposed Grain-80 cipher utilizes <1% chip area and operates at 542.7 MHz on Artix-7 field programmable gate array (FPGA). The proposed Grain-80 cipher improves the operating frequency by 14.85% over conventional Grain-80 cipher. The Grain-80 cipher obtains the throughput of 4.35 Gbps and 8.69 Gbps for unrolling factors 8 and 16, respectively. Lastly, the proposed Grain-80 cipher is compared with existing Grain-80 ciphers with improved throughput and hardware efficiency.</span>

A modular technique of Booth encoding and Vedic multiplier for low-area and high-speed applications

A technique for efficiently multiplying two signed numbers using limited area and high speed is presented in this paper. This work uses both the Booth and Vedic multiplication sutra methodologies to enhance the speed and reduction in the area by using two VLSI architectures of radix encoding techniques—Radix-4 and Radix-8—with the Vedic multiplier. The functionality of the proposed methods is tested using an Artix-7 Field Programmable Gate Array (FPGA-XC7A100T-CSG324) in Xilinx Vivado 2019.1 and ASIC 45 nm technology. Two methods of Booth encoding using Vedic multiplier (Urdhva-Tiryakbhyam sutra) were used to develop, and examine the benefits of rapid computational multiplier. The results of the proposed multiplier for Booth-Vedic-Radix-4 encoding (BVR-4) decrease area by 89% and improve Area-Delay Product (ADP) by 72% for a 16-bit multiplier when subjected to other existing multipliers. The Booth-Vedic-Radix-8 (BVR-8) method shows that there will be an 89% reduction in area and an improvement in ADP by 72% for the 16-bit multiplier. The performance is evaluated regarding area occupancy (i.e., LUTs number) and propagation delay (output time). In terms of resource utilization, the proposed BVR-4 and BVR-8 multipliers outperform all the current designs with a marginal effect on speed and area for narrower bit-width ranges.

An efficient asynchronous spatial division multiplexing router for network-on-chip on the hardware platform

The quasi-delay-insensitive (QDI) based asynchronous network-on-chip (ANoC) has several advantages over clock-based synchronous network-on-chips (NoCs). The asynchronous router uses a virtual channel (VC) as a primary flow-control mechanism however, the spatial division multiplexing (SDM) based mechanism performs better over input traffics over VC. This manuscript uses an asynchronous spatial division multiplexing (ASDM) based router for NoC architecture on a field-programmable gate array (FPGA) platform. The ASDM router is configurable to different bandwidths and VCs. The ASDM router mainly contains input-output (I/O) buffers, a switching allocator, and a crossbar unit. The 4-phase 1-of-4 dual-rail protocol is used to construct the I/O buffers. The performance of the ASDM router is analyzed in terms of lower urinary tract symptoms (LUTs) (chip area), delay, latency, and throughput parameters. The work is implemented using Verilog-HDL with Xilinx ISE 14.7 on artix-7 FPGA. The ASDM router achieves % chip area and obtains 0.8 ns of latency with a throughput of 800 Mfps. The proposed router is compared with existing asynchronous approaches with improved latency and throughput metrics.

MF-RALU: design of an efficient multi-functional reversible arithmetic and logic unit for processor design on the field programmable gate array platform

Most modern computer applications use reversible logic gates to solve power dissipation issues. This manuscript uses an efficient multi-functional reversible arithmetic and logical unit (MF-RALU) to perform 30 operations. The 32-bit MF-RALU includes arithmetic, logical, complement, shifters, multiplexers, different adders, and multipliers. The multi-bit reversible multiplexers are used to construct the MF-RALU structure. The Reduced instruction set computer (RISC) processor is designed to realize the functionality of the MF-RALU. The MF-RALU can perform its operation in a single clock cycle. The 1-bit RALU is developed and compared with existing approaches with improvements in performance metrics. The 32-bit reversible arithmetic units (RAUs) and reversible logical units (RLUs) are constructed using 1-bit RALU. The MF-RALU and RISC processor are synthesized individually in the Vivado environment using Verilog-HDL and implemented on Artix-7 field programmable gate array (FPGA). The MF-RALU utilizes a <11% chip area and consumes 332 mW total power. The RISC processor utilizes a <3% chip area and works at 483 MHZ frequency by consuming 159 mW of total power on Artix-7 FPGA.

An efficient ultra-wideband digital transceiver for wireless applications on the field-programmable gate array platform

<p>The ultra-wideband (UWB) technology is a promising short-range communication technology for most wireless applications. The UWB works at higher frequencies and is affected by interferences with the same frequency standards. This manuscript has designed an efficient and low-cost implementation of IEEE 802.15.4a-based UWB-digital transceiver (DTR). The design module contains UWB transmitter (TX), channel, and UWB-receiver (RX) units. Convolutional encoding and modulation units like burst position modulation and binary phase-shift keying modulation are used to construct the UWB-TX. The synchronization and Viterbi decoder units are used to recover the original data bits and are affected by noise in UWB-RX. The UWB-DTR is synthesized using Xilinx ISE<sup>®</sup> environment with Verilog hardware description language (HDL) and implemented on Artix-7 field-programmable gate array (FPGA). The UWB-DTR utilizes less than 2% (slices and look-up table/LUTs), operates at 268 MHz, and consumes 91 mW of total power on FPGA. The transceiver achieves a 6.86 Mbps data rate, which meets the IEEE 802.15.4a standard. The UWB-DTR module obtains the bit error rate (BER) of 2×10<sup>-4</sup> by transmitting 105 data bits. The UWB-DTR module is compared with similar physical layer (PHY) transceivers with improvements in chip area (slices), power, data rate, and BER. </p>

Neutron-Irradiation Testing of FPGA-Embedded Hadron Fluence Sensors

Hadron fluence monitors based on static random access memories (SRAMs) are being used at CERN and have been proposed for proton therapy facilities. Some of the limitations of the state of the art are related to the usage of separate components for sensing upsets and for reading them out, as these increase the power consumption, the board complexity and its size. Moreover, in some cases, due to radiation-tolerance requirements, the readout logic is fixed and it cannot be updated once the system has been implemented. In this work, we show how to overcome the mentioned limitations by using an SRAM-based field programmable gate array (FPGA) for implementing both the sensitive element (the configuration SRAM) and the readout logic (the firmware in the fabric). In fact, we describe the implementation of a compact, reprogrammable, low-power, actively self-reading hadron fluence sensor realized by means of a Xilinx Artix-7 FPGA. Moreover, we present the customized radiation-hardening-by-design techniques adopted for the readout logic. We irradiated two sensor prototypes at the Jožef Stefan Institute’s TRIGA Mark II research reactor, under different neutron spectra and flux conditions. We discuss our results, which include the measurements of the radiation tolerance, the single event upset cross section of the configuration SRAM, the sensitivity to thermal neutrons, and the failure cross section of the readout logic.

Implementation of a secure wireless communication system using true random number generator for internet of things

This paper describes the design and implementation of an internet of thing (IoT)-based application that uses a true random number generator (TRNG) with an all digital phase locked loop (ADPLL) for secure wireless communication. Field programmable gate array (FPGA) boards were used on the transmitter and receiver sides and were interfaced with Esp8266 chips to wirelessly send and receive encrypted sensor data. The MQ-2 gas sensor and tracking sensor were connected to the FPGA board on the transmitter side, where data from the sensors was encrypted using the exclusive-OR (XOR) function and the TRNG architecture. The system can be controlled by users through a web browser served by the ThingSpeak cloud. The Artix-7 FPGA device is used to implement the proposed wireless communication system, for which design and synthesis were done using the Xilinx Vivado 2015.2 tool. The proposed system uses a low amount of power and is suitable for a standalone, highly secure TRNG-based IoT application. The National Institute of Standard and Testing (NIST SP 800-22) test showed that ADPLL with finite impulse response (FIR) filter-based TRNGs are better for encrypting IoT devices for secure wireless communication.

Deep reinforcement learning-based pairwise DNA sequence alignment method compatible with embedded edge devices

Sequence alignment is an essential component of bioinformatics, for identifying regions of similarity that may indicate functional, structural, or evolutionary relationships between the sequences. Genome-based diagnostics relying on DNA sequencing have benefited hugely from the boom in computing power in recent decades, particularly due to cloud-computing and the rise of graphics processing units (GPUs) and other advanced computing platforms for running advanced algorithms. Translating the success of such breakthroughs in diagnostics to affordable solutions for low-cost healthcare requires development of algorithms that can operate on the edge instead of in the cloud, using low-cost and low-power electronic systems such as microcontrollers and field programmable gate arrays (FPGAs). In this work, we present EdgeAlign, a deep reinforcement learning based method for performing pairwise DNA sequence alignment on stand-alone edge devices. EdgeAlign uses deep reinforcement learning to train a deep Q-network (DQN) agent for performing sequence alignment on fixed length sub-sequences, using a sliding window that is scanned over the length of the entire sequence. The hardware resource-consumption for implementing this scheme is thus independent of the lengths of the sequences to be aligned, and is further optimized using a novel AutoML based method for neural network model size reduction. Unlike other algorithms for sequence alignment reported in literature, the model demonstrated in this work is highly compact and deployed on two edge devices (NVIDIA Jetson Nano Developer Kit and Digilent Arty A7-100T, containing Xilinx XC7A35T Artix-7 FPGA) for demonstration of alignment for sequences from the publicly available Influenza sequences at the National Center for Biotechnology Information (NCBI) Virus Data Hub.

Design of efficient reversible floating-point arithmetic unit on field programmable gate array platform and its performance analysis

<p><span>The reversible logic gates are used to improve the power dissipation in modern computer applications. The floating-point numbers with reversible features are added advantage to performing complex algorithms with high-performance computations. This manuscript implements an efficient reversible floating-point arithmetic (RFPA) unit, and its performance metrics are realized in detail. The RFP adder/subtractor (A/S), RFP multiplier, and RFP divider units are designed as a part of the RFP arithmetic unit. The RFPA unit is designed by considering basic reversible gates. The mantissa part of the RFP multiplier is created using a 24x24 Wallace tree multiplier. In contrast, the reciprocal unit of the RFP divider is designed using Newton Raphson’s method. The RFPA unit and its submodules are executed in parallel by utilizing one clock cycle individually. The RFPA unit and its submodules are synthesized separately on the Vivado IDE environment and obtained the implementation results on Artix-7 field programmable gate array (FPGA). The RFPA unit utilizes only 18.44% slice look-up tables (LUTs) by consuming the 0.891 W total power on Artix-7 FPGA. The RFPA unit sub-models are compared with existing approaches with better performance metrics and chip resource utilization improvements.</span></p>

Range-enhanced packet classification to improve computational performance on field programmable gate array

<span lang="EN-US">Multi-filed packet classification is a powerful classification engine that classifies input packets into different fields based on predefined rules. As the demand for the internet increases, efficient network routers can support many network features like quality of services (QoS), firewalls, security, multimedia communications, and virtual private networks. However, the traditional packet classification methods do not fulfill today’s network functionality and requirements efficiently. In this article, an efficient range enhanced packet classification (REPC) module is designed using a range bit-vector encoding method, which provides a unique design to store the precomputed values in memory. In addition, the REPC supports range to prefix features to match the packets to the corresponding header fields. The synthesis and implementation results of REPC are analyzed and tabulated in detail. The REPC module utilizes 3% slices on Artix-7 field programmable gate array (FPGA), works at 99.87 Gbps throughput with a latency of 3 clock cycles. The proposed REPC is compared with existing packet classification approaches with better hardware constraints improvements.</span>

Design and performance analysis of human body communication digital transceiver for wireless body area network applications

Wireless body area network (WBAN) is a prominent technology for resolving health-care concerns and providing high-speed continuous monitoring and real-time help. Human body communication (HBC) is an IEEE 802.15.6 physical layer standard for short-range communications that is not reliant on radio frequency (RF). Most WBAN applications can benefit from the HBC's low-latency and low-power architectural features. In this manuscript, an efficient digital HBC transceiver (TR) hardware architecture is designed as per IEEE 802.15.6 standard to overcome the drawbacks of the RF-wireless communication standards like signal leakage, on body antenna and power consumption. The design is created using a frequency selective digital transmission scheme for transmitter and receiver modules. The design resources are analyzed using different field programmable gate array (FPGA) families. The HBC TR utilizes <1% slices, consumes 101 mW power, and provides a throughput of 24.31 Mbps on Artix-7 FPGA with a latency of 10.5 clock cycles. In addition, the less than 10-4bit error rate of HBC is achieved with a 9.52 Mbps data rate. The proposed work is compared with existing architectures with significant improvement in performance parameters like chip area, power, and data rate.

Systematic Unsupervised Recycled Field-Programmable Gate Array Detection

With the expansion of the semiconductor supply chain, the use of recycled field-programmable gate arrays (FPGAs) has become a serious concern. Several methods for detecting recycled FPGAs by analyzing the ring oscillator (RO) frequencies have been proposed; however, most assume the known fresh FPGAs (KFFs) as the training data in machine-learning-based classification. In this study, we propose a novel recycled FPGA detection method based on an unsupervised anomaly detection scheme when there are few or no KFFs available. As the RO frequencies in the neighboring logic blocks on an FPGA are similar because of systematic process variation, our method compares the RO frequencies and does not require KFFs. The proposed method efficiently identifies recycled FPGAs through outlier detection using direct density ratio estimation. Experiments using Xilinx Artix-7 FPGAs demonstrate that the proposed method successfully distinguishes recycled FPGAs from 35 fresh FPGAs. In contrast, a conventional recycled FPGA detection method results in certain misclassification.

Energy-Efficient UART Design on FPGA Using Dynamic Voltage Scaling for Green Communication in Industrial Sector

In the present scheme of the world, the problem of shortage of power is seen across the world which can be a vulnerability to various communication securities. The scope of proposed research is that it is a step towards completing green communication technology concepts. In order to improve energy efficiency in communication networks, we designed UART using different nanometers of FPGA, which consumes the least amount of energy. This shortage is happening because of expanding of industries across the world and the rapid growth of the population. Therefore, to save the power for our upcoming generation, the globe is moving towards the concept and ideas of green communication and power-/energy-efficient gadget. In this work, a power-efficient universal asynchronous receiver transmitter (UART) is implemented on 28 nm Artix-7 field-programmable gate array (FPGA). The objective of this work is to reduce the power utilization of UART with the FPGA device in industries. To do this, the same authors have used voltage scaling techniques and compared the results with the existing FPGA works.

Throughput/Area Optimized Architecture for Elliptic-Curve Diffie-Hellman Protocol

This paper presents a high-speed and low-area accelerator architecture for shared key generation using an elliptic-curve Diffie-Hellman protocol over GF(2233). Concerning the high speed, the proposed architecture employs a two-stage pipelining and a Karatsuba finite field multiplier. The use of pipelining shortens the critical path which ultimately improves the clock frequency. Similarly, the employment of a Karatsuba multiplier decreases the required number of clock cycles. Moreover, an efficient rescheduling of point addition and doubling operations avoids data hazards that appear due to pipelining. Regarding the low area, the proposed architecture computes finite field squaring and inversion operations using the hardware resources of the Karatsuba multiplier. Furthermore, two dedicated controllers are used for efficient control functionalities. The implementation results after place-and-route are provided on Virtex-7, Spartan-7, Artix-7 and Kintex-7 FPGA (field-programmable gate arrays) devices. The utilized FPGA slices are 5102 (on Virtex-7), 5634 (on Spartan-7), 5957 (on Artix-7) and 6102 (on Kintex-7). In addition to this, the time required for one shared-key generation is 31.08 (on Virtex-7), 31.68 (on Spartan-7), 31.28 (on Artix-7) and 32.51 (on Kintex-7). For performance comparison, a figure-of-merit in terms of throughputarea is utilized which shows that the proposed architecture is 963.3 and 2.76 times faster as compared to the related architectures. In terms of latency, the proposed architecture is 302.7 and 132.88 times faster when compared to the most relevant state-of-the-art approaches. The achieved results and performance comparison prove the significance of presented architecture in all those shared key generation applications which require high speed with a low area.

Development of pipelined polynomial multiplier modulo irreducible polynomials for cryptosystems

In this paper, we consider a schematic solution of the pipeline multiplier modulo, where multiplication begins with the analysis of the lowest order of the polynomial multiplier, which can serve as an operating unit for high-speed encryption and decryption of data by hardware implementation of cryptosystems based on a non-positional polynomial notation. The functional diagram of the pipeline and the structure of its logical blocks, as well as an example of performing the operation of multiplying polynomials modulo, are given. The correct functioning of the developed circuit was checked by modeling in the Vivado Design Suite computer-aided design for the implementation of the multiplication device on the development/evaluation kit Artix-7 based on the Spartan 6 field-programmable gate array series by Xilinx. The effectiveness of the proposed hardware pipeline multiplier in modulo is confirmed by the Verilog Testbench timing diagram implemented for the evaluation kit Artix-7 field-programmable gate array. In addition, the developed pipelined modulo multiplier takes no more than 0.02 % of the resources of the used field-programmable gate array for a given length of input data. Compared to the matrix multiplication method, a pipelined modulo multiplier can handle a large data stream without waiting for the result of the previous multiplication step. The modulo pipelined multiplier depth depends on the bit width of the input data. The developed pipeline device can be used in digital computing devices operating in a polynomial system of residue classes, as well as for high-speed data encryption in blocks of cipher processors operating on the basis of a non-positional polynomial number system.

Functional validation of highly synthesizable voltage comparator on FPGA

In the present work, a methodology has been proposed to design analog/mixed-signal circuits using digital-in-concept circuits in digital technology. This paper presents a digital standard cell-based, scalable, highly synthesizable analog voltage comparator designed in 180 nm CMOS technology at the supply voltage of 1.8 V. The digital design requires less design effort and is highly immune to process, voltage and temperature variations. The proposed comparator has been designed and simulated in the Cadence virtuoso analog design environment. It is observed from the simulation results that the total power dissipation and propagation delay is 143.2 μW and 1.07 ns, respectively. The offset voltage of the proposed comparator is 4.98 mV. Also, as a proof-of-concept, the feasibility of the proposed analog voltage comparator is carried out on Artix-7 field-programmable gate array (FPGA) using Xilinx Basys-3 FPGA kit and other off-the-shelf components. The proposed digital-in-concept comparator is suitable for low power and high-speed SoC application with reduced design effort and lesser time-to-market.

Field Programmable Gate Array based elliptic curve Menezes‐Qu‐Vanstone key agreement protocol realization using Physical Unclonable Function and true random number generator primitives

The trust, authenticity and integrity of Internet-of-Things (IoT) systems are heavily reliant on Physical Unclonable Functions (PUFs) and True random number generators (TRNGs). The PUF and TRNG produce device intrinsic digital signatures and random binary sequences, which are used for cryptographic key generation, key agreement/exchange, device authentication, cloning prevention etc. This article reports an efficient Field Programmable Gate Array (FPGA)-based realization of elliptic curve Menezes-Qu-Vanstone (ECMQV)-authenticated key agreement protocol using PUF and TRNG with very competitive area-throughput trade-offs. The key agreement protocols, which establish a shared secret key between two IoT devices, make use of PUF and TRNG primitives for the long- and short-term secret keys generation while the elliptic curve is employed for public key generated from the corresponding secret key. The performance of the protocol is investigated on FPGAs. The authors' implementation of the ECMQV protocol takes 1.802 ms using 18852 slices on Artix-7 FPGA.

Modular reduction with step-by-step using of several bits of the reducible number

Although public key cryptography is known to solve the problem of physically secure key exchange, the main drawback of this system is its low performance during encrypting and decrypting data. One of the ways to solve this issue is to increase the speed of the modular reduction operation, one of the basic operations of asymmetric cryptoalgorithms. A new method of step-by-step reduction by the N-bit module P using several bits of the 2Nbit reducible number A in one step is proposed in this paper. The method is based on using multiples of the P and reducing modulo at each step not the entire initial number, but its parts (A1, A2… Ai), which allows to reduce the bit capacity of A. A structural diagram of the hardware implementation of this method are developed. The main unit of the modular reduction device is a block of partial remainder formers, in which the partial remainder is computed using multiples of the P. The circuits are modeled in the Vivado Design Suite computer aided design (CAD) on base Artix-7 Fieldprogrammable gate array (FPGA) device from Xilinx. Optimization of hardware costs is achieved by applying the same comparison circuits to compare different multiples of P with Ai