Novel techniques for Montgomery modular multiplication algorithms for public key cryptosystems

In this paper, techniques for fast implementations of Montgomery's modular multiplication algorithm are described. These use non-redundant multi-bit recoding. These are compared regarding the area and computation requirements and compared with conventional implementation.

This paper presents a new efficient implementation approach of elliptic curve cryptosystem based on a novel finite field multiplication and a high performance scalar multiplication algorithm for wireless network authentication. In this new finite field multiplication, CLNZ sliding window method is used on the signed-digit multiplier in order to reduce the multiplication steps. In addition, in scalar multiplication algorithm of proposed implementation approach, point addition and point doubling operation compute in parallel. Also window technique and signed-digit representation are used in order to reduce the number of point operation. So the multiplication cost in the proposed implementation approach reduced considerably. Using this new implementation approach, the security of the elliptic curve cryptosystem increased considerably. The results show that the proposed finite field multiplication reduces the number of multiplication steps at about 40%–82.4% for d=2–10 in compare with Montgomery modular multiplication algorithm. In addition, the efficiency of the proposed implementation approach of elliptic curve cryptosystem enhances about 88%-97% in compare with the implementation approach of traditional window NAF elliptic curve scalar multiplication algorithm and enhances about 77% in compare with the implementation approach of window NAF elliptic curve scalar multiplication algorithm (based on interleaving) where w=4 and 8, and d=8.

Modular multiplication is an integral part of RSA cryptosystems and its performance heavily determines the performance of the encryption hardware. This paper provides a hardware implementation of Montgomery's modular multiplication algorithm using iterative architecture. The propsed design is implemented in Verilog HDL and simulated functionally using ModelSim Altera 10.1E. The synthesis is performed using Altera Quartus II 9.1 with target FPGA board Altera DE2-70. The proposed design consumes 17540 logic elements with 15480 LUT and takes 2048 clock cycles to perform multiplication process. Based on trade-off parameter AT2 measure, the proposed design offers the best performance among other designs.

This paper presents a new and improved word-based processor array architecture for unified and scalable radix2 Montgomery modular multiplication algorithm. In this architecture, the multiplicand and the modulus words are allocated to each processing element rather than pipelined between the processing elements as in the previous architecture extracted by Ç. Koç̧, and also the multiplier bits are fed serially to the first processing element of the processor array every odd clock cycle. Moreover, this architecture was modified to reduce the critical path delay and area by replacing the two levels of carry save adder (CSA) logic by modified 4-to-2 CSA that use only one level of dual field adder logic (DFA) taking advantage of processing two operand words by the same processing element (PE) of the processor array. An ASIC Implementation of the proposed architecture shows that it can perform 1024-bit modular multiplication (for word size w = 32) in about 17.07 μs. Also, the results show that it has smaller Area ×Time values compared to all existing designs by ratios ranging from 11.6 % to 47.8 % which makes it suitable for implementations where both area and performance are of concern. Moreover, it has higher throughput (1.8-39.5 %) than most of the published unified and scalable architectures except the architecture extracted by Harris. It has slightly higher throughput (4.5 %) than the proposed one.

The functionality of embedded systems is ever growing. The computational power of embedded systems is growing to match this demand, with embedded multiprocessor systems becoming more common. The limitations of embedded systems are not always related to chip size but are commonly due to energy and/or power constraints. While it can be possible to embed a more powerful Multiprocessor System on Chip (MPSoC), it is not always possible to provide an energy or power supply that meets its demands within the device’s size and weight requirements. Power management through Dynamic Voltage and Frequency Scaling (DVFS) enables the device to be run at less than its maximum voltage and frequency, allowing high computational capability when necessary while conserving power at other times. Embedded systems commonly perform real-time functionality. A real-time application has an associated formal model to verify that it meets its timing requirements. This formal model is used to perform a worst-case timing analysis to ensure that the application meets its requirements. These models incorporate the worst-case timing of the application’s computation and communication. Timing changes due to power management must also be taken into account, complicating the verification process. The drive for evermore functionality has led to mixed time-criticality systems, in which multiple applications of various timing criticalities share the same (hardware) resources. This complicates the verification process further as the timing interference due to shared resource contention must be taken into account. A monolithic verification effort is therefore traditionally required after system integration and must be carried out again if any modifications are made that affect the timing of any of the applications. The problem that we aim to solve in this work is to enable real-time applications to perform independent execution and power management without violating their timing requirements or invalidating the timing verification of concurrently executing applications. To solve this problem, we contribute the Composable and Predictable Microkernel (CoMik) to composably and predictably virtualise processors. When used in combination with composable and predictable memory controllers and interconnect (as provided by the Composable and Predictable System-on-Chip (CompSOC) platform), these virtual processors cannot interfere with each other’s timing by even a single cycle. If whatever executes on the virtual processors (e.g. an Operating System (OS) or an application directly) has a real-time requirement, it can be verified independently of whatever executes on the concurrent virtual processors and other virtual resources. To enable formally analysable application execution, we contribute the Predictable Operating System (POSe) that enables dataflow applications to be executed on the (virtualised) processor. We contribute a combined application and platform dataflow graph, including an algorithm to automate this process. When annotated with worst-case timings, the combined application and system graph is then used to verify that the application meets its timing requirement. If the application’s performance is better than its requirement (e.g. when the input or platform behaviour are better than worst case), its performance can be reduced using DVFS to achieve a reduction in power consumption. We contribute an off-line convex optimisation that uses the combined application and platform dataflow model to derive static run-time operating frequency levels to achieve low power consumption. The off-line technique is able to exploit static slack in the schedule, but not dynamic run-time slack due to variations in task execution times. Before dynamic slack can be used it must be possible to observe it. For this purpose, CoMik provides independent power, energy and timing accounting per virtual processor. This enables each virtual processor to be assigned individual power and energy budgets and POSe applications to be assigned timing budgets. We contribute a description and model of how energy and power budgets can be distributed between multiple virtual processors, enabling whatever executes on the virtual processor to perform composable independent power-management without affecting the ability of other virtual processors from using their entire budget allocation. Using CoMik’s accounting infrastructure, we also demonstrate how the quality of applications can be dynamically scaled to assist meeting timing, energy or power requirements. We further contribute a distributed dynamic power management policy that enables dataflow applications that are mapped onto multiple (virtual) processors to make distributed dynamic slack observations and local power-management decisions. We demonstrate the applicability of the presented techniques on an implemented Field Programmable Gate Array (FPGA) prototype of a CompSOC hardware platform instance, using an H.263 decoder as a case-study application. We show that our techniques do not only work in theory, but that they are also implementable and implemented.

We propose an efficient multi-exponentiation algorithm based on the modified Booth' algorithm and Montgomery's modular multiplication algorithm. The multi-exponentiation algorithm can be used to implement fast modern cryptosystems. Owing to the reduced number of multiplications, this algorithm is about 10% faster than Pekmestzi's algorithms. The proposed algorithm can be implemented in hardware as a small component. The component can then be used to form an efficient modular multi-exponentiation module by combining it with an efficient Montgomery modular multiplication module.

In order to make the typical Montgomery's algorithm suitable for various lengths RSA, this paper proposes a scheme for an expandable RSA coprocessor. This design allows for cascading the hardware of Process Elements when larger modulo are required. The proposed design uses optimized Montgomery's modular multiplication Algorithm. The proposed architecture has distinctive features, i.e. not only the computation speed is significantly fast but also the hardware overhead is drastically decreased.

RSA is a widely acceptable and well used algorithm in many security applications. Its main mathematical function is the demanding, in terms of speed, operation of modular exponentiation. In this paper a systolic, scalable, redundant carry-save modular multiplier and an RSA encryption architecture are proposed using the Montgomery modular multiplication algorithm. By completely avoiding the transformations from redundant to non-redundant numbers at the intermediate stages of the architectures, the need for addition is eliminated and very interesting results, in terms of clock frequency, throughput and chip covered area, are achieved.

Modular multiplication is a crucial operation in public-key cryptography systems such as RSA and ECC. In this study, we analyze and improve the iteration steps of the classic Montgomery modular multiplication (MMM) algorithm and propose an interleaved pipeline (IP) structure, which meets the high-performance and low-cost requirements for Internet of Things devices. Compared to the classic pipeline structure, the IP does not require a multiplexing processing element (PE), which helps shorten the data path of intermediate results. We further introduce a disruption in the critical path to complete an iterative step of the MMM algorithm in two clock cycles. Our proposed hardware architecture is implemented on Xilinx Virtex-7 Series FPGA, using DSP48E1, to realize the multiplier. The implemented results show that the modular multiplication of 1024 bits by 2048 bits requires 1.03 μs and 2.13 μs, respectively. Moreover, our area–time–product analysis reveals a favorable outcome compared to the state-of-the-art designs across a 1024-bit and 2048-bit modulus.

This paper presents a field-programmable gate array (FPGA)-based emulator for rapidly prototyping a non-volatile (NV)-FPGA-based system on chip (SoC) where an FPGA, a central processing unit, and other peripheral modules are integrated into a single chip to address the difficulty in estimating the usefulness of an NV-FPGA-based SoC on a target system. By mimicking the NV-memory device as a sequential logic circuit, a virtual NV-FPGA SoC is implemented on a commercial FPGA board, and its system-level functional verification is performed. As a typical design example, the proposed hardware is implemented on a Digilent Basys 3 board. A fundamental arithmetic logic circuit is implemented on the NV-FPGA, and its logical behavior is the same as that of the original circuit, and NV-logic-specific functions, i.e. store and recall operations are successfully confirmed. Additionally, a 96.3% power reduction is estimated by the NV-FPGA SoC compared to that of a volatile FPGA SoC system under 1.83% of activity ratio.

Field-Programmable Gate Arrays (FPGAs) have become a mainstay in the digital electronics world both for the ease of implementation as well as their inherent usefulness in incrementally refining hardware designs. When moving to an Application Specific Integrated Circuit (ASIC) or System on a Chip (SoC), verification becomes a very time consuming process, with virtually no room for error. As a result, a variety of methods have been devised to decrease the risk when creating an ASIC or SoC. We describe a hardware and software framework for testing real-time vision algorithms for lowering the uncertainty in FPGA and SoC development, while reducing the SoC verification time. The framework benefits from hardware and software verification, ease of reconfiguration for testing multiple vision algorithms, and an iterative hardware/ software co-design.

Three-dimensional integrated circuits (3-D ICs) offer significant improvements over two-dimensional circuits, and promise a solution to the severe problems that are being, and will be, encountered as monolithic process geometries are reduced to below 65 nm. Several methods associated with the fabrication of 3-D ICs are discussed in this paper, and the techniques developed by Tezzaron Semiconductor Corp., are described in detail. Four successful 3-D ICs are described, along with the anticipated benefits of applying 3-D design to future system-on-chip (SoC) devices.

Improving the computational efficiency of modular operations for embedded systems

Large integer modular multiplication and modular addition/subtraction are basic operations in elliptic curve cryptography (ECC). Traditionally, systolic array was used to implement modular multiplication. It is not efficient because of data dependency between neighbor processing elements (PEs). In this paper, we propose a compact modular multiplication and addition unit (MMAU) specific for ECC by dividing an improved Montgomery modular multiplication (MMM) algorithm into three parts, mapping them correspondingly to a 3-stage pipelined circuit and adding a small amount of hardware to the third stage of the pipeline for implementing modular addition/subtraction without affecting the performance of MMM. The MMAU computes one modular multiplication and one modular addition/subtraction in every 3(n+7) clock cycles for n-bit operands. When the pipeline is fully utilized, the MMAU can generate a result at every n+7 cycles. The experimental results show that our scheme achieves a higher throughput with less resource compared with existed ones.

The modular multiplication operation is the most time-consuming operation for number-theoretic cryptographic algorithms involving large integers, such as RSA and Diffie-Hellman. Implementations reveal that more than 75 percent of the time is spent in the modular multiplication function within the RSA for more than 1,024-bit moduli. There are fast multiplier architectures to minimize the delay and increase the throughput using parallelism and pipelining. However such designs are large in terms of area and low in efficiency. In this paper, we integrate the fast Fourier transform (FFT) method into the McLaughlin’s framework, and present an improved FFT-based Montgomery modular multiplication (MMM) algorithm achieving high area-time efficiency. Compared to the previous FFT-based designs, we inhibit the zero-padding operation by computing the modular multiplication steps directly using cyclic and nega-cyclic convolutions. Thus, we reduce the convolution length by half. Furthermore, supported by the number-theoretic weighted transform, the FFT algorithm is used to provide fast convolution computation. We also introduce a general method for efficient parameter selection for the proposed algorithm. Architectures with single and double butterfly structures are designed obtaining low area-latency solutions, which we implemented on Xilinx Virtex-6 FPGAs. The results show that our work offers a better area-latency efficiency compared to the state-of-the-art FFT-based MMM architectures from and above 1,024-bit operand sizes. We have obtained area-latency efficiency improvements up to 50.9 percent for 1,024-bit, 41.9 percent for 2,048-bit, 37.8 percent for 4,096-bit and 103.2 percent for 7,680-bit operands. Furthermore, the operating latency is also outperformed with high clock frequency for length-64 transform and above.