FPGA Implementation of a Novel Multifunction Modulo (2ⁿ ± 1) Multiplier Using Radix-4 Booth Encoding Scheme

Abstract

The residue number system is widely used in applications such as communication systems, cryptography, digital filters, digital signal processors, fault-tolerant detection, and so on. This paper proposes a multifunction modulo (2ⁿ ± 1) multiplier based on the radix-4 Booth encoding scheme that can operate both modulo (2ⁿ − 1) and modulo (2ⁿ + 1) multipliers using the same hardware structure with only one control signal. A novel modulo (2ⁿ ± 1) multiplier based on radix-4 Booth encoding is proposed that can achieve superior performance, with low power, fast operation, high area efficiency, and low area-delay product (ADP) and power-delay product (PDP) compared with similar modified Booth-encoding methods. In addition, by integrating the separate modulo functions of the modulo (2ⁿ − 1) multiplier and modulo (2ⁿ + 1) multiplier into a single multifunction modulo (2ⁿ ± 1) multiplier, the proposed method can save up to 52.59% (n = 16) of hardware area, up to 5.45% (n = 32) of delay time, up to 49.05% (n = 16) of dynamic power, up to 50.92% (n = 32) of ADP, and up to 50.02% (n = 32) of PDP compared with the original separate circuits merged together. Furthermore, the operation ranges of the multiplicand and multiplier of the proposed modulo (2ⁿ + 1) multiplier and modulo (2ⁿ − 1) multiplier are {0, 2ⁿ + 1} and {0, 2ⁿ}, respectively, which are wider than for other reported hardware structures. The hardware area, power consumption, and delay time are simulated and verified using Verilog HDL and Xilinx FPGA (Field Programmable Gate Array) Vivado tools. The Xilinx Artix-7 XC7A35T-CSG324-1 FPGA chipset is adopted in the proposed work.

1. Introduction

The residue number system (RNS) [1,2,3,4,5,6,7,8,9,10,11] has been widely used in the field of cryptography [10] for purposes such as Fermat number transform [12], elliptic curve ciphers [4,13], Caesar ciphers [4], IDEA (International Data Encryption Algorithm) [13] block ciphers and asymmetric encryption [10] algorithms. The parallel carry-free feature of the RNS will speed up arithmetic operation when the RNS is applied over a binary-based system, thereby greatly reducing the propagation delay and hardware cost of the system. Therefore, the RNS has found wide application in communication systems, digital signal processors [6], digital filters [3], finite-impulse response (FIR) filters [3,7], fast Fourier transform (FFT) [7], fault-tolerant detection [11], error correction coding [11,14] and other fields [12]. Parallel-prefix adders [15] can provide a very efficient modulo arithmetic operation in RNS mathematical computation. Parallel-prefix adders [15] such as Kogge Stone adder, Sklansky adder, Ladner Fischer adder and Ling’s adder are some of the most commonly used circuit architectures in modular operation.

Modulo (2ⁿ + 1) multipliers [2,13,16,17,18,19,20,21], modulo (2ⁿ − 1) multipliers [1,8,21,22,23,24], modulo (2ⁿ + 1) adders [15,25,26] and modulo (2ⁿ − 1) subtractors [26], which are ROM (Read Only Memory)-less modular arithmetic operations, are most commonly used. Many modular multiplication methods have been reported, such as diminished-1modulo multiplication [12,15,16,27,28], weighted modulo multiplication [2,19,26,29], and Booth-encoder multiplication [8,30]. Diminished-1 and weighted representations are commonly used in modulo (2ⁿ − 1) and modulo (2ⁿ + 1) multipliers. For the diminish-1 and weighted representation, n bits are required for the numbers [0, 2ⁿ − 1] and n + 1 bits are needed for the numbers [0, 2ⁿ], respectively. For diminished-1 modular multiplication [12,15,16,27,28], 1 needs to be subtracted from both the multiplicand and multiplier during operation; therefore, the multiplicand and multiplier cannot be 0 and their operation range is only [1, 2ⁿ − 1]. Thus, it is important to use an additional conversion circuit and a zero detection circuit, as well as to add back one adder after processing the sum of all the partial products. However, this will increase the hardware area, power consumption, and delay time in the design of diminished-1 modulo multiplication. To address these shortcomings, the weighted modular multiplication architecture was proposed [2,19,26,29], which has an improved operation range of [0, 2ⁿ] and greatly reduces the delay time and circuit area. It is worth noting that the zero-detection circuit is not required for weighted modular multiplication. Booth-encoder modular multiplication, which uses two bits as the multiplier in operation, can further reduce the area and delay compared to weighted modular multiplication. The partial product of radix-4 Booth-encoder modular multiplication [8,30], which uses three bits as the multiplier, is reduced by nearly half compared to that of conventional Booth-encoder modular multiplication. This will considerably reduce the use of the full adder, thereby minimizing the circuit area and delay time.

This study proposes an FPGA implementation of multifunction modulo (2ⁿ ± 1) multiplier based on radix-4 Booth encoding that can operate both the modulo (2ⁿ + 1) multiplier and modulo (2ⁿ − 1) multiplier in the same hardware. Juang and Huang [8] presented a modified Booth-encoder-based multifunction modulo (2ⁿ ± 1) multiplier using the structure of a modified Booth-coded 1′s complement modulo (2ⁿ − 1) multiplier [1] and a modified Booth-based modulo (2ⁿ + 1) multiplier [18] and combined them into a multifunction modulo (2ⁿ ± 1) multiplier. However, their hardware structure still has room for improvement in terms of the power consumption, delay time, hardware area, and operation range. The architecture of the radix-4 Booth-encoding-based multifunction modulo (2ⁿ ± 1) multiplier proposed in this paper can achieve significant improvements in power consumption, latency, hardware area, and operation range. The proposed modular multiplication can greatly reduce the hardware cost, delay time, and power consumption, while having wider operation ranges compared to that of Juang and Huang’s method [8]. In this article, the circuit design and implementation are carried out using an FPGA (Field Programmable Gate Array) [3]. FPGA is a popular and important electronic device that can be easily corrected, reusable and reprogrammable by a designer. The advantages of FPGA are that it is easier to be implemented and configured by logic circuit, has a shorter time to develop a prototype circuit and has a lower price than conventional ASICs (Application Specific Integrated circuits). Based on these advantages, FPGA is widely used for circuit prototyping and product development. In the proposed FPGA-based multifunction modulo (2ⁿ ± 1) multiplier using the radix-4 Booth scheme, the abovementioned delay time, hardware area and power are simulated and verified using Xilinx FPGA (Field Programmable Gate Array) Vivado tools. The target chip is implemented using a Xilinx Artix-7 XC7A35T-CSG324-1 FPGA in the proposed work.

The rest of this paper is structured as follows. Section 2 describes the methods reported in the literature. Section 3 presents the proposed radix-4 Booth-encoding-based multifunction modulo (2ⁿ ± 1) multiplier design, which is the area-delay-power efficient. Section 4 presents the results of the proposed method in comparison with those of various other methods. Finally, Section 5 summarizes the study.

2. Related Work

This section reviews radix-4 Booth-encoding-based modulo (2ⁿ + 1) and modified Booth-encoding-based modulo (2ⁿ − 1) multipliers, as well as multifunction modulo (2ⁿ ± 1) multipliers based on the modified Booth encoding scheme [8]. Unlike the traditional Booth encoder, which uses double-code interpretation, the modified Booth encoder, also called radix-4 Booth encoding, adopts a three-code interpretation. First, 0 is added after the least significant bit of the multiplier and 0 is also added in front of the most significant bit of the multiplier. This three-code interpretation method substantially reduces the partial product, which almost halves the hardware area. As a result, the significant reduction in the use of the full adder facilitates faster calculations. The modulo (2ⁿ + 1) multipliers and modulo (2ⁿ − 1) multipliers are the two most commonly used in the RNS, and these will be reviewed in the following paragraphs.

The operating range of the modified Booth-coded 1′s complement modulo (2ⁿ − 1) multiplier [1] is [0, 2ⁿ − 1]. The block diagram of modulo multiplication using modified Booth encoding is shown in Figure 1. Multiplier B first performs improved Booth encoding (BE) and multiplicand A operates a Booth selector (BS) according to the encoded value. Subsequently, the addition array of partial products is generated and these partial products are summed by the full and half adders. Finally, the result is obtained by performing end-around-carry. The relative formula found using the modified Booth-encoder-based 1’s complement modulo (2ⁿ − 1) multiplier is given by

$$B={\displaystyle \sum _{i=0}^{n-1}{2}^{2i}(b_{2i-1}+b_{2i}-2b_{2i+1})}$$

(1)

$$A\times B={\displaystyle \sum _{i=0}^{n-1}A\times [2^{2i}(b_{2i-1}+b_{2i}-2b_{2i+1})]}={\displaystyle \sum _{i=0}^{n-1}P{P}_i}$$

(2)

The system block of modified Booth-coded 1′s complement modulo (2ⁿ − 1) multipliers is shown in Figure 2, where n is set to 8; therefore, it is the block diagram of the modified Booth-encoding-based modulo 63 multiplier [1]. In Figure 2b,c, the ×1 and ×2 of the Booth encoder and Booth selector are used to multiply by 1 and 2, respectively. The sign signal is used to determine positive or negative based on the modified Booth encoding regulation. Multiplier B adopts a three-code interpretation in operation, so the logic circuit diagram of BE in Figure 2b can be deduced from the operation in Equation (1). When three numbers, b_2i−1, b_2i, and b_2i+1, are inputted, the output can be obtained from the signals multiplied by 1 (×1), multiplied by 2 (×2), and the sign signal. The sign signal is used to determine whether the multiplication output of multiplicand will be positive or negative. When the value of sign is 0, the output of the multiplicand is a positive value; when it is 1, the output of the multiplicand is a negative value. According to Equation (1), multiplier B multiplied by multiplicand A will provide Equation (2), where PP_i is its partial product and PP_i = A × 2²ⁱ(b_2i−1 + b_2i − 2b_2i+1).

For modulo (2ⁿ + 1) multiplication using modified Booth encoding, a modified Booth-encoding-based modulo (2ⁿ + 1) multiplier was proposed [2]. This work adopted weighted architecture and did not require additional subtraction and addition operations, such as the diminished-1 architecture as well as additional conversion circuits and a zero detection circuit. However, an additional weighting circuit is required to obtain the final result. Even so, the performance of this method is still better than that of modified Booth-encoding-based modulo multiplication using the diminished-1 architecture. The block diagram of the modified Booth-based modulo 65 (2ⁿ + 1) multiplier [2], with n set to 8 is shown in Figure 3. Here, the Booth encoder, Booth selector, half adder, and full adder in the hardware structure are abbreviated as BE, BS, HA, and FA, respectively. In this work [2], the multiplicand A and multiplier B are expressed in binary as A = a_n a_n−1 a_n−2… a₀ and B = b_n b_n−1 b_n−2… b₀. The final result Q [2] can be expressed as Equation (3):

$$Q={\left|{\displaystyle \sum _{i=0}^{\frac{n}{2}-1}{\left|(A_{n-1}{}_{:0}+1)2^{2i}{b}_i^{MB}\right|}_{2^n+1}-a_{n}B-B}\right|}_{2^n+1}$$

(3)

where $b_i^{MB}=b_{2i-1}+b_{2i}+b_{2i+1} , {\left|(A_{n-1}{}_{:0}+1)2^{2i}{b}_i^{MB}\right|}_{2^n+1}=P{P}_i+1+z_{2i}{2}^{2i}$.

According to Equation (3), the BE and BS are shown in Figure 4a,b. Equation (3) can be deduced and calculated according to [2] and, finally, Equations (4) and (5) can be obtained:

$$Q={\left|C+S+1\right|}_{2^n+1}$$

(4)

$${\left|C+S\right|}_{2^n+1}={\left|{\displaystyle \sum _{i=0}^{\frac{n}{2}-1}p{p}_i+p{p}_1*+Z_n*+\overline{B_L}+\frac{n}{2}}\right|}_{2^n+1}$$

(5)

In these equations, C is carry, S is sum, Z_n* and $\stackrel{-}{B_L}$ are the compensation values of the weighting circuit used to obtain accurate output values, and pp₁* is taken by the inverter of pp_i,0−2i−1 (for i > 0). Finally, Equations (4) and (5) are summed using the inverted end-around-carry (IEAC) carry-save adder (CSA) tree to obtain the final output value Q.

From the above descriptions of the modified Booth-encoding modulo (2ⁿ − 1) multiplier and modulo (2ⁿ + 1) multiplexer, it is clear that the two architectures have many similar circuits. The similar parts of the two architectures can be leveraged as a basis for common use. A certain method can be used to distinguish the different operations so that a circuit with a slightly larger hardware can be used to complete the two kinds of modulus operations. Juang and Huang [8] also adopted the modified Booth-coded 1 complement modulo (2ⁿ − 1) multiplier [1] and modified Booth-encoding-based modulo (2ⁿ + 1) multiplier [18] and combined them into the multifunction modulo (2ⁿ ± 1) multiplier using the same hardware. However, their hardware structure still has room for improvement in terms of power consumption, delay time, hardware area, and operation range. The architecture of the radix-4 Booth-encoding-based multifunction modulo (2ⁿ ± 1) multiplier proposed in this paper can achieve significant improvements in power consumption, latency, and hardware area. Moreover, the structure of the proposed low-power, area-saving, and fast modulo (2ⁿ ± 1) multiplier can operate the modulo (2ⁿ + 1) and modulo (2ⁿ − 1) multipliers with only one control signal sharing the same hardware structure. Meanwhile, the operation ranges of the multiplicand and multiplier of the proposed modulo (2ⁿ + 1) and modulo (2ⁿ − 1) multipliers are {0, 2ⁿ + 1} and {0, 2ⁿ}, respectively. These ranges are wider than those of the hardware structure reported by Juang and Huang [8]. The system architecture of the proposed FPGA implementation of multifunction modulo (2ⁿ ± 1) multiplier based on radix-4 Booth encoding will be presented in the following section.

3. Proposed Modulo (2ⁿ ± 1) Multiplier Based on Radix-4 Booth Encoding

The system architecture of the proposed modulo (2ⁿ ± 1) multiplier based on radix-4 Booth-encoding is shown in Figure 5. The proposed modulo (2ⁿ ± 1) multipliers can support modulo (2ⁿ + 1) and modulo (2ⁿ − 1) multiplication functions in the same circuit hardware. Compared with the traditional modulo (2ⁿ + 1) or modulo (2ⁿ − 1) multipliers, the proposed multipliers can use the same architecture to distinguish between two different operations by selecting signal T. When T = 1, the modulo (2ⁿ + 1) multiplier is selected and, when T = 0, the modulo (2ⁿ − 1) multiplier is selected. The operation ranges of the multiplicand and multiplier are {0, 2ⁿ + 1} and {0, 2ⁿ} for the modulo (2ⁿ + 1) multiplier and modulo (2ⁿ − 1) multipliers, respectively. The proposed modulo (2ⁿ ± 1) multiplier system architecture, shown in Figure 5, includes a partial product circuit unit, a correct circuit unit, an IEAC CSA tree, and a parallel prefix modulo (2ⁿ ± 1) adder. For the partial product circuit unit, the circuit architectures of BE and BS use the same circuit reported in [2], as shown in Figure 4. For each BS row array in the partial product, in the specific BS block with a₇ input, the output signal is XOR with selection signal T, and the output of the XOR gate will feed into the next BS block stage. To increase the operating range of the modulo (2⁸ + 1) and modulo (2⁸ − 1) multipliers (for modulo (2ⁿ + 1) and modulo (2ⁿ − 1) multipliers, n is set to 8), as shown in Figure 5, the OR gate is used to connect b₈ and b₇ and the output of this OR gate will be subjected to an AND gate with selection signal T for the BE input of the first row and second row in the partial product block. An appropriate circuit unit is used to produce the compensation values for obtaining accurate output values for modular multiplication operations using modified Booth-encoding schemes. In the IEAC part, for the structure of the modulo (2ⁿ − 1) multiplexer, the original value of end-around-carry is retained. For the structure of the modulo (2ⁿ + 1) multiplexer, the original value of end-around-carry will be reversed. The proposed improved parallel prefix adder of modulo (2ⁿ ± 1) multipliers based on radix-4 Booth encoding (n = 8) is shown in Figure 6. Assuming that the T signal passes through the XOR gate, when the signal is 0, the modulo (2ⁿ − 1) adder is calculated, and when the signal is 1, the modulo (2ⁿ + 1) adder is calculated.

As an example of the proposed multifunction modulo (2ⁿ ± 1) multiplier based on radix-4 Booth encoding, n = 8 is chosen. Figure 7a,b show the operation process of the proposed modulo (2ⁿ ± 1) multiplier. For the modulo (2ⁿ − 1) multiplication operation with T = 0, A = 256, and B = 256, the final result is 1 (Figure 7a) and, for the modulo (2ⁿ + 1) multiplication operation with T = 1, A = 256, and B = 256, the final result will be 1 (Figure 7b).

In the next section, the proposed FPGA implementation of radix-4 Booth-encoding-based multifunction modulo (2ⁿ ± 1) multipliers are experimentally analyzed, and the results are compared with those of other methods in terms of the operation range, area, delay time, dynamic power, area-delay product (ADP), and power-delay product (PDP).

4. Experimental Results and Comparison

The experimental results obtained using the proposed structure of multifunction modulo (2ⁿ ± 1) multipliers based on radix-4 Booth encoding, as explained in Section 3, are discussed in this section. The term ‘merge circuits’ is used to represent the incorporated circuits that can support two different types of modular multiplication functions adopted by the modulo (2ⁿ − 1) [1] and modulo (2ⁿ + 1) [2] multipliers. The proposed multifunction modulo (2ⁿ ± 1) multipliers can support the aforementioned modular multiplication functions in the same circuit hardware.

Table 1. Comparison of operation ranges of the proposed method and other methods.

Item		Merged Circuits [1,2]		Modulo (2n ± 1) [8]		Proposed Work
		Modulo (2n − 1) [1]	Modulo (2n + 1) [2]	Modulo (2n − 1)	Modulo (2n + 1)	Modulo (2n − 1)	Modulo (2n + 1)
Operation Range	Multiplicand	{0, 2n − 1}	{0, 2n + 1}	{0, 2n}	{1, 2n + 1}	{0, 2n}	{0, 2n + 1}
	Multiplier	{0, 2n − 1}	{0, 2n + 1}	{0, 2n}	{0, 2n}	{0, 2n}	{0, 2n + 1}

compares the operation ranges of the proposed method and other reported methods. The operation ranges of the multiplicand and multiplier of the proposed modulo (2ⁿ + 1) and modulo (2ⁿ − 1) multipliers are {0, 2ⁿ + 1} and {0, 2ⁿ}, respectively; these ranges are wider than those of Juang and Huang’s [8] reported hardware structure. The integration of the single modulo functions of the modulo (2ⁿ + 1) multipliers and modulo (2ⁿ − 1) multipliers into a multifunction modulo (2ⁿ ± 1) multiplier can save up to 52.59% (n = 16) of the hardware area, as shown in

Table 2. Comparison of the hardware areas of the proposed method and other methods.

Item	Modulo (2n − 1) [1]	Modulo (2n + 1) [2]	Merged Circuits [1,2]		Modulo (2n ± 1) [8]		Proposed Work
n	Area (LUT)	Area (LUT)	Area (LUT)	Area Saving	Area (LUT)	Area Saving	Area (LUT)	Area Saving
8	84	103	187	0	113	+39.57%	107	+42.78%
16	355	358	713	0	382	+46.42%	338	+52.59%
32	1406	1281	2687	0	1498	+44.25%	1284	+52.21%

. As shown in

Table 3. Comparison of the delay times of the proposed method and other methods.

Item	Modulo (2n − 1) [1]	Modulo (2n + 1) [2]	Merged Circuits [1,2]		Modulo (2n ± 1) [8]		Proposed Work
n	Delay (ns)	Delay (ns)	Delay (ns)	Delay Saving	Delay (ns)	Delay Saving	Delay (ns)	Delay Saving
8	13.203	15.332	15.332 *	0	15.954	+4.06%	15.351	−0.12%
16	19.346	19.457	19.457 *	0	21.219	+9.06%	20.729	−6.54%
32	30.532	28.565	30.532 *	0	33.376	+9.31%	28.868	+5.45%

*: Taking the worst case of delay time between [1,2].

, compared with that of the original single circuit performing the function of the modulo (2ⁿ − 1) multiplier [1] or modulo (2ⁿ + 1) multiplier [2] exclusively, the delay time of the proposed method slightly increased for n = 8 and 16 but slightly decreased for n = 32. The delay time of the proposed method is considerably lower than that of Juang and Huang’s [8] method. For the merged circuits, as shown in Table 3, it should be noted that the merged delay time of the modulo (2ⁿ − 1) [1] and modulo (2ⁿ + 1) [2] multipliers would be selected to be the worst case for the two of them. As

Table 4. Comparison of the dynamic powers of the proposed method and other methods.

Item	Modulo (2n − 1) [1]	Modulo (2n + 1) [2]	Merged Circuits [1,2]		Modulo (2n ± 1) [8]		Proposed Work
n	Power (W)	Power (W)	Power (W)	Power Saving	Power (W)	Power Saving	Power (W)	Power Saving
8	0.042	0.044	0.086	0	0.048	+44.19%	0.048	+44.19%
16	0.105	0.105	0.21	0	0.114	+45.71%	0.107	+49.05%
32	0.282	0.283	0.565	0	0.309	+45.31%	0.289	+48.85%

shows, the dynamic power consumption of the proposed study can be reduced by 44.19% (n = 8) to 49.05% (n = 16). As

Table 5. Comparison of the ADPs of the proposed study and other methods.

Item	Merged Circuits [1,2]		Modulo (2n ± 1) [8]		Proposed Work
n	ADP	ADP Saving	ADP	ADP Saving	ADP	ADP Saving
8	2688.25	0	1802.8	+32.94%	1642.6	+38.90%
16	13,830.93	0	8105.7	+41.93%	7006.4	+49.34%
32	75,519.76	0	49,997.2	+33.80%	37,066.5	+50.92%

shows, the ADP can be reduced by 38.90% (n = 8) to 50.92% (n = 32). As

Table 6. Comparison of the PDP of the proposed study and other methods.

Item	Merged Circuits [1,2]		Modulo (2n ± 1) [8]		Proposed Work
n	PDP	PDP Saving	PDP	PDP Saving	PDP	PDP Saving
8	1.2291	0	0.7658	+37.70%	0.7368	+40.05%
16	4.0743	0	2.4190	+40.63%	2.2180	+45.56%
32	16.6939	0	10.3132	+38.22%	8.3429	+50.02%

shows, the PDP can be reduced by up to 50.02% (n = 32). It should be noted that the aforementioned ADP and PDP of the merged circuits are defined as the sums of the individual ADPs and PDPs, respectively, of the modulo (2ⁿ − 1) [1] multipliers and modulo (2ⁿ + 1) [2] multipliers. In Table 1, Table 2, Table 3, Table 4, Table 5 and Table 6, it is clear that the proposed modulo (2ⁿ ± 1) multiplier based on radix-4 Booth encoding outperforms other modified Booth methods by exhibiting a wider operation range, lower power consumption, faster operation, better area efficiency, and lower ADP and PDP. The hardware area, power consumption, and delay time are simulated, synthesized, and verified using Verilog HDL and Xilinx FPGA Vivado tools. The Xilinx Artix-7 XC7A35T-CSG324-1 FPGA chipset is adopted. The operating clock frequency, voltage and junction temperature in the simulation are 100 MHz, 3.3 V and 25.4 °C, respectively.

5. Conclusions

This study improved the performance of Juang and Huang’s modified Booth-encoder-based multifunction modulo (2ⁿ ± 1) multiplier [8] by basing it on a radix-4 Booth encoding scheme to have a wider operation range, lower power consumption, faster operation, better area efficiency, and lower area-delay product (ADP) and power-delay product (PDP). It can support the multiplication functions of both modulo (2ⁿ + 1) and modulo (2ⁿ − 1) multiplexers in the same circuit hardware. By integrating their single modulo functions into a multifunction modulo (2ⁿ ± 1) multiplier, it can save up to 50.92% (n = 32) of ADP and 50.02% (n = 32) of PDP compared with merged separate circuits. This multifunction modulo multiplier can be applied to communication systems, cryptography, digital filters, and digital signal processors.