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Issue No. 01 - Jan. (2016 vol. 65)
ISSN: 0018-9340
pp: 147-160
Donald Donglong Chen , Department of Electronic Engineering, City University of Hong Kong, Hong Kong
Gavin Xiaoxu Yao , Department of Electronic Engineering, City University of Hong Kong, Hong Kong
Ray C.C. Cheung , Department of Electronic Engineering, City University of Hong Kong, Hong Kong
Derek Pao , Department of Electronic Engineering, City University of Hong Kong, Hong Kong
Cetin Kaya Koc , Department of Computer Science, University of California Santa Barbara, Santa Barbara, CA
ABSTRACT
Modular multiplication is the core operation in public-key cryptographic algorithms such as RSA and the Diffie-Hellman algorithm. The efficiency of the modular multiplier plays a crucial role in the performance of these cryptographic methods. In this paper, improvements to FFT-based Montgomery Modular Multiplication (FFTM $^3$ ) using carry-save arithmetic and pre-computation techniques are presented. Moreover, pseudo-Fermat number transform is used to enrich the supported operand sizes for the FFTM$^3$ . The asymptotic complexity of our method is $O(l\; \log\, l\; \log\; \log l)$ , which is the same as the Schönhage-Strassen multiplication algorithm (SSA). A systematic procedure to select suitable parameter set for the FFTM$^3$ is provided. Prototypes of the improved FFTM$^3$ multiplier with appropriate parameter sets are implemented on Xilinx Virtex-6 FPGA. Our method can perform 3,100-bit and 4,124-bit modular multiplications in 6.74 and 7.78 $\mu$ s, respectively. It offers better computation latency and area-latency product compared to the state-of-the-art methods for operand size of 3,072-bit and above.
INDEX TERMS
Polynomials, Transforms, Algorithm design and analysis, Spectral analysis, Hardware, Convolution, Complexity theory,field-programmable gate array (FPGA), Schonhage-Strassen Algorithm, number theoretic transform (NTT), Montgomery modular multiplication, parallel computation,field-programmable gate array (FPGA), Schönhage-Strassen algorithm, number theoretic transform (NTT), Montgomery modular multiplication, parallel computation
CITATION
Donald Donglong Chen, Gavin Xiaoxu Yao, Ray C.C. Cheung, Derek Pao, Cetin Kaya Koc, "Parameter Space for the Architecture of FFT-Based Montgomery Modular Multiplication", IEEE Transactions on Computers, vol. 65, no. , pp. 147-160, Jan. 2016, doi:10.1109/TC.2015.2417553
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