Journal of the Korea Institute of Information and Communication Engineering (한국정보통신학회논문지)

The Korea Institute of Information and Commucation Engineering (한국정보통신학회)
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A 521-bit high-performance modular multiplier using 3-way Toom-Cook multiplication and fast reduction algorithm

3-way Toom-Cook 곱셈과 고속 축약 알고리듬을 이용한 521-비트 고성능 모듈러 곱셈기

  • Yang, Hyeon-Jun (Department of Electronic Engineering, Kumoh National Institute of Technology) ;
  • Shin, Kyung-Wook (School of Electronic Engineering, Kumoh National Institute of Technology)
  • Received : 2021.10.13
  • Accepted : 2021.11.10
  • Published : 2021.12.31
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Abstract

This paper describes a high-performance hardware implementation of modular multiplication used as a core operation in elliptic curve cryptography. A 521-bit high-performance modular multiplier for NIST P-521 curve was designed by adopting 3-way Toom-Cook integer multiplication and fast reduction algorithm. Considering the property of the 3-way Toom-Cook algorithm in which the result of integer multiplication is multiplied by 1/3, modular multiplication was implemented on the Toom-Cook domain where the operands were multiplied by 3. The modular multiplier was implemented in the xczu7ev FPGA device to verify its hardware operation, and hardware resources of 69,958 LUTs, 4,991 flip-flops, and 101 DSP blocks were used. The maximum operating frequency on the Zynq7 FPGA device was 50 MHz, and it was estimated that about 4.16 million modular multiplications per second could be achieved.

본 논문은 타원곡선 암호에 핵심 연산으로 사용되는 모듈러 곱셈의 고성능 하드웨어 구현에 대해 기술한다. NIST P-521 곡선에 적합한 521-비트 고성능 모듈러 곱셈기를 3-way Toom-Cook 정수 곱셈과 고속 축약 알고리듬을 적용하여 설계하였다. 정수곱셈 결과에 3이 곱해져 출력되는 3-way Toom-Cook 알고리듬의 속성을 고려하여, 피연산자에 1/3을 곱한 Toom-Cook 도메인 상에서 모듈러 곱셈이 연산되도록 구현하였다. 모듈러 곱셈기를 xczu7ev FPGA 디바이스에 구현하여 하드웨어 동작을 검증하였으며, 69,958개의 LUT와 4,991개의 플립플롭 그리고 101개의 DSP 블록의 하드웨어 자원이 사용되었다. Zynq7 FPGA 디바이스에서 최대 동작주파수는 50 MHz으로 예측되었으며, 초당 약 416만 번의 모듈러 곱셈을 연산할 수 있는 것으로 평가되었다.

Keywords

Acknowledgement

·This work was supported by Basic Science Research Program through the National Research Foundation of Korea (NRF) funded by the Ministry of Education (No. 2020R1I1A3A04038083) ·This paper was supported by Korea Institute for Advancement of Technology (KIAT) grant funded by the Korea Government (MOTIE) (P0017011, HRD Program for Industrial Innovation) ·The authors are thankful to IDEC for EDA tool support.

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