Montgomery multiplication and squaring for Optimal Prime Fields

Hwajeong Seo, Zhe Liu, Yasuyuki Nogami, Jongseok Choi, Howon Kim

Research output: Contribution to journalArticlepeer-review

3 Citations (Scopus)


Optimal Prime Fields (OPFs) are considered to be one of the best choices for lightweight elliptic curve cryptography implementations on resource-constraint embedded processors. In this paper, we revisit the efficient modular arithmetic over the special prime fields, and present improved implementations of modular multiplication and squaring for OPFs, called Optimal Prime Field Coarsely Integrated Operand Caching (OPF-CIOC) and Coarsely Integrated Sliding Block Doubling (OPF-CISBD) methods. The OPF-CIOC and OPF-CISBD methods follow the general ideas of (consecutive) operand caching and sliding block doubling techniques, respectively. The methods have been carefully optimized and redesigned for Montgomery multiplication and squaring in an integrated fashion. We then evaluate the practical performance of proposed methods on representative 8-bit AVR processor. Experimental results show that the proposed OPF-CIOC and OPF-CISBD methods outperform the previous best known results in ACNS'14 by a factor of 8% and 32%. Furthermore, our methods are implemented in a regular way which helps to reduce the leakage of side-channel information.

Original languageEnglish
Pages (from-to)276-291
Number of pages16
JournalComputers and Security
Publication statusPublished - Jul 1 2015


  • Consecutive operand caching
  • Embedded processors
  • Montgomery multiplication
  • Operand caching
  • Optimal Prime Fields
  • Public key cryptography
  • Sliding block doubling

ASJC Scopus subject areas

  • General Computer Science
  • Law


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