Unified dual-radix architecture for scalable montgomery multiplications in GF(P) and GF(2n)

Kazuyuki Tanimura, Ryuta Nara, Shunitsu Kohara, Youhua Shi, Nozomu Togawa, Masao Yanagisawa, Tatsuo Ohtsuki

Research output: Contribution to journalArticlepeer-review


Modular multiplication is the most dominant arithmetic operation in elliptic curve cryptography (ECC), that is a type of publickey cryptography. Montgomery multiplier is commonly used to compute the modular multiplications and requires scalability because the bit length of operands varies depending on its security level. In addition, ECC is performed in GF(P) or GF(2n), and unified architecture for multipliers in GF(P) and GF(2n) is required. However, in previous works, changing frequency is necessary to deal with delay-time difference between GF ( P) and GF(2n) multipliers because the critical path of the GF(P) multiplier is longer. This paper proposes unified dual-radix architecture for scalable Montgomery multiplications in GF(P) and GF(2n). This proposed architecture unifies four parallel radix-216 multipliers in GF(P) and a radix-264 multiplier in GF(2n) into a single unit. Applying lower radix to GF(P) multiplier shortens its critical path and makes it possible to compute the operands in the two fields using the same multiplier at the same frequency so that clock dividers to deal with the delay-time difference are not required. Moreover, parallel architecture in GF(P) reduces the clock cycles increased by dual-radix approach. Consequently, the proposed architecture achieves to compute a GF(P) 256-bit Montgomery multiplication in 0.28 μs. The implementation result shows that the area of the proposal is almost the same as that of previous works: 39 kgates.

Original languageEnglish
Pages (from-to)2304-2317
Number of pages14
JournalIEICE Transactions on Fundamentals of Electronics, Communications and Computer Sciences
Issue number9
Publication statusPublished - 2009 Sept


  • Dual-radix
  • Elliptic curve cryptography
  • Modular multiplication
  • Montgomery multiplication
  • Scalability
  • Unified

ASJC Scopus subject areas

  • Signal Processing
  • Computer Graphics and Computer-Aided Design
  • Electrical and Electronic Engineering
  • Applied Mathematics


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