Elliptic Curve Cryptography with Efficiently Computable Endomorphisms and Its Hardware Implementations for the Internet of Things

Research output: Contribution to journalArticle


  • Zhe Liu
  • Johann Großschädl
  • Zhi Hu
  • Kimmo Järvinen
  • Husen Wang
  • Ingrid Verbauwhede

Research units

  • Nanjing University of Aeronautics and Astronautics
  • University of Waterloo
  • University of Luxembourg
  • Central South University
  • KU Leuven


Verification of an ECDSA signature requires a double scalar multiplication on an elliptic curve. In this work, we study the computation of this operation on a twisted Edwards curve with an efficiently computable endomorphism, which allows reducing the number of point doublings by approximately 50 percent compared to a conventional implementation. In particular, we focus on a curve defined over the 207-bit prime field $\mathbb {F}-p$ with $p = 2 {207}-5{,}131$. We develop several optimizations to the operation and we describe two hardware architectures for computing the operation. The first architecture is a small processor implemented in 0.13 $\mu$ m CMOS ASIC and is useful in resource-constrained devices for the Internet of Things (IoT) applications. The second architecture is designed for fast signature verifications by using FPGA acceleration and can be used in the server-side of these applications. Our designs offer various trade-offs and optimizations between performance and resource requirements and they are valuable for IoT applications.


Original languageEnglish
Article number7727929
Pages (from-to)773-785
Number of pages13
JournalIEEE Transactions on Computers
Issue number5
Publication statusPublished - 1 May 2017
MoE publication typeA1 Journal article-refereed

    Research areas

  • elliptic curve cryptography, Internet-of-Things, multiple-precision arithmetic, signature verification, VLSI designs

ID: 12965302