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

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

Research output: Contribution to journalArticleScientificpeer-review

44 Citations (Scopus)

Abstract

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
Volume66
Issue number5
DOIs
Publication statusPublished - 1 May 2017
MoE publication typeA1 Journal article-refereed

Keywords

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

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