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 language English 7727929 773-785 13 IEEE Transactions on Computers 66 5 https://doi.org/10.1109/TC.2016.2623609 Published - 1 May 2017 A1 Journal article-refereed

Keywords

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