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See detailElliptic Curve Cryptography with Efficiently Computable Endomorphisms and Its Hardware Implementations for the Internet of Things
Liu, Zhe UL; Groszschädl, Johann UL; Hu, Zhi et al

in IEEE Transactions on Computers (2017), 66(5), 773-785

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 ... [more ▼]

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 Fp with p = 2^207 - 5131. 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 μm 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. [less ▲]

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See detailEfficient Implementation of ECDH Key Exchange for MSP430-Based Wireless Sensor Networks
Liu, Zhe UL; Seo, Hwajeong; Hu, Zhi et al

in Bao, Feng; Miller, Steven; Zhou, Jianying (Eds.) et al ASIACCS'15: Proceedings of the 10th ACM Symposium on Information, Computer and Communications Security, April 14-17, 2015, Singapore (2015, April)

Public-Key Cryptography (PKC) is an indispensable building block of modern security protocols, and, thus, essential for secure communication over insecure networks. Despite a significant body of research ... [more ▼]

Public-Key Cryptography (PKC) is an indispensable building block of modern security protocols, and, thus, essential for secure communication over insecure networks. Despite a significant body of research devoted to making PKC more "lightweight," it is still commonly perceived that software implementations of PKC are computationally too expensive for practical use in ultra-low power devices such as wireless sensor nodes. In the present paper we aim to challenge this perception and present a highly-optimized implementation of Elliptic Curve Cryptography (ECC) for the TI MSP430 series of 16-bit microcontrollers. Our software is inspired by MoTE-ECC and supports scalar multiplication on two families of elliptic curves, namely Montgomery and twisted Edwards curves. However, in contrast to MoTE-ECC, we use pseudo-Mersenne prime fields as underlying algebraic structure to facilitate inter-operability with existing ECC implementations. We introduce a novel "zig-zag" technique for multiple-precision squaring on the MSP430 and assess its execution time. Similar to MoTE-ECC, we employ the Montgomery model for variable-base scalar multiplications and the twisted Edwards model if the base point is fixed (e.g. to generate an ephemeral key pair). Our experiments show that the two scalar multiplications needed to perform an ephemeral ECDH key exchange can be accomplished in 4.88 million clock cycles altogether (using a 159-bit prime field), which sets a new speed record for ephemeral ECDH on a 16-bit processor. We also describe the curve generation process and analyze the execution time of various field and point arithmetic operations on curves over a 159-bit and a 191-bit pseudo-Mersenne prime field. [less ▲]

Detailed reference viewed: 155 (18 UL)