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Parallel Implementation of SM2 Elliptic Curve Cryptography on Intel Processors with AVX2 ; ; et al in Liu, Joseph K.; Cui, Hui (Eds.) Information Security and Privacy, 25th Australasian Conference, ACISP 2020, Perth, WA, Australia, November 30 - December 2, 2020, Proceedings (2020, November) This paper presents an efficient and secure implementation of SM2, the Chinese elliptic curve cryptography standard that has been adopted by the International Organization of Standardization (ISO) as ISO ... [more ▼] This paper presents an efficient and secure implementation of SM2, the Chinese elliptic curve cryptography standard that has been adopted by the International Organization of Standardization (ISO) as ISO/IEC 14888-3:2018. Our SM2 implementation uses Intel’s Advanced Vector Extensions version 2.0 (AVX2), a family of three-operand SIMD instructions operating on vectors of 8, 16, 32, or 64-bit data elements in 256-bit registers, and is resistant against timing attacks. To exploit the parallel processing capabilities of AVX2, we studied the execution flows of Co-Z Jacobian point arithmetic operations and introduce a parallel 2-way Co-Z addition, Co-Z conjugate addition, and Co-Z ladder algorithm, which allow for fast Co-Z scalar multiplication. Furthermore, we developed an efficient 2-way prime-field arithmetic library using AVX2 to support our Co-Z Jacobian point operations. Both the field and the point operations utilize branch-free (i.e. constant-time) implementation techniques, which increase their ability to resist Simple Power Analysis (SPA) and timing attacks. Our software for scalar multiplication on the SM2 curve is, to our knowledge, the first constant-time implementation of the Co-Z based ladder that leverages the parallelism of AVX2. [less ▲] Detailed reference viewed: 107 (4 UL)Fast ECDH Key Exchange Using Twisted Edwards Curves with an Efficiently Computable Endomorphism Groszschädl, Johann ; Liu, Zhe ; et al in Proceedings of the 8th International Workshop on Secure Internet of Things 2019 (SIoT 2019) (2019, September) It is widely accepted that public-key cryptosystems play a major role in the security arena of the Internet of Things (IoT), but they need to be implemented efficiently to not deplete the scarce resources ... [more ▼] It is widely accepted that public-key cryptosystems play a major role in the security arena of the Internet of Things (IoT), but they need to be implemented efficiently to not deplete the scarce resources of battery-operated devices such as wireless sensor nodes. This paper describes a highly-optimized software implementation of scalar multiplication for Elliptic Curve Diffie-Hellman (ECDH) key exchange on resource-limited IoT devices that achieves fast execution times along with reasonably small code size and RAM consumption. Our software uses a special class of elliptic curves, namely twisted Edwards curves with an efficiently computable endomorphism similar to that of the so- called Gallant-Lambert-Vanstone (GLV) curves. This allows us to combine the main advantage of the GLV model, which is an efficiently-computable endomorphism to speed up variable-base scalar multiplication, with the fast and complete addition rules of the (twisted) Edwards model. We implemented variable-base scalar multiplication for static ECDH on two such curves, one over a 159-bit and the second over a 207-bit pseudo-Mersenne prime field, respectively, and evaluated their execution time on a 16-bit MSP430F1611 processor. The arithmetic operations in the prime field do not contain operand-dependent conditional statements (in particular no "if-then-else" clauses) and also the scalar multiplication follows a fixed execution path for a given (static) scalar. A variable-base scalar multiplication on curves over the 159 and 207-bit field takes about 2.63 and 4.84 million clock cycles, respectively, on an MSP430F1611 processor. These results compare favorably with the Montgomery ladder on the equivalent Montgomery curves, which is almost 50% slower. [less ▲] Detailed reference viewed: 40 (5 UL)Elliptic Curve Cryptography with Efficiently Computable Endomorphisms and Its Hardware Implementations for the Internet of Things Liu, Zhe ; Groszschädl, Johann ; 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 ▲] Detailed reference viewed: 130 (3 UL)Efficient Implementation of ECDH Key Exchange for MSP430-Based Wireless Sensor Networks Liu, Zhe ; ; 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: 269 (19 UL) |
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