Hardware Implementations of Elliptic Curve Cryptography Using Shift-Sub Based Modular Multiplication Algorithms-Reference-Cited by-同舟云学术

Hardware Implementations of Elliptic Curve Cryptography Using Shift-Sub Based Modular Multiplication Algorithms

Published:2023-11-10 Issue:4 Volume:7 Page:57
ISSN:2410-387X
Container-title:Cryptography
language:en
Short-container-title:Cryptography

Author:

Li Yamin¹^ORCID

Affiliation:

1. Computer Architecture Laboratory, Faculty of Computer and Information Sciences, Hosei University, Tokyo 184-8584, Japan

Abstract

Elliptic curve cryptography (ECC) over prime fields relies on scalar point multiplication realized by point addition and point doubling. Point addition and point doubling operations consist of many modular multiplications of large operands (256 bits for example), especially in projective and Jacobian coordinates which eliminate the modular inversion required in affine coordinates for every point addition or point doubling operation. Accelerating modular multiplication is therefore important for high-performance ECC. This paper presents the hardware implementations of modular multiplication algorithms, including (1) interleaved modular multiplication (IMM), (2) Montgomery modular multiplication (MMM), (3) shift-sub modular multiplication (SSMM), (4) SSMM with advance preparation (SSMMPRE), and (5) SSMM with CSAs and sign detection (SSMMCSA) algorithms, and evaluates their execution time (the number of clock cycles and clock frequency) and required hardware resources (ALMs and registers). Experimental results show that SSMM is 1.80 times faster than IMM, and SSMMCSA is 3.27 times faster than IMM. We also present the ECC hardware implementations based on the Secp256k1 protocol in affine, projective, and Jacobian coordinates using the IMM, SSMM, SSMMPRE, and SSMMCSA algorithms, and investigate their cost and performance. Our ECC implementations can be applied to the design of hardware security module systems.

Publisher

MDPI AG

Subject

Applied Mathematics,Computational Theory and Mathematics,Computer Networks and Communications,Computer Science Applications,Software

Link

https://www.mdpi.com/2410-387X/7/4/57/pdf

Reference19 articles.

1. Elliptic curve cryptosystems;Koblitz;Math. Comput.,1987

2. Miller, V.S. (1986). Proceedings of the Advances in Cryptology—CRYPTO’85 Proceedings, Springer. Available online: https://link.springer.com/content/pdf/10.1007/3-540-39799-X_31.pdf?pdf=inline%20link.

3. Hankerson, D., Menezes, A., and Vanstone, S. (2004). Guide to Elliptic Curve Cryptography, Springer.

4. Certicom_Corp (2010). Standards for Efficient Cryptography. SEC 2: Recommended Elliptic Curve Domain Parameters, Certicom Corp. Available online: http://www.secg.org/sec2-v2.pdf.

5. Barker, E., Chen, L., Roginsky, A., Vassilev, A., and Davis, R. (2023, October 15). SP 800-56A Rev. 3, Recommendation for Pair-Wise Key-Establishment Schemes Using Discrete Logarithm Cryptography, Available online: https://nvlpubs.nist.gov/nistpubs/SpecialPublications/NIST.SP.800-56Ar3.pdf.