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Some New Methods to Generate Short Addition Chains

Authors:
Yuanchao Ding , School of Cyber Science and Technology, Beihang University, Beijing, China; State Key Laboratory of Cryptology, P.O. Box, Beijing, China
Hua Guo , School of Cyber Science and Technology, Beihang University, Beijing, China; State Key Laboratory of Cryptology, P.O. Box, Beijing, China
Yewei Guan , School of Cyber Science and Technology, Beihang University, Beijing, China; State Key Laboratory of Software Development Environment, Beihang University, Beijing, China
Hutao Song , School of Cyber Science and Technology, Beihang University, Beijing, China; State Key Laboratory of Software Development Environment, Beihang University, Beijing, China
Xiyong Zhang , Beijing Institute of Satellite Information Engineering, Beijing, China
Jianwei Liu , School of Cyber Science and Technology, Beihang University, Beijing, China
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DOI: 10.46586/tches.v2023.i2.270-285
URL: https://tches.iacr.org/index.php/TCHES/article/view/10284
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Abstract: Modular exponentiation and scalar multiplication are important operations in most public-key cryptosystems, and their efficient computation is essential to cryptosystems. The shortest addition chain is one of the most important mathematical concepts to realize the optimization of computation. However, finding a shortest addition chain of length r is generally regarded as an NP-hard problem, whose time complexity is comparable to O(r!). This paper proposes some novel methods to generate short addition chains. We firstly present a Simplified Power-tree method by deeply deleting the power-tree whose time complexity is reduced to O(r2). In this paper, a Cross Window method and its variant are introduced by improving the Window method. The Cross Window method uses the cross correlation to deal with the windows and its pre-computation is optimized by a new Addition Sequence Algorithm. The theoretical analysis is conducted to show the correctness and effectiveness. Meanwhile, our experiments show that the new methods can obtain shorter addition chains compared to the existing methods. The Cross Window method with the Addition Sequence algorithm can attain 44.74% and 9.51% reduction of the addition chain length, in the best case, compared to the Binary method and the Window method respectively.
BibTeX
@article{tches-2023-33038,
  title={Some New Methods to Generate Short Addition Chains},
  journal={IACR Transactions on Cryptographic Hardware and Embedded Systems},
  publisher={Ruhr-Universit├Ąt Bochum},
  volume={2023, Issue 2},
  pages={270-285},
  url={https://tches.iacr.org/index.php/TCHES/article/view/10284},
  doi={10.46586/tches.v2023.i2.270-285},
  author={Yuanchao Ding and Hua Guo and Yewei Guan and Hutao Song and Xiyong Zhang and Jianwei Liu},
  year=2023
}