d-MUL: Optimizing and Implementing a Multidimensional Scalar Multiplication Algorithm over Elliptic Curves

  • Huseyin Hisil
  • Aaron Hutchinson
  • Koray KarabinaEmail author
Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 11348)


This paper aims to answer whether d-MUL, the multidimensional scalar point multiplication algorithm, can be implemented efficiently. d-MUL is known to access costly matrix operations and requires memory access frequently. In the first part of the paper, we derive several theoretical results on the structure and the construction of the addition chains in d-MUL. These results are interesting on their own right. In the second part of the paper, we exploit our theoretical results, and propose an optimized variant of d-MUL. Our implementation results show that d-MUL can be very practical for small d, and it remains as an interesting algorithm to further explore for parallel implementation and cryptographic applications.


d-MUL Elliptic curve scalar multiplication Differential addition chain Isochronous implementation 



The authors would like to thank reviewers for their comments and corrections. Research reported in this paper was supported by the Army Research Office under the award number W911NF-17-1-0311. The content is solely the responsibility of the authors and does not necessarily represent the official views of the Army Research Office.


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Copyright information

© Springer Nature Switzerland AG 2018

Authors and Affiliations

  • Huseyin Hisil
    • 1
  • Aaron Hutchinson
    • 2
  • Koray Karabina
    • 2
    Email author
  1. 1.Yasar UniversityİzmirTurkey
  2. 2.Florida Atlantic UniversityBoca RatonUSA

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