Paper 2020/144

Double-Base Chains for Scalar Multiplications on Elliptic Curves

Wei Yu, Saud Al Musa, and Bao Li


Double-base chains (DBCs) are widely used to speed up scalar multiplications on elliptic curves. We present three results of DBCs. First, we display a structure of the set containing all DBCs and propose an iterative algorithm to compute the number of DBCs for a positive integer. This is the first polynomial time algorithm to compute the number of DBCs for positive integers. Secondly, we present an asymptotic lower bound on average Hamming weights of DBCs $\frac{\log n}{8.25}$ for a positive integer $n$. This result answers an open question about the Hamming weights of DBCs. Thirdly, we propose a new algorithm to generate an optimal DBC for any positive integer. The time complexity of this algorithm is $\mathcal{O}\left(\left(\log n\right)^2 \log\log n\right)$ bit operations and the space complexity is $\mathcal{O}\left(\left(\log n\right)^{2}\right)$ bits of memory. This algorithm accelerates the recoding procedure by more than $6$ times compared to the state-of-the-art Bernstein, Chuengsatiansup, and Lange's work. The Hamming weights of optimal DBCs are over $60$\% smaller than those of NAFs. Scalar multiplication using our optimal DBC is about $13$\% faster than that using non-adjacent form on elliptic curves over large prime fields.

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Publication info
A minor revision of an IACR publication in EUROCRYPT 2020
Elliptic curve cryptographyScalar multiplicationDouble-base chainHamming weight
Contact author(s)
yuwei_1_yw @ 163 com
yuwei @ iie ac cn
2021-02-19: last of 2 revisions
2020-02-10: received
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      author = {Wei Yu and Saud Al Musa and Bao Li},
      title = {Double-Base Chains  for Scalar Multiplications on Elliptic Curves},
      howpublished = {Cryptology ePrint Archive, Paper 2020/144},
      year = {2020},
      note = {\url{}},
      url = {}
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