On the gap between terms in an addition chain

Theophilus Agama

Paper 2025/049

On the gap between terms in an addition chain

Theophilus Agama

, African Institute for Mathematical Sciences

Abstract

In this paper, we study the distribution of the \textit{gap} between terms in an addition chain. In particular, we show that if $1,2,\ldots,s_{\delta(n)}=n$ is an addition chain of length $\delta(n)$ leading to $n$, then $$\underset{1\leq l\leq \delta(n)}{\mathrm{sup}}(s_{l+k}-s_l)\gg k\frac{n}{\delta(n)}$$ and $$\underset{1\leq l\leq \delta(n)}{\mathrm{inf}}(s_{l+k}-s_l)\ll k\frac{n}{\delta(n)}$$ for fixed $k\geq 1$.

Note: Current version of the paper has addressed the concerns of the editor. Potential applications to cryptography has been highlighted and briefly explained.

Metadata

Available format(s): PDF
Category: Applications
Publication info: Preprint.
Keywords: gap
Contact author(s): theophilus @ aims edu gh
History: 2025-01-14: approved; 2025-01-13: received; See all versions
Short URL: https://ia.cr/2025/049
License: CC BY

BibTeX

@misc{cryptoeprint:2025/049,
      author = {Theophilus Agama},
      title = {On the gap between terms in an addition chain},
      howpublished = {Cryptology {ePrint} Archive, Paper 2025/049},
      year = {2025},
      url = {https://eprint.iacr.org/2025/049}
}