[Resource Topic] 2025/049: On the gap between terms in an addition chain

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Title:
On the gap between terms in an addition chain

Authors: Theophilus Agama

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.

ePrint: https://eprint.iacr.org/2025/049

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