Abstract
An addition chain for a natural number n is a sequence \({1=a_0 < a_1 < \cdots < a_r=n}\) of numbers such that for each 0 < i ≤ r, a i = a j + a k for some 0 ≤ k ≤ j < i. The minimal length of an addition chain for n is denoted by ℓ(n). If j = i − 1, then step i is called a star step. We show that there is a minimal length addition chain for n such that the last four steps are stars. Then we conjecture that there is a minimal length addition chain for n such that the last \({\lfloor\frac{\ell(n)}{2}\rfloor}\)-steps are stars. We verify that the conjecture is true for all numbers up to 218. An application of the result and the conjecture to generate a minimal length addition chain reduce the average CPU time by 23–29% and 38–58% respectively, and memory storage by 16–18% and 26–45% respectively for m-bit numbers with 14 ≤ m ≤ 22.
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Communicated by C.H. Cap.
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Bahig, H.M. Star reduction among minimal length addition chains. Computing 91, 335–352 (2011). https://doi.org/10.1007/s00607-010-0122-z
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DOI: https://doi.org/10.1007/s00607-010-0122-z