On the distribution of addition chains
Fuente:
arXiv
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| Formato: | Preprint |
| Publicado: |
2020
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| _version_ | 1866918461384425472 |
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| author | Agama, Theophilus |
| author_facet | Agama, Theophilus |
| contents | Addition chains are a classical construction for fast exponentiation and related computation problems. In this paper, we study a chain for a fixed integer $n$ by decomposing each generator into a \emph{determiner} and a \emph{regulator} (gap). This viewpoint leads to explicit identities for two aggregate statistics of the chain: the sum of the determiners and the sum of the chain elements. We then derive the corresponding lower bounds by using the positivity of the regulators. In parallel, we establish an identity for the reciprocal sum of the chain, showing how the harmonic profile of the chain can also be written in terms of the same gap sequence. These identities provide a unified way to compare addition chains of the same target and length. The paper concludes with a balancing problem that asks for the chain(s) that minimize the difference between the arithmetic sum and the harmonic sum, together with a structural decomposition of that optimization objective. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2004_05221 |
| institution | arXiv |
| publishDate | 2020 |
| record_format | arxiv |
| spellingShingle | On the distribution of addition chains Agama, Theophilus Number Theory 11Y16 Addition chains are a classical construction for fast exponentiation and related computation problems. In this paper, we study a chain for a fixed integer $n$ by decomposing each generator into a \emph{determiner} and a \emph{regulator} (gap). This viewpoint leads to explicit identities for two aggregate statistics of the chain: the sum of the determiners and the sum of the chain elements. We then derive the corresponding lower bounds by using the positivity of the regulators. In parallel, we establish an identity for the reciprocal sum of the chain, showing how the harmonic profile of the chain can also be written in terms of the same gap sequence. These identities provide a unified way to compare addition chains of the same target and length. The paper concludes with a balancing problem that asks for the chain(s) that minimize the difference between the arithmetic sum and the harmonic sum, together with a structural decomposition of that optimization objective. |
| title | On the distribution of addition chains |
| topic | Number Theory 11Y16 |
| url | https://arxiv.org/abs/2004.05221 |