A Note on Vectorial Boolean Functions as Embeddings
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arXiv
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| Auteurs principaux: | , |
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| Format: | Preprint |
| Publié: |
2024
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| _version_ | 1866913862622642176 |
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| author | Musukwa, Augustine Sala, Massimiliano |
| author_facet | Musukwa, Augustine Sala, Massimiliano |
| contents | Let $F$ be a vectorial Boolean function from $\mathbb{F}_2^n$ to $\mathbb{F}_2^m$, with $m \geq n$. We define $F$ as an embedding if $F$ is injective. In this paper, we examine the component functions of $F$, focusing on constant and balanced components. Our findings reveal that at most $2^m - 2^{m-n}$ components of $F$ can be balanced, and this maximum is achieved precisely when $F$ is an embedding, with the remaining $2^{m-n}$ components being constants. Additionally, for partially-bent embeddings, we demonstrate that there are always at least $2^n - 1$ balanced components when $n$ is even, and $2^{m-1} + 2^{n-1} - 1$ balanced components when $n$ is odd. A relation with APN functions is shown. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2406_06429 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | A Note on Vectorial Boolean Functions as Embeddings Musukwa, Augustine Sala, Massimiliano Cryptography and Security 06E30, 94A60, 14G50 Let $F$ be a vectorial Boolean function from $\mathbb{F}_2^n$ to $\mathbb{F}_2^m$, with $m \geq n$. We define $F$ as an embedding if $F$ is injective. In this paper, we examine the component functions of $F$, focusing on constant and balanced components. Our findings reveal that at most $2^m - 2^{m-n}$ components of $F$ can be balanced, and this maximum is achieved precisely when $F$ is an embedding, with the remaining $2^{m-n}$ components being constants. Additionally, for partially-bent embeddings, we demonstrate that there are always at least $2^n - 1$ balanced components when $n$ is even, and $2^{m-1} + 2^{n-1} - 1$ balanced components when $n$ is odd. A relation with APN functions is shown. |
| title | A Note on Vectorial Boolean Functions as Embeddings |
| topic | Cryptography and Security 06E30, 94A60, 14G50 |
| url | https://arxiv.org/abs/2406.06429 |