Constructing Boolean Functions Using Blended Representations
In this paper, we study blended representations of Boolean functions, and construct the following two classes of Boolean functions. Two bounds on the <inline-formula> <tex-math notation="LaTeX">$r$ </tex-math></inline-formula>-order nonlinearity were given by Carlet...
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| Format: | Article |
| Language: | English |
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IEEE
2019-01-01
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| Series: | IEEE Access |
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| Online Access: | https://ieeexplore.ieee.org/document/8784181/ |
| _version_ | 1811274548950073344 |
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| author | Qichun Wang Caihong Nie Youle Xu |
| author_facet | Qichun Wang Caihong Nie Youle Xu |
| author_sort | Qichun Wang |
| collection | DOAJ |
| description | In this paper, we study blended representations of Boolean functions, and construct the following two classes of Boolean functions. Two bounds on the <inline-formula> <tex-math notation="LaTeX">$r$ </tex-math></inline-formula>-order nonlinearity were given by Carlet in the IEEE Transactions on Information Theory, vol. 54. In general, the second bound is better than the first bound. But it was unknown whether it is always better. Recently, Mesnager <italic>et al.</italic> constructed a class of Boolean functions where the second bound is strictly worse than the first bound, for <inline-formula> <tex-math notation="LaTeX">$r=2$ </tex-math></inline-formula>. However, it is still an open problem for <inline-formula> <tex-math notation="LaTeX">$r\geq 3$ </tex-math></inline-formula>. Using the blended representation, we construct a class of Boolean functions based on the trace function and show that the second bound can also be strictly worse than the first bound, for <inline-formula> <tex-math notation="LaTeX">$r=3$ </tex-math></inline-formula>. The second class is based on the hidden weighted bit function, which seems to have the best cryptographic properties among all currently known functions. |
| first_indexed | 2024-04-12T23:20:59Z |
| format | Article |
| id | doaj.art-df48742dd99b482bb9008d8e1ee3cf75 |
| institution | Directory Open Access Journal |
| issn | 2169-3536 |
| language | English |
| last_indexed | 2024-04-12T23:20:59Z |
| publishDate | 2019-01-01 |
| publisher | IEEE |
| record_format | Article |
| series | IEEE Access |
| spelling | doaj.art-df48742dd99b482bb9008d8e1ee3cf752022-12-22T03:12:32ZengIEEEIEEE Access2169-35362019-01-01710702510703110.1109/ACCESS.2019.29324238784181Constructing Boolean Functions Using Blended RepresentationsQichun Wang0https://orcid.org/0000-0003-3474-4115Caihong Nie1Youle Xu2College of Science, Hunan University of Science and Engineering, Yongzhou, ChinaSchool of Mathematical Sciences, Nanjing Normal University, Nanjing, ChinaSchool of Computer Science and Technology, Nanjing Normal University, Nanjing, ChinaIn this paper, we study blended representations of Boolean functions, and construct the following two classes of Boolean functions. Two bounds on the <inline-formula> <tex-math notation="LaTeX">$r$ </tex-math></inline-formula>-order nonlinearity were given by Carlet in the IEEE Transactions on Information Theory, vol. 54. In general, the second bound is better than the first bound. But it was unknown whether it is always better. Recently, Mesnager <italic>et al.</italic> constructed a class of Boolean functions where the second bound is strictly worse than the first bound, for <inline-formula> <tex-math notation="LaTeX">$r=2$ </tex-math></inline-formula>. However, it is still an open problem for <inline-formula> <tex-math notation="LaTeX">$r\geq 3$ </tex-math></inline-formula>. Using the blended representation, we construct a class of Boolean functions based on the trace function and show that the second bound can also be strictly worse than the first bound, for <inline-formula> <tex-math notation="LaTeX">$r=3$ </tex-math></inline-formula>. The second class is based on the hidden weighted bit function, which seems to have the best cryptographic properties among all currently known functions.https://ieeexplore.ieee.org/document/8784181/Boolean functionsblended representationsnonlinearityalgebraic immunityhigher-order nonlinearity |
| spellingShingle | Qichun Wang Caihong Nie Youle Xu Constructing Boolean Functions Using Blended Representations IEEE Access Boolean functions blended representations nonlinearity algebraic immunity higher-order nonlinearity |
| title | Constructing Boolean Functions Using Blended Representations |
| title_full | Constructing Boolean Functions Using Blended Representations |
| title_fullStr | Constructing Boolean Functions Using Blended Representations |
| title_full_unstemmed | Constructing Boolean Functions Using Blended Representations |
| title_short | Constructing Boolean Functions Using Blended Representations |
| title_sort | constructing boolean functions using blended representations |
| topic | Boolean functions blended representations nonlinearity algebraic immunity higher-order nonlinearity |
| url | https://ieeexplore.ieee.org/document/8784181/ |
| work_keys_str_mv | AT qichunwang constructingbooleanfunctionsusingblendedrepresentations AT caihongnie constructingbooleanfunctionsusingblendedrepresentations AT youlexu constructingbooleanfunctionsusingblendedrepresentations |