The differential spectrum and boomerang spectrum of a class of locally-APN functions
Abstract In this paper, we study the boomerang spectrum of the power mapping $$F(x)=x^{k(q-1)}$$ over $${\mathbb {F}}_{q^2}$$, where $$q=p^m$$, p is a prime, m is a positive integer and $$\gcd (k,q+1)=1$$. We first determine the differential spectrum of F(x) and show that F(x) is locally-APN. This e...
Ausführliche Beschreibung
Gespeichert in:
| Autor*in: |
Hu, Zhao [verfasserIn] |
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| Format: |
Artikel |
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| Sprache: |
Englisch |
| Erschienen: |
2023 |
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| Schlagwörter: |
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| Anmerkung: |
© The Author(s), under exclusive licence to Springer Science+Business Media, LLC, part of Springer Nature 2022. Springer Nature or its licensor (e.g. a society or other partner) holds exclusive rights to this article under a publishing agreement with the author(s) or other rightsholder(s); author self-archiving of the accepted manuscript version of this article is solely governed by the terms of such publishing agreement and applicable law. |
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| Übergeordnetes Werk: |
Enthalten in: Designs, codes and cryptography - Springer US, 1991, 91(2023), 5 vom: 06. Jan., Seite 1695-1711 |
|---|---|
| Übergeordnetes Werk: |
volume:91 ; year:2023 ; number:5 ; day:06 ; month:01 ; pages:1695-1711 |
| Links: |
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| DOI / URN: |
10.1007/s10623-022-01161-w |
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| 520 | |a Abstract In this paper, we study the boomerang spectrum of the power mapping $$F(x)=x^{k(q-1)}$$ over $${\mathbb {F}}_{q^2}$$, where $$q=p^m$$, p is a prime, m is a positive integer and $$\gcd (k,q+1)=1$$. We first determine the differential spectrum of F(x) and show that F(x) is locally-APN. This extends a result of (IEEE Trans. Inf. Theory 57(12):8127-8137, 2011) from $$(p,k)=(2,1)$$ to general (p, k). We then determine the boomerang spectrum of F(x) by making use of its differential spectrum, which shows that the boomerang uniformity of F(x) is 4 if $$p=2$$ and m is odd and otherwise it is 2. Our results not only generalize the results in Hasan et al. (Des Codes Cryptogr 89:2627–2636, 2021) and Yan et al. (Adv Math Commun 16(4):1111–1120, 2022) but also extend the example $$x^{45}$$ over $${\mathbb F}_{2^8}$$ in Hasan et al. (Des Codes Cryptogr 89:2627–2636, 2021) into an infinite class of power mappings with boomerang uniformity 2. | ||
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10.1007/s10623-022-01161-w doi (DE-627)OLC2134575204 (DE-He213)s10623-022-01161-w-p DE-627 ger DE-627 rakwb eng 004 VZ 17,1 ssgn Hu, Zhao verfasserin aut The differential spectrum and boomerang spectrum of a class of locally-APN functions 2023 Text txt rdacontent ohne Hilfsmittel zu benutzen n rdamedia Band nc rdacarrier © The Author(s), under exclusive licence to Springer Science+Business Media, LLC, part of Springer Nature 2022. Springer Nature or its licensor (e.g. a society or other partner) holds exclusive rights to this article under a publishing agreement with the author(s) or other rightsholder(s); author self-archiving of the accepted manuscript version of this article is solely governed by the terms of such publishing agreement and applicable law. Abstract In this paper, we study the boomerang spectrum of the power mapping $$F(x)=x^{k(q-1)}$$ over $${\mathbb {F}}_{q^2}$$, where $$q=p^m$$, p is a prime, m is a positive integer and $$\gcd (k,q+1)=1$$. We first determine the differential spectrum of F(x) and show that F(x) is locally-APN. This extends a result of (IEEE Trans. Inf. Theory 57(12):8127-8137, 2011) from $$(p,k)=(2,1)$$ to general (p, k). We then determine the boomerang spectrum of F(x) by making use of its differential spectrum, which shows that the boomerang uniformity of F(x) is 4 if $$p=2$$ and m is odd and otherwise it is 2. Our results not only generalize the results in Hasan et al. (Des Codes Cryptogr 89:2627–2636, 2021) and Yan et al. (Adv Math Commun 16(4):1111–1120, 2022) but also extend the example $$x^{45}$$ over $${\mathbb F}_{2^8}$$ in Hasan et al. (Des Codes Cryptogr 89:2627–2636, 2021) into an infinite class of power mappings with boomerang uniformity 2. Boomerang spectrum Differential spectrum Locally-APN function Li, Nian (orcid)0000-0003-4913-7844 aut Xu, Linjie aut Zeng, Xiangyong aut Tang, Xiaohu aut Enthalten in Designs, codes and cryptography Springer US, 1991 91(2023), 5 vom: 06. Jan., Seite 1695-1711 (DE-627)130994197 (DE-600)1082042-5 (DE-576)029154375 0925-1022 nnns volume:91 year:2023 number:5 day:06 month:01 pages:1695-1711 https://doi.org/10.1007/s10623-022-01161-w lizenzpflichtig Volltext GBV_USEFLAG_A SYSFLAG_A GBV_OLC SSG-OLC-MAT SSG-OPC-MAT AR 91 2023 5 06 01 1695-1711 |
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10.1007/s10623-022-01161-w doi (DE-627)OLC2134575204 (DE-He213)s10623-022-01161-w-p DE-627 ger DE-627 rakwb eng 004 VZ 17,1 ssgn Hu, Zhao verfasserin aut The differential spectrum and boomerang spectrum of a class of locally-APN functions 2023 Text txt rdacontent ohne Hilfsmittel zu benutzen n rdamedia Band nc rdacarrier © The Author(s), under exclusive licence to Springer Science+Business Media, LLC, part of Springer Nature 2022. Springer Nature or its licensor (e.g. a society or other partner) holds exclusive rights to this article under a publishing agreement with the author(s) or other rightsholder(s); author self-archiving of the accepted manuscript version of this article is solely governed by the terms of such publishing agreement and applicable law. Abstract In this paper, we study the boomerang spectrum of the power mapping $$F(x)=x^{k(q-1)}$$ over $${\mathbb {F}}_{q^2}$$, where $$q=p^m$$, p is a prime, m is a positive integer and $$\gcd (k,q+1)=1$$. We first determine the differential spectrum of F(x) and show that F(x) is locally-APN. This extends a result of (IEEE Trans. Inf. Theory 57(12):8127-8137, 2011) from $$(p,k)=(2,1)$$ to general (p, k). We then determine the boomerang spectrum of F(x) by making use of its differential spectrum, which shows that the boomerang uniformity of F(x) is 4 if $$p=2$$ and m is odd and otherwise it is 2. Our results not only generalize the results in Hasan et al. (Des Codes Cryptogr 89:2627–2636, 2021) and Yan et al. (Adv Math Commun 16(4):1111–1120, 2022) but also extend the example $$x^{45}$$ over $${\mathbb F}_{2^8}$$ in Hasan et al. (Des Codes Cryptogr 89:2627–2636, 2021) into an infinite class of power mappings with boomerang uniformity 2. Boomerang spectrum Differential spectrum Locally-APN function Li, Nian (orcid)0000-0003-4913-7844 aut Xu, Linjie aut Zeng, Xiangyong aut Tang, Xiaohu aut Enthalten in Designs, codes and cryptography Springer US, 1991 91(2023), 5 vom: 06. Jan., Seite 1695-1711 (DE-627)130994197 (DE-600)1082042-5 (DE-576)029154375 0925-1022 nnns volume:91 year:2023 number:5 day:06 month:01 pages:1695-1711 https://doi.org/10.1007/s10623-022-01161-w lizenzpflichtig Volltext GBV_USEFLAG_A SYSFLAG_A GBV_OLC SSG-OLC-MAT SSG-OPC-MAT AR 91 2023 5 06 01 1695-1711 |
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10.1007/s10623-022-01161-w doi (DE-627)OLC2134575204 (DE-He213)s10623-022-01161-w-p DE-627 ger DE-627 rakwb eng 004 VZ 17,1 ssgn Hu, Zhao verfasserin aut The differential spectrum and boomerang spectrum of a class of locally-APN functions 2023 Text txt rdacontent ohne Hilfsmittel zu benutzen n rdamedia Band nc rdacarrier © The Author(s), under exclusive licence to Springer Science+Business Media, LLC, part of Springer Nature 2022. Springer Nature or its licensor (e.g. a society or other partner) holds exclusive rights to this article under a publishing agreement with the author(s) or other rightsholder(s); author self-archiving of the accepted manuscript version of this article is solely governed by the terms of such publishing agreement and applicable law. Abstract In this paper, we study the boomerang spectrum of the power mapping $$F(x)=x^{k(q-1)}$$ over $${\mathbb {F}}_{q^2}$$, where $$q=p^m$$, p is a prime, m is a positive integer and $$\gcd (k,q+1)=1$$. We first determine the differential spectrum of F(x) and show that F(x) is locally-APN. This extends a result of (IEEE Trans. Inf. Theory 57(12):8127-8137, 2011) from $$(p,k)=(2,1)$$ to general (p, k). We then determine the boomerang spectrum of F(x) by making use of its differential spectrum, which shows that the boomerang uniformity of F(x) is 4 if $$p=2$$ and m is odd and otherwise it is 2. Our results not only generalize the results in Hasan et al. (Des Codes Cryptogr 89:2627–2636, 2021) and Yan et al. (Adv Math Commun 16(4):1111–1120, 2022) but also extend the example $$x^{45}$$ over $${\mathbb F}_{2^8}$$ in Hasan et al. (Des Codes Cryptogr 89:2627–2636, 2021) into an infinite class of power mappings with boomerang uniformity 2. Boomerang spectrum Differential spectrum Locally-APN function Li, Nian (orcid)0000-0003-4913-7844 aut Xu, Linjie aut Zeng, Xiangyong aut Tang, Xiaohu aut Enthalten in Designs, codes and cryptography Springer US, 1991 91(2023), 5 vom: 06. Jan., Seite 1695-1711 (DE-627)130994197 (DE-600)1082042-5 (DE-576)029154375 0925-1022 nnns volume:91 year:2023 number:5 day:06 month:01 pages:1695-1711 https://doi.org/10.1007/s10623-022-01161-w lizenzpflichtig Volltext GBV_USEFLAG_A SYSFLAG_A GBV_OLC SSG-OLC-MAT SSG-OPC-MAT AR 91 2023 5 06 01 1695-1711 |
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10.1007/s10623-022-01161-w doi (DE-627)OLC2134575204 (DE-He213)s10623-022-01161-w-p DE-627 ger DE-627 rakwb eng 004 VZ 17,1 ssgn Hu, Zhao verfasserin aut The differential spectrum and boomerang spectrum of a class of locally-APN functions 2023 Text txt rdacontent ohne Hilfsmittel zu benutzen n rdamedia Band nc rdacarrier © The Author(s), under exclusive licence to Springer Science+Business Media, LLC, part of Springer Nature 2022. Springer Nature or its licensor (e.g. a society or other partner) holds exclusive rights to this article under a publishing agreement with the author(s) or other rightsholder(s); author self-archiving of the accepted manuscript version of this article is solely governed by the terms of such publishing agreement and applicable law. Abstract In this paper, we study the boomerang spectrum of the power mapping $$F(x)=x^{k(q-1)}$$ over $${\mathbb {F}}_{q^2}$$, where $$q=p^m$$, p is a prime, m is a positive integer and $$\gcd (k,q+1)=1$$. We first determine the differential spectrum of F(x) and show that F(x) is locally-APN. This extends a result of (IEEE Trans. Inf. Theory 57(12):8127-8137, 2011) from $$(p,k)=(2,1)$$ to general (p, k). We then determine the boomerang spectrum of F(x) by making use of its differential spectrum, which shows that the boomerang uniformity of F(x) is 4 if $$p=2$$ and m is odd and otherwise it is 2. Our results not only generalize the results in Hasan et al. (Des Codes Cryptogr 89:2627–2636, 2021) and Yan et al. (Adv Math Commun 16(4):1111–1120, 2022) but also extend the example $$x^{45}$$ over $${\mathbb F}_{2^8}$$ in Hasan et al. (Des Codes Cryptogr 89:2627–2636, 2021) into an infinite class of power mappings with boomerang uniformity 2. Boomerang spectrum Differential spectrum Locally-APN function Li, Nian (orcid)0000-0003-4913-7844 aut Xu, Linjie aut Zeng, Xiangyong aut Tang, Xiaohu aut Enthalten in Designs, codes and cryptography Springer US, 1991 91(2023), 5 vom: 06. Jan., Seite 1695-1711 (DE-627)130994197 (DE-600)1082042-5 (DE-576)029154375 0925-1022 nnns volume:91 year:2023 number:5 day:06 month:01 pages:1695-1711 https://doi.org/10.1007/s10623-022-01161-w lizenzpflichtig Volltext GBV_USEFLAG_A SYSFLAG_A GBV_OLC SSG-OLC-MAT SSG-OPC-MAT AR 91 2023 5 06 01 1695-1711 |
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the differential spectrum and boomerang spectrum of a class of locally-apn functions |
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The differential spectrum and boomerang spectrum of a class of locally-APN functions |
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Abstract In this paper, we study the boomerang spectrum of the power mapping $$F(x)=x^{k(q-1)}$$ over $${\mathbb {F}}_{q^2}$$, where $$q=p^m$$, p is a prime, m is a positive integer and $$\gcd (k,q+1)=1$$. We first determine the differential spectrum of F(x) and show that F(x) is locally-APN. This extends a result of (IEEE Trans. Inf. Theory 57(12):8127-8137, 2011) from $$(p,k)=(2,1)$$ to general (p, k). We then determine the boomerang spectrum of F(x) by making use of its differential spectrum, which shows that the boomerang uniformity of F(x) is 4 if $$p=2$$ and m is odd and otherwise it is 2. Our results not only generalize the results in Hasan et al. (Des Codes Cryptogr 89:2627–2636, 2021) and Yan et al. (Adv Math Commun 16(4):1111–1120, 2022) but also extend the example $$x^{45}$$ over $${\mathbb F}_{2^8}$$ in Hasan et al. (Des Codes Cryptogr 89:2627–2636, 2021) into an infinite class of power mappings with boomerang uniformity 2. © The Author(s), under exclusive licence to Springer Science+Business Media, LLC, part of Springer Nature 2022. Springer Nature or its licensor (e.g. a society or other partner) holds exclusive rights to this article under a publishing agreement with the author(s) or other rightsholder(s); author self-archiving of the accepted manuscript version of this article is solely governed by the terms of such publishing agreement and applicable law. |
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Abstract In this paper, we study the boomerang spectrum of the power mapping $$F(x)=x^{k(q-1)}$$ over $${\mathbb {F}}_{q^2}$$, where $$q=p^m$$, p is a prime, m is a positive integer and $$\gcd (k,q+1)=1$$. We first determine the differential spectrum of F(x) and show that F(x) is locally-APN. This extends a result of (IEEE Trans. Inf. Theory 57(12):8127-8137, 2011) from $$(p,k)=(2,1)$$ to general (p, k). We then determine the boomerang spectrum of F(x) by making use of its differential spectrum, which shows that the boomerang uniformity of F(x) is 4 if $$p=2$$ and m is odd and otherwise it is 2. Our results not only generalize the results in Hasan et al. (Des Codes Cryptogr 89:2627–2636, 2021) and Yan et al. (Adv Math Commun 16(4):1111–1120, 2022) but also extend the example $$x^{45}$$ over $${\mathbb F}_{2^8}$$ in Hasan et al. (Des Codes Cryptogr 89:2627–2636, 2021) into an infinite class of power mappings with boomerang uniformity 2. © The Author(s), under exclusive licence to Springer Science+Business Media, LLC, part of Springer Nature 2022. Springer Nature or its licensor (e.g. a society or other partner) holds exclusive rights to this article under a publishing agreement with the author(s) or other rightsholder(s); author self-archiving of the accepted manuscript version of this article is solely governed by the terms of such publishing agreement and applicable law. |
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Abstract In this paper, we study the boomerang spectrum of the power mapping $$F(x)=x^{k(q-1)}$$ over $${\mathbb {F}}_{q^2}$$, where $$q=p^m$$, p is a prime, m is a positive integer and $$\gcd (k,q+1)=1$$. We first determine the differential spectrum of F(x) and show that F(x) is locally-APN. This extends a result of (IEEE Trans. Inf. Theory 57(12):8127-8137, 2011) from $$(p,k)=(2,1)$$ to general (p, k). We then determine the boomerang spectrum of F(x) by making use of its differential spectrum, which shows that the boomerang uniformity of F(x) is 4 if $$p=2$$ and m is odd and otherwise it is 2. Our results not only generalize the results in Hasan et al. (Des Codes Cryptogr 89:2627–2636, 2021) and Yan et al. (Adv Math Commun 16(4):1111–1120, 2022) but also extend the example $$x^{45}$$ over $${\mathbb F}_{2^8}$$ in Hasan et al. (Des Codes Cryptogr 89:2627–2636, 2021) into an infinite class of power mappings with boomerang uniformity 2. © The Author(s), under exclusive licence to Springer Science+Business Media, LLC, part of Springer Nature 2022. Springer Nature or its licensor (e.g. a society or other partner) holds exclusive rights to this article under a publishing agreement with the author(s) or other rightsholder(s); author self-archiving of the accepted manuscript version of this article is solely governed by the terms of such publishing agreement and applicable law. |
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