The second-order zero differential spectra of some functions over finite fields

[2309.04219] The second-order zero differential spectra of some functions over finite fields Skip to main content Search Submit Donate Log in Computer Science > Information Theory arXiv:2309.04219 (cs) [Submitted on 8 Sep 2023 ( v1 ), last revised 27 Jan 2025 (this version, v2)] Title: The second-order zero differential spectra of some functions over finite fields Authors: Kirpa Garg , Sartaj Ul Hasan , Constanza Riera , Pantelimon Stanica View PDF HTML (experimental) Abstract: It was shown by Boukerrou et al.~[IACR Trans. Symmetric Cryptol. 1 (2020), 331--362] that the $F$-boomerang uniformity (which is the same as the second-order zero differential uniformity in even characteristic) of perfect nonlinear functions is~$0$ on $\F_{p^n}$ ($p$ prime) and the one of almost perfect nonlinear functions on $\F_{2^n}$ is~$0$. It is natural to inquire what happens with APN or other low differential uniform functions in even and odd characteristics. Here, we explicitly determine the second-order zero differential spectra of several maps with low differential uniformity. In particular, we compute the second-order zero differential spectra for some almost perfect nonlinear (APN) functions over finite fields of odd characteristic, pushing further the study started in Boukerrou et al. and continued in Li et al.~[Cryptogr. Commun. 14.3 (2022), 653--662], and it turns out that our considered functions also have low second-order zero differential uniformity. Moreover, we study the second-order zero differential spectra of certain functions with low differential uniformity over finite fields of even characteristic. We connect this new concept to the sum-freedom and vanishing flats concepts and find some counts for the number of vanishing flats via our methods. We provide detailed analyses on several equations over finite fields that may have an interest outside of the scope of our paper. Comments: 30 pages, to appear in Cryptography and communication Subjects: Information Theory (cs.IT) ; Number Theory (math.NT) MSC classes: 12E20, 11T06, 94A60 Cite as: arXiv:2309.04219 [cs.IT] (or arXiv:2309.04219v2 [cs.IT] for this version) https://doi.org/10.48550/arXiv.2309.04219 Focus to learn more arXiv-issued DOI via DataCite Journal reference: Cryptography and Communication, Vol. 17, pp. 665-694, 2025 Related DOI : https://doi.org/10.1007/s12095-025-00776-9 Focus to learn more DOI(s) linking to related resources Submission history From: Sartaj Ul Hasan [ view email ] [v1] Fri, 8 Sep 2023 09:08:34 UTC (19 KB) [v2] Mon, 27 Jan 2025 06:32:52 UTC (25 KB) Full-text links: Access Paper: View PDF HTML (experimental) TeX Source view license Additional Features Audio Summary Current browse context: cs.IT < prev | next > new | recent | 2023-09 Change to browse by: cs math math.IT math.NT References & Citations SciX Google Scholar Semantic Scholar export BibTeX citation Bookmark Bibliographic Tools Bibliographic and Citation Tools Bibliographic Explorer Toggle Bibliographic Explorer ( What is the Explorer? ) Connected Papers Toggle Connected Papers ( What is Connected Papers? ) Litmaps Toggle Litmaps ( What is Litmaps? ) scite.ai Toggle scite Smart Citations ( What are Smart Citations? ) Code, Data, Media Code, Data and Media Associated with this Article alphaXiv Toggle alphaXiv ( What is alphaXiv? ) Links to Code Toggle CatalyzeX Code Finder for Papers ( What is CatalyzeX? ) DagsHub Toggle DagsHub ( What is DagsHub? ) GotitPub Toggle Gotit.pub ( What is GotitPub? ) Huggingface Toggle Hugging Face ( What is Huggingface? ) ScienceCast Toggle ScienceCast ( What is ScienceCast? ) Demos Demos Replicate Toggle Replicate ( What is Replicate? ) Spaces Toggle Hugging Face Spaces ( What is Spaces? ) Spaces Toggle TXYZ.AI ( What is TXYZ.AI? ) Related Papers Recommenders and Search Tools Link to Influence Flower Influence Flower ( What are Influence Flowers? ) Core recommender toggle CORE Recommender ( What is CORE? ) Author Venue Institution Topic About arXivLabs arXivLabs: experimental projects with community collaborators arXivLabs is a framework that allows collaborators to develop and share new arXiv features directly on our website. Both individuals and organizations that work with arXivLabs have embraced and accepted our values of openness, community, excellence, and user data privacy. arXiv is committed to these values and only works with partners that adhere to them. Have an idea for a project that will add value for arXiv's community? Learn more about arXivLabs . Which authors of this paper are endorsers? | Disable MathJax ( What is MathJax? ) We gratefully acknowledge support from our major funders , member institutions , and all contributors. About · Help · Contact · Subscribe · Copyright · Privacy · Accessibility · Operational Status (opens in new tab)

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