dc.contributor.author Kaleyski, Nikolay Stoyanov dc.date.accessioned 2021-08-18T09:33:35Z dc.date.available 2021-08-18T09:33:35Z dc.date.issued 2021-08-24 dc.date.submitted 2021-08-17T01:57:07Z dc.identifier container/62/2c/2f/9c/622c2f9c-51fd-4ffe-9ffa-995d8eca4a55 dc.identifier.isbn 9788230848784 dc.identifier.isbn 9788230849026 dc.identifier.uri https://hdl.handle.net/11250/2770059 dc.description.abstract This dissertation is dedicated to the properties, construction and analysis of APN and AB functions. Being cryptographically optimal, these functions lack any general structure or patterns, which makes their study very challenging. Despite intense work since at least the early 90's, many important questions and conjectures in the area remain open. We present several new results, many of which are directly related to important longstanding open problems; we resolve some of these problems, and make significant progress towards the resolution of others. en_US More concretely, our research concerns the following open problems: i) the maximum algebraic degree of an APN function, and the Hamming distance between APN functions (open since 1998); ii) the classification of APN and AB functions up to CCZ-equivalence (an ongoing problem since the introduction of APN functions, and one of the main directions of research in the area); iii) the extension of the APN binomial $x^3 + \beta x^{36}$ over $F_{2^{10}}$ into an infinite family (open since 2006); iv) the Walsh spectrum of the Dobbertin function (open since 2001); v) the existence of monomial APN functions CCZ-inequivalent to ones from the known families (open since 2001); vi) the problem of efficiently and reliably testing EA- and CCZ-equivalence (ongoing, and open since the introduction of APN functions). In the course of investigating these problems, we obtain i.a. the following results: 1) a new infinite family of APN quadrinomials (which includes the binomial $x^3 + \beta x^{36}$ over $F_{2^{10}}$); 2) two new invariants, one under EA-equivalence, and one under CCZ-equivalence; 3) an efficient and easily parallelizable algorithm for computationally testing EA-equivalence; 4) an efficiently computable lower bound on the Hamming distance between a given APN function and any other APN function; 5) a classification of all quadratic APN polynomials with binary coefficients over $F_{2^n}$ for $n \le 9$; 6) a construction allowing the CCZ-equivalence class of one monomial APN function to be obtained from that of another; 7) a conjecture giving the exact form of the Walsh spectrum of the Dobbertin power functions; 8) a generalization of an infinite family of APN functions to a family of functions with a two-valued differential spectrum, and an example showing that this Gold-like behavior does not occur for infinite families of quadratic APN functions in general; 9) a new class of functions (the so-called partially APN functions) defined by relaxing the definition of the APN property, and several constructions and non-existence results related to them. dc.language.iso eng en_US dc.publisher The University of Bergen en_US dc.relation.haspart Paper 1: N.S. Kaleyski. Changing APN functions at two points. Cryptography and Communications 11, 1165–1184 (2019). The article is available in the thesis file. The article is also available at: https://doi.org/10.1007/s12095-019-00366-6 en_US dc.relation.haspart Paper 2: L. Budaghyan, C. Carlet, T. Helleseth, N. S. Kaleyski. On the Distance Between APN Functions. IEEE Transactions on Information Theory, vol. 66, no. 9, pp. 5742-5753, Sept. 2020. The article is available at: https://hdl.handle.net/11250/2756701 en_US dc.relation.haspart Paper 3: Y. Yu, N. S. Kaleyski, L. Budaghyan, Y. Li. Classification of quadratic APN functions with coefficients in F2 for dimensions up to 9. Finite Fields and Their Applications, Volume 68, 2020, 101733. The article is available at: https://hdl.handle.net/11250/2756786 en_US dc.relation.haspart Paper 4: L. Budaghyan, T. Helleseth, N. Kaleyski. A New Family of APN Quadrinomials. IEEE Transactions on Information Theory, vol. 66, no. 11, pp. 7081-7087, Nov. 2020. The article is available at: https://hdl.handle.net/11250/2756784 en_US dc.relation.haspart Paper 5: N. S. Kaleyski. Deciding EA-equivalence via invariants. Cryptography and Communications, 2021. The submitted version is available in the thesis file. The published version is available at: https://doi.org/10.1007/s12095-021-00513-y en_US dc.relation.haspart Paper 6: D. Davidova, N. S. Kaleyski. Generalization of a class of APN binomials to Gold-like functions. In: Bajard J.C., Topuzo˘glu A. (eds) Arithmetic of Finite Fields. WAIFI 2020. Lecture Notes in Computer Science, vol. 12542. The article is available in the thesis file. The article is also available at: https://doi.org/10.1007/978-3-030-68869-1_11 en_US dc.relation.haspart Paper 7: L. Budaghyan, N. S. Kaleyski, S. Kwon, C. S. Riera, P. Stanica. Partially APN Boolean functions and classes of functions that are not APN infinitely often. Cryptography and Communications 12, 527–545 (2020). The article is available in the thesis file. The article is also available at: https://doi.org/10.1007/s12095-019-00372-8 en_US dc.relation.haspart Paper 8: L. Budaghyan, N. S. Kaleyski, C. S. Riera, P. Stanica. Partially APN functions with APN-like polynomial representations. Designs, Codes and Cryptography 88, 1159–1177 (2020). The article is available at: https://hdl.handle.net/11250/2756790 en_US dc.relation.haspart Paper 9: L. Budaghyan, M. Calderini, C. Carlet, D. Davidova, N. S. Kaleyski. On two fundamental problems on APN power functions. Cryptology ePrint Archive: Report 2020/1359. The article is not available in BORA. The preprint version is available at: https://eprint.iacr.org/2020/1359.pdf en_US dc.rights In copyright dc.rights.uri http://rightsstatements.org/page/InC/1.0/ dc.title Towards a deeper understanding of APN functions and related longstanding problems en_US dc.type Doctoral thesis en_US dc.date.updated 2021-08-17T01:57:07Z dc.rights.holder Copyright the Author. All rights reserved en_US dc.contributor.orcid https://orcid.org/0000-0002-9695-1454 dc.description.degree Doktorgradsavhandling fs.unitcode 12-12-0
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