Low differential uniform functions from algebraic and combinatoric structures
| Author(s) | Bergman, Emily | |
| Date Accessioned | 2021-01-26T14:54:11Z | |
| Date Available | 2021-01-26T14:54:11Z | |
| Publication Date | 2020 | |
| SWORD Update | 2020-10-13T19:03:18Z | |
| Abstract | This dissertation splits into two major parts. First, given a function, f over Fq and nonzero a in Fq we define the difference function as Df(x; a) := f (x + a) - f (x) - f (a). The differential uniformity (DU) of f is d:= max #f{c : c in Fq | Df (c; a) = b}. Functions with lower differential uniformity are more resistant to differential cryptanalysis and as such are more desirable for use in substitution boxes. Inspired by the work in [23], we investigate the differential uniformity of functions of the form f (x) = x * x using multiplication from algebraic objects with the objective of constructing functions with low differential uniformity. The second major part concerns the classification of planar monomials over fields of size Fp3 . Previous work on this problem has been completed for fields of size p, p2, and p4 with p odd [48], [21], [26] and the problem has been reduced considerably for fields of size p^{2k} with p > 5 and k > 2 through the work in [26]. We make significant progress on the p3 case. The thesis can be broken down as follows. In Chapter 1, we give the preliminary material that we will need throughout this work including: background on finite fields and polynomials over those fields; the framework for algebraic objects like S-sets, (pre)semifields, and nearfields; and the background of differential uniformity. The main result for Chapter 2, Theorem 2.1.1, explores the notion of creating a new function with low DU by replacing the coordinate functions of known functions. We use mutually orthogonal systems to create bounds for the differential uniformity of these new functions and we discuss known results in terms of this new methodology. The results of our investigation for low DU functions from algebraic objects are included in Chapter 3. We find functions corresponding to specific Kantor presemifields that are at most 4 DU in Theorem 3.3.2 and give a bound for the differential uniformity of functions from other Kantor presemifields in Theorem 3.3.3. In Theorem 3.4.3, we give a lower bound for the differential uniformity for the function f (x) = x * x where the multiplication is from the regular planar nearfield N(2; q2) with q odd. This constitutes the first major part. The second major part of the thesis is contained in Chapter 4. The classification of planar monomials for fields of size p3 falls in three parts. Proposition 4.2.1 and Proposition 4.2.2 fully resolve two of those cases while Section 4.3 outlines the current status of case three. Finally, in Chapter 5 we give discuss open problems that have come from this work. | en_US |
| Advisor | Coulter, Robert S. | |
| Degree | Ph.D. | |
| Department | University of Delaware, Department of Mathematical Sciences | |
| DOI | https://doi.org/10.58088/k681-6v08 | |
| Unique Identifier | 1232922444 | |
| URL | https://udspace.udel.edu/handle/19716/28542 | |
| Language | en | |
| Publisher | University of Delaware | en_US |
| URI | https://login.udel.idm.oclc.org/login?url=https://www.proquest.com/dissertations-theses/low-differential-uniform-functions-algebraic/docview/2455533005/se-2?accountid=10457 | |
| Keywords | Differential uniformity | en_US |
| Keywords | Nearfields | en_US |
| Keywords | Semifields | en_US |
| Title | Low differential uniform functions from algebraic and combinatoric structures | en_US |
| Type | Thesis | en_US |