Selected Aspects of Multiple-valued Bent Functions

Claudio Moraga

Technical University of Dortmund, Germany

Bent functions were introduced by O. Rothaus in 1976, as the most non-linear Boolean functions, and captured very soon greatest interest from the Cryptography Community. Extensions of Bent functions to non-binary domains were done independently by P.V. Kumar et al. in 1985 and M. Luis and C. Moraga in 1989. Although in the non-binary domain bent functions continued to be most non-linear, their possible cryptographic properties did not have a priority. Cryptography was already strongly anchored in the binary domain. However non-binary or (p-valued) bent functions presented new highly challenging mathematical problems; among others: their generation in different mathematical structures, their characterization, formal representation, and the computation of the cardinality of their several known classes.
In the spectral domain, multiple-valued bent functions are characterized by a Vilenkin-Chrestenson circular spectrum, such that all coefficients have the same absolute value pn/2. The best known classes of p-valued bent functions are the Gamma class, generated based on the tensor sum of bent functions and with the use of spectral invariant operations the p-valued version of the Maiorana- McFarland (binary) class, and an extension of this class. In the p-valued domain there are bent functions for both even and odd number of arguments (meanwhile in the binary case, n is restricted to be even.) A matrix-based version of the Maiorana-McFarland generation method for p-valued bent functions was introduced in allowing a general spectral proof of bentness for all functions of the class. Furthermore, work with the Maiorana-McFarland class allowed to determine a lower bound on the number of p-valued bent functions for every n. For example there are 708,588 ternary bent functions of four arguments. Although this number seems to be large, it should be recalled that there are 381.= 4.43•10x38 ternary functions of four arguments.
In the Lecture, the Gamma and Maiorana-McFarlansd classes will be explained, it will be shown that they are disjoint, and examples of generation and characterization of p-valued bent functions will be given. .

The Survival Signature for System Reliability

Frank Coolen

Durham University, United Kingdom

Quantification of reliability of systems is of obvious importance in many application areas. The key mathematical concept for such reliability quantification is the system structure function, which describes whether or not the system functions given the state of all of its components. For many basic inferences on system reliability, however, one only needs a summary of the structure function. One such a summary is Samaniego's system signature, which has led to many journal publications in recent years, but its practical value is limited as it can only be applied to systems whose components all have exchangeable failure times, so-called `identical components'. To overcome this disadvantage, we presented the survival signature in 2012: this summary of the structure function is closely related to Samaniego's signature for systems with identical components, but straightforwardly generalized to systems with multiple types of components. We present an introductory overview of the survival signature and discuss recent advances, including its use with imprecise probabilities, consideration of resilience of systems through the possibility of swapping components, and computational aspects including its use for very efficient simulation of system failure times.

Challenges in Medical Image Analysis

Max A. Viergever

University Medical Center Utrecht, The Netherlands

In order to disambiguate the title of this presentation, the talk is not aimed at providing a visionary display of possible breakthroughs in the field of medical image analysis. Much the opposite, it concerns a well established - although still relatively new - approach to evaluating medical image analysis algorithms, sometimes also referred to as "grand challenges", in which algorithms that try and solve a specific problem in medical image analysis are compared, usually in a competitive fashion. The presentation will describe what challenges are, why they are organized, how they have been and may be set up, and give examples of challenges covering several topics in medical image analysis.

Towards uniform approaches to create quantum Grover oracles for practical problems

Marek Perkowski

Portland State University, USA

It is well-known that the "Unsorted Database" quantum algorithm by Grover gives quadratic speedup to several important combinatorial and enumerative problems, such as: SAT, Graph Coloring, Maximum Cliques, Travelling Salesman and others. Recently, quantum programming languages such as Quipper start to be used to design, verify and simulate practical quantum algorithms for important problems in Quantum Machine Learning. So far, however, no methodologies have been created to program Grover Oracles for particular classes of problems. In contrast, such methodologies have been created for classical Constraint Satisfaction Problems. The goal of this invited talk is to show results of some initial research towards creating systematic methodologies to program quantum computers that solve search problems in Artificial Intelligence, Logic Design and Machine Learning. Our methods are based on unified oracle blocks for such representations as set partition algebra, cube calculus and optimal mappings. For instance, several important problems in CAD and Machine Learning can be solved using only two basic operations on set partitions; π₁ ≤ π₂ and π₁ x π₂ .