Formal Framework for Property-driven Obfuscations

We study the existence and the characterization of function transformers that minimally or maximally modify a function in order to reveal or conceal a certain property. Based on this general formal framework we develop a strategy for the design of the maximal obfuscating transformation that conceals a given property while revealing the desired observational behaviou

On calculating residuated approximations and the structure of finite lattices of small width

The concept of a residuated mapping relates to the concept of Galois connections; both arise in the theory of partially ordered sets. They have been applied in mathematical theories (e.g., category theory and formal concept analysis) and in theoretical computer science. The computation of residuated approximations between two lattices is influenced by lattice properties, e.g. distributivity. In previous work, it has been proven that, for any mapping f : L → [special characters omitted] between two complete lattices L and [special characters omitted], there exists a largest residuated mapping ρf dominated by f, and the notion of the shadow σ f of f is introduced. A complete lattice [special characters omitted] is completely distributive if, and only if, the shadow of any mapping f : L → [special characters omitted] from any complete lattice L to [special characters omitted] is residuated. Our objective herein is to study the characterization of the skeleton of a poset and to initiate the creation of a structure theory for finite lattices of small widths. We introduce the notion of the skeleton L˜ of a lattice L and apply it to find a more efficient algorithm to calculate the umbral number for any mapping from a ∼ finite lattice to a complete lattice. We take a maximal autonomous chain containing x as an equivalent class [x] of x. The lattice L˜ is based on the sets {[x] | x ∈ L}. The umbral number for any mapping f : L → [special characters omitted] between two complete lattices is related to the property of L˜. Let L be a lattice satisfying the condition that [x] is finite for all x ∈ L; such an L is called ∼ finite. We define Lo = {[special characters omitted][x] | x ∈ L} and fo = [special characters omitted]. The umbral number for any isotone mapping f is equal to the umbral number for fo, and [special characters omitted] for any ordinal number α. Let [special characters omitted] be the maximal umbral number for all isotone mappings f : L → [special characters omitted] between two complete lattices. If L is a ∼ finite lattice, then [special characters omitted]. The computation of [special characters omitted] is less than or equal to that of [special characters omitted], we have a more efficient method to calculate the umbral number [special characters omitted]. The previous results indicate that the umbral number [special characters omitted] determined by two lattices is determined by their structure, so we want to find out the structure of finite lattices of small widths. We completely determine the structure of lattices of width 2 and initiate a method to illuminate the structure of lattices of larger width

Characterizing A Property-Driven Obfuscation Strategy

n recent years, code obfuscation has attracted both researchers and software developers as a useful technique for protecting secret properties of proprietary programs. The idea of code obfuscation is to modify a program, while preserving its functionality, in order to make it more difficult to analyze. Thus, the aim of code obfuscation is to conceal certain properties to an attacker, while revealing its intended behavior. However, a general methodology for deriving an obfuscating transforma- tion from the properties to conceal and reveal is still missing. In this work, we start to address this problem by studying the existence and the characterization of function transformers that minimally or maximally modify a program in order to reveal or conceal a certain property. Based on this general formal framework, we are able to provide a characterization of the maximal obfuscating strategy for transformations concealing a given property while revealing the desired observational behavior. To conclude, we discuss the applicability of the proposed characterization by showing how some common obfuscation techniques can be interpreted in this framework. Moreover, we show how this approach allows us to deeply understand what are the behavioral properties that these transformations conceal, and therefore protect, and which are the ones that they reveal, and therefore disclose