Knapsack sets for cryptography

The basic goal of this study is to extend old and propose new ways to generate knapsack sets suitable for use in public key cryptography. The knapsack problem and its cryptographic use are reviewed in the introductory chapter. Terminology is based on common cryptographic vocabulary. For example, solving the knapsack problem (which is here a subset sum problem) is termed decipherment. Chapter 1 also reviews the most famous knapsack cryptosystem, the Merkle Hellman system. It is based on a superincreasing knapsack and uses modular multiplication as a trapdoor transformation. The insecurity caused by these two properties exemplifies the two general categories of attacks against knapsack systems. These categories provide the motivation for Chapters 2 and 4. Chapter 2 discusses the density of a knapsack and the dangers of having a low density. Chapter 3 interrupts for a while the more abstract treatment by showing examples of small injective knapsacks and extrapolating conjectures on some characteristics of knapsacks of larger size, especially their density and number. The most common trapdoor technique, modular multiplication, is likely to cause insecurity, but as argued in Chapter 4, it is difficult to find any other simple trapdoor techniques. This discussion also provides a basis for the introduction of various categories of non injectivity in Chapter 5. Besides general ideas of non injectivity of knapsack systems, Chapter 5 introduces and evaluates several ways to construct such systems, most notably the "exceptional blocks" in superincreasing knapsacks and the usage of "too small" a modulus in the modular multiplication as a trapdoor technique. The author believes that non injectivity is the most promising direction for development of knapsack cryptosystema. Chapter 6 modifies two well known knapsack schemes, the Merkle Hellman multiplicative trapdoor knapsack and the Graham Shamir knapsack. The main interest is in aspects other than non injectivity, although that is also exploited. In the end of the chapter, constructions proposed by Desmedt et. al. are presented to serve as a comparison for the developments of the subsequent three chapters. Chapter 7 provides a general framework for the iterative construction of injective knapsacks from smaller knapsacks, together with a simple example, the "three elements" system. In Chapters 8 and 9 the general framework is put into practice in two different ways. Modularly injective small knapsacks are used in Chapter 9 to construct a large knapsack, which is called the congruential knapsack. The addends of a subset sum can be found by decrementing the sum iteratively by using each of the small knapsacks and their moduli in turn. The construction is also generalized to the non injective case, which can lead to especially good results in the density, without complicating the deciphering process too much. Chapter 9 presents three related ways to realize the general framework of Chapter 7. The main idea is to join iteratively small knapsacks, each element of which would satisfy the superincreasing condition. As a whole, none of these systems need become superincreasing, though the development of density is not better than that. The new knapsack systems are injective but they can be deciphered with the same searching method as the non injective knapsacks with the "exceptional blocks" in Chapter 5. The final Chapter 10 first reviews the Chor Rivest knapsack system, which has withstood all cryptanalytic attacks. A couple of modifications to the use of this system are presented in order to further increase the security or make the construction easier. The latter goal is attempted by reducing the size of the Chor Rivest knapsack embedded in the modified system. '

The knapsack problem approach in solving partial hedging problems of options

En option är ett finansiellt kontrakt som ger dess innehavare en rättighet (men medför ingen skyldighet) att sälja eller köpa någonting (till exempel en aktie) till eller från säljaren av optionen till ett visst pris vid en bestämd tidpunkt i framtiden. Den som säljer optionen binder sig till att gå med på denna framtida transaktion ifall optionsinnehavaren längre fram bestämmer sig för att inlösa optionen. Säljaren av optionen åtar sig alltså en risk av att den framtida transaktion som optionsinnehavaren kan tvinga honom att göra visar sig vara ofördelaktig för honom. Frågan om hur säljaren kan skydda sig mot denna risk leder till intressanta optimeringsproblem, där målet är att hitta en optimal skyddsstrategi under vissa givna villkor. Sådana optimeringsproblem har studerats mycket inom finansiell matematik. Avhandlingen "The knapsack problem approach in solving partial hedging problems of options" inför en ytterligare synpunkt till denna diskussion: I en relativt enkel (ändlig och komplett) marknadsmodell kan nämligen vissa partiella skyddsproblem beskrivas som så kallade kappsäcksproblem. De sistnämnda är välkända inom en gren av matematik som heter operationsanalys. I avhandlingen visas hur skyddsproblem som tidigare lösts på andra sätt kan alternativt lösas med hjälp av metoder som utvecklats för kappsäcksproblem. Förfarandet tillämpas även på helt nya skyddsproblem i samband med så kallade amerikanska optioner.

Security and Privacy in a Ubiquitous Information Screen

We expose the ubiquitous interaction between an information screen and its’ viewers mobile devices, highlights the communication vulnerabilities, suggest mitigation strategies and finally implement these strategies to secure the communication. The screen infers information preferences’ of viewers within its vicinity transparently from their mobile devices over Bluetooth. Backend processing then retrieves up-to-date versions of preferred information from content providers. Retrieved content such as sporting news, weather forecasts, advertisements, stock markets and aviation schedules, are systematically displayed on the screen. To maximise users’ benefit, experience and acceptance, the service is provided with no user interaction at the screen and securely upholding preferences privacy and viewers anonymity. Compelled by the personal nature of mobile devices, their contents privacy, preferences confidentiality, and vulnerabilities imposed by screen, the service’s security is fortified. Fortification is predominantly through efficient cryptographic algorithms inspired by elliptic curves cryptosystems, access control and anonymity mechanisms. These mechanisms are demonstrated to attain set objectives within reasonable performance.