Partition games

We introduce CUT, the class of 2-player partition games. These are NIM type games, played on a finite number of heaps of beans. The rules are given by a set of positive integers, which specifies the number of allowed splits a player can perform on a single heap. In normal play, the player with the last move wins, and the famous Sprague-Grundy theory provides a solution. We prove that several rulesets have a periodic or an arithmetic periodic Sprague-Grundy sequence (i.e. they can be partitioned into a finite number of arithmetic progressions of the same common difference). This is achieved directly for some infinite classes of games, and moreover we develop a computational testing condition, demonstrated to solve a variety of additional games. Similar results have previously appeared for various classes of games of take-and-break, for example octal and hexadecimal; see e.g. Winning Ways by Berlekamp, Conway and Guy (1982). In this context, our contribution consists of a systematic study of the subclass `break-without-take'

Taking and Breaking Games

V této práci analyzujeme několik otevřených problémů v oblasti nestranných i stranných her typu Taking and Breaking. Pro nestranné odčítací hry dokážeme existenci hry s aperiodickou nim-sekvencí a periodickou sekvencí výhra-prohra. Analyzujeme ekvivalenční třídy těchto her a nalézáme řešení jedné z těchto tříd. Také představujeme novou hru typu Taking and Breaking, kterou z velké části vyřešíme. V oblasti stranných her provedeme analýzu několika odčítacích her a her typu Pure Breaking. Pro tyto hry také představíme obecnou techniku testování aritmetické periodicity. Pro automatické řešení nestranných her typu Taking and Breaking navrhujeme několik algoritmů. Práci uzavíráme důkazem PSPACE-těžkosti nestranné zobecněné odčítací hry a EXPTIME-těžkosti této hry ve stranné variantě.In this thesis, we examine several open problems in taking and breaking games under the impartial and partizan setting. We prove the existence of an impartial subtraction game with aperiodic nim-sequence and periodic outcome sequence. We also analyze the equivalence classes of subtraction games and provide a solution to one of these classes. We introduce a new taking and breaking game and partially solve it. Then we solve several partizan subtraction games and partizan pure breaking games and describe a general technique for testing arithmetic periodicity of these games. Moreover, we design some game solving algorithms for impartial taking and breaking games. We prove PSPACE-hardness for a generalized subtraction game under the impartial setting and EXPTIME-hardness under the partizan setting

Master index

Pla general, del mural ceràmic que decora una de les parets del vestíbul de la Facultat de Química de la UB. El mural representa diversos símbols relacionats amb la química

Glosarium Matematika

273 p.; 24 cm

All-Silicon-Based Photonic Quantum Random Number Generators

Random numbers are fundamental elements in different fields of science and technology such as computer simulation like Monte Carlo-method simulation, statistical sampling, cryptography, games and gambling, and other areas where unpredictable results are necessary. Random number generators (RNG) are generally classified as “pseudo”-random number generators (PRNG) and "truly" random number generators (TRNG). Pseudo random numbers are generated by computer algorithms with a (random) seed and a specific formula. The random numbers produced in this way (with a small degree of unpredictability) are good enough for some applications such as computer simulation. However, for some other applications like cryptography they are not completely reliable. When the seed is revealed, the entire sequence of numbers can be produced. The periodicity is also an undesirable property of PRNGs that can be disregarded for most practical purposes if the sequence recurs after a very long period. However, the predictability still remains a tremendous disadvantage of this type of generators. Truly random numbers, on the other hand, can be generated through physical sources of randomness like flipping a coin. However, the approaches exploiting classical motion and classical physics to generate random numbers possess a deterministic nature that is transferred to the generated random numbers. The best solution is to benefit from the assets of indeterminacy and randomness in quantum physics. Based on the quantum theory, the properties of a particle cannot be determined with arbitrary precision until a measurement is carried out. The result of a measurement, therefore, remains unpredictable and random. Optical phenomena including photons as the quanta of light have various random, non-deterministic properties. These properties include the polarization of the photons, the exact number of photons impinging a detector and the photon arrival times. Such intrinsically random properties can be exploited to generate truly random numbers. Silicon (Si) is considered as an interesting material in integrated optics. Microelectronic chips made from Si are cheap and easy to mass-fabricate, and can be densely integrated. Si integrated optical chips, that can generate, modulate, process and detect light signals, exploit the benefits of Si while also being fully compatible with electronic. Since many electronic components can be integrated into a single chip, Si is an ideal candidate for the production of small, powerful devices. By complementary metal-oxide-semiconductor (CMOS) technology, the fabrication of compact and mass manufacturable devices with integrated components on the Si platform is achievable. In this thesis we aim to model, study and fabricate a compact photonic quantum random number generator (QRNG) on the Si platform that is able to generate high quality, "truly" random numbers. The proposed QRNG is based on a Si light source (LED) coupled with a Si single photon avalanche diode (SPAD) or an array of SPADs which is called Si photomultiplier (SiPM). Various implementations of QRNG have been developed reaching an ultimate geometry where both the source and the SPAD are integrated on the same chip and fabricated by the same process. This activity was performed within the project SiQuro—on Si chip quantum optics for quantum computing and secure communications—which aims to bring the quantum world into integrated photonics. By using the same successful paradigm of microelectronics—the study and design of very small electronic devices typically made from semiconductor materials—, the vision is to have low cost and mass manufacturable integrated quantum photonic circuits for a variety of different applications in quantum computing, measure, sensing, secure communications and services. The Si platform permits, in a natural way, the integration of quantum photonics with electronics. Two methodologies are presented to generate random numbers: one is based on photon counting measurements and another one is based on photon arrival time measurements. The latter is robust, masks all the drawbacks of afterpulsing, dead time and jitter of the Si SPAD and is effectively insensitive to ageing of the LED and to its emission drifts related to temperature variations. The raw data pass all the statistical tests in national institute of standards and technology (NIST) tests suite and TestU01 Alphabit battery without a post processing algorithm. The maximum demonstrated bit rate is 1.68 Mbps with the efficiency of 4-bits per detected photon. In order to realize a small, portable QRNG, we have produced a compact configuration consisting of a Si nanocrystals (Si-NCs) LED and a SiPM. All the statistical test in the NIST tests suite pass for the raw data with the maximum bit rate of 0.5 Mbps. We also prepared and studied a compact chip consisting of a Si-NCs LED and an array of detectors. An integrated chip, composed of Si p+/n junction working in avalanche region and a Si SPAD, was produced as well. High quality random numbers are produced through our robust methodology at the highest speed of 100 kcps. Integration of the source of entropy and the detector on a single chip is an efficient way to produce a compact RNG. A small RNG is an essential element to guarantee the security of our everyday life. It can be readily implemented into electronic devices for data encryption. The idea of "utmost security" would no longer be limited to particular organs owning sensitive information. It would be accessible to every one in everyday life

Implementation and study of a true random number generator

Securing information has been a concern throughout history. Especially nowadays since many user applications such as smart cards or Internet connections deal with sensible data. To protect this information dfferent cryptography protocols are used. These are algorithms that encapsulate the data by ciphering it. However, this is done by programming an application to run a digital mathematical function. This means that it is also possible to program malign applications to decode the cipher. In order to avoid this it is necessary to add unpredictability or randomness to the encoding process which can be done by employing a Random Number Generator. A RNG can be implemented in both software and hardware; however, a truly unpredictable sequence is not achieved through a digital process governed by mathematical formulae. This results in most RNGs producing a form of pseudo-randomness. A True Random Number Generator must be implemented on a technology that allows it to harvest entropy from an unpredictable or even chaotic physical process. This is why TRNGs are designed and implemented for hardware. In fact, it is possible to gather entropy through integrated circuits like ASICs or FPGAs. The objective of this project is to design and implement a TRNG on FPGA technology because its pre-defined logic blocks that only require a small amount of resources make it an appealing solution. First, an analysis of typical RNG designs is presented to understand the between a pseudo-RNG and a TRNG. Once this is stablished, the specific ways of designing TRNGs for integrated circuits are delved into. Moreover, the need for evaluation of the quality of randomness is also stated. This is ensured by a battery of tests that study the statistical properties of the output of a RNG. Secondly, the TRNG design proposals by B ohl on which this project is based on are introduced and analyzed before creating the design and implementation. Afterwards, the four experiments performed are explained. It was decided to first test the behavior of the TRNG at different frequencies to decide which provided randomness with the best quality. Afterwards, the TRNG was placed in different areas of the FPGA at the optimal frequency to test the variability of the device. A third experiment consisted of comparing these results in more devices to further study the variability. The final experiment consisted on forcing a reset of the circuit to ensure that the TRNG was resilient against this type of attacks. Last but not least, the results are summarized and several future developments are presented. After this the legal aspects and management of the project are explained.La protección de información ha sido una constante preocupación a lo largo de la historia. Especialmente hoy en día debido a las muchas aplicaciones que manejan datos confidenciales como tarjetas inteligentes o conexiones a Internet. Para proteger esta información diferentes protocolos criptográficos son usados. Estos son algoritmos que cifran los datos para encapsularlos. Sin embargo, esto se hace programando una aplicación que corre una formula matemática digital. Esto significa que también es posible programar aplicaciones maliciosas para decodificar el cifrado. Para poder evitar esto es necesario añadir aleatoriedad o un elemento impredecible al proceso de codificación. Esto puede hacerse empleando un Generador de Números Aleatorios cuyas siglas en inglés son RNG. Es posible implementar un RNG tanto en software como en hardware; sin embargo, una secuencia realmente impredecible no se puede generar a través de un proceso digital basado en la computación de fórmulas matemáticas. Esto es lo que hace que la mayoría de RNGs produzcan una especie de pseudo-aleatoriedad. Un Generador de Números Realmente Aleatorios (True Random Number Generator o TRNG) debe ser implementado en una tecnología que le permita extraer entropía de un proceso físico impredecible o caótico. Es por esto que los TRNG se implementan en hardware. De hecho, es posible obtener entropía a través de circuitos integrados como ASICs o FPGAs. El objetivo de este proyecto es diseñar e implementar un TRNG en tecnología FPGA puesto que sus bloques lógicos prede finidos que solo necesitan unos recursos reducidos la convierten en una solución atractiva. Se empieza por presentar un análisis de los diseños de RNG típicos para comprender la diferencia entre generadores pseudo aleatorios y TRNGs. Tras esto, se especifica la forma en la que los TRNGs se diseñan para circuitos integrados. Además, se expone la necesidad de evaluar la calidad de la aleatoriedad que se genera. Esta se comprueba a través de una batería de tests que estudian las propiedades estadísticas del output del TRNG. A continuación, las propuestas de diseño de TRNGs de Böhl en las que este proyecto se basa son introducidas y analizadas seguidas del diseño e implementación propios. Tras lo cual se explican los cuatro experimentos realizados. Primero se decidió comprobar el comportamiento del TRNG a diferentes frecuencias con el fin de determinar a cuál de ellas se producía la aleatoriedad de mayor calidad. Segundo, el TRNG fue posicionado en diferentes áreas de la FPGA a la frecuencia óptima para evaluar la variabilidad de la placa. El tercer experimento explora aún más la variabilidad al realizar el experimento anterior en otras placas. El último experimento consistió en forzar un reset del circuito para comprobar la resistencia TRNG ante ataque de este tipo. Finalmente, los resultados obtenidos se presentan resumidos junto con varias propuestas de mejoras futuras. Tras ello se muestran los aspectos legales del proyecto y su gestión.Ingeniería en Tecnologías de Telecomunicació

Dynamic block encryption with self-authenticating key exchange

One of the greatest challenges facing cryptographers is the mechanism used for key exchange. When secret data is transmitted, the chances are that there may be an attacker who will try to intercept and decrypt the message. Having done so, he/she might just gain advantage over the information obtained, or attempt to tamper with the message, and thus, misguiding the recipient. Both cases are equally fatal and may cause great harm as a consequence. In cryptography, there are two commonly used methods of exchanging secret keys between parties. In the first method, symmetric cryptography, the key is sent in advance, over some secure channel, which only the intended recipient can read. The second method of key sharing is by using a public key exchange method, where each party has a private and public key, a public key is shared and a private key is kept locally. In both cases, keys are exchanged between two parties. In this thesis, we propose a method whereby the risk of exchanging keys is minimised. The key is embedded in the encrypted text using a process that we call `chirp coding', and recovered by the recipient using a process that is based on correlation. The `chirp coding parameters' are exchanged between users by employing a USB flash memory retained by each user. If the keys are compromised they are still not usable because an attacker can only have access to part of the key. Alternatively, the software can be configured to operate in a one time parameter mode, in this mode, the parameters are agreed upon in advance. There is no parameter exchange during file transmission, except, of course, the key embedded in ciphertext. The thesis also introduces a method of encryption which utilises dynamic blocks, where the block size is different for each block. Prime numbers are used to drive two random number generators: a Linear Congruential Generator (LCG) which takes in the seed and initialises the system and a Blum-Blum Shum (BBS) generator which is used to generate random streams to encrypt messages, images or video clips for example. In each case, the key created is text dependent and therefore will change as each message is sent. The scheme presented in this research is composed of five basic modules. The first module is the key generation module, where the key to be generated is message dependent. The second module, encryption module, performs data encryption. The third module, key exchange module, embeds the key into the encrypted text. Once this is done, the message is transmitted and the recipient uses the key extraction module to retrieve the key and finally the decryption module is executed to decrypt the message and authenticate it. In addition, the message may be compressed before encryption and decompressed by the recipient after decryption using standard compression tools