ONLY PROBLEMS, NOT SOLUTIONS!

The development of mathematics continues in a rapid rhythm, some unsolved problems are elucidated and simultaneously new open problems to be solved appear

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

A SET OF NEW SMARANDACHE FUNCTIONS, SEQUENCES AND CONJECTURES IN NUMBER THEORY

The Smarandache's universe is undoubtedly very fascinating and is halfway between the number theory and the recreational mathematics. Even though sometime this universe has a very simple structure from number theory standpoint, it doesn't cease to be deeply mysterious and interesting

Automated theory formation in pure mathematics

The automation of specific mathematical tasks such as theorem proving and algebraic manipulation have been much researched. However, there have only been a few isolated attempts to automate the whole theory formation process. Such a process involves forming new concepts, performing calculations, making conjectures, proving theorems and finding counterexamples. Previous programs which perform theory formation are limited in their functionality and their generality. We introduce the HR program which implements a new model for theory formation. This model involves a cycle of mathematical activity, whereby concepts are formed, conjectures about the concepts are made and attempts to settle the conjectures are undertaken.HR has seven general production rules for producing a new concept from old ones and employs a best first search by building new concepts from the most interesting old ones. To enable this, HR has various measures which estimate the interestingness of a concept. During concept formation, HR uses empirical evidence to suggest conjectures and employs the Otter theorem prover to attempt to prove a given conjecture. If this fails, HR will invoke the MACE model generator to attempt to disprove the conjecture by finding a counterexample. Information and new knowledge arising from the attempt to settle a conjecture is used to assess the concepts involved in the conjecture, which fuels the heuristic search and closes the cycle.The main aim of the project has been to develop our model of theory formation and to implement this in HR. To describe the project in the thesis, we first motivate the problem of automated theory formation and survey the literature in this area. We then discuss how HR invents concepts, makes and settles conjectures and how it assesses the concepts and conjectures to facilitate a heuristic search. We present results to evaluate HR in terms of the quality of the theories it produces and the effectiveness of its techniques. A secondary aim of the project has been to apply HR to mathematical discovery and we discuss how HR has successfully invented new concepts and conjectures in number theory

DEFINITIONS, SOLVED AND UNSOLVED PROBLEMS, CONJECTURES, AND THEOREMS IN NUMBER THEORY AND GEOMETRY

Florentin Smarandache, an American mathematician of Romanian descent has generated a vast variety of mathematical problems. Some problems are easy, others medium, but many are interesting or unsolved and this is the reason why the present book appears. Here, of course, there are problems from various types. Solving these problems is addictive like eating pumpkin seed: having once started, one cannot help doing it over and over again

DEFINITIONS, SOLVED AND U SOLVED PROBLEMS, CONJECTURES, AND THEOREMS IN NUMBER THEORY AND GEOMETRY

Florentin Smarandache, an American mathematician of Romanian descent, has generated a vast variety of mathematical problems. Some problems are easy, others medium, but many are interesting or unsolved and this is the reason why the present book appears. Here, of course, there are problems from various types. Solving these problems is addictive like eating pumpkin seed: having once started, one cannot help doing it over and over again