Pseudorandom Numbers and Elliptic Curves over Finite Fields

Abstract

Random and pseudorandom numbers are extensively used in simulation and statistical modeling systems, in controlling computational processes, and in computer games. Many mathematical optimization methods and game theory apply random and pseudorandom elements. Pseudorandom binary sequences are also widely used in information security algorithms. Pseudorandom number generation is the art and science of deterministically generating a sequence of numbers that is hard to differentiate from a true random sequence. This thesis studies some methods of random number generation. In the first section we describe the most commonly used pseudorandom number generators to provide the necessary background. Elliptic curves are rich mathematical structures which have shown themselves to be incredibly useful in a wide range of applications. Most of the products and standards that use public-key cryptography for encryption and digital signatures use RSA. The key length for secure RSA use has increased over recent years, and this has put a heavier processing load on applications using RSA. Elliptic curve cryptography can provide the same level and type of security as RSA but with much shorter keys. In the third, fourth and fifth chapters new pseudorandom number generators are developed using elliptic curves over finite fields and the existing generators. The emphasis will be on the length of the sequences produced by such generators and the statistical properties to ensure their usage in cryptographic application. An interesting property of numbers is that almost all numbers become palindromes quickly after repeated reversal and addition of its digits. But there are some numbers which are an exception to this. These numbers are called Lychrel numbers. In the next chapter two algorithms are presented to generate secret keys with palindrome and Lychrel numbers. In the last chapter the deployment of secret keys for security purpose in stream ciphers and stegnography are studied.