Empirical Testing of Pseudo Random Number Generators Based on Elliptic Curves
Degree project Empirical testing of pseudo random number generators based on elliptic curves Abstract An introduction on random numbers, their history and applications is given, along with explanations of different methods currently used to generate them. Such generators can be of different kinds, and in particular they can be based on physical systems or algorithmic procedures. The latter type of procedures gives rise to pseudo-random number generators. Specifically, several such generators which are based on elliptic curves are examined. Therefore, in order to ease understanding, a basic primer on elliptic curves over fields and the operations arising from their group structure is also provided. Empirical tests to verify randomness of generated sequences are then considered. Afterwards, there are some statistical considerations and observations about theoretical properties of the generators at hand, useful in order to use them optimally. Finally, several randomly generated curves are created and used to produce pseudo-random se- quences which are then tested by means of the previously described generators. In the end, an analysis of the results is attempted and some final considerations are made. Keywords: elliptic curves, cryptography, pseudo random, number generation, PRNG, TRNG, linear congruential generator, power generator, Naor-Reingold generator, empirical testing, frequency test, serial test, run test, poker test, autocorrelation test Acknowledgements I would like to thank in particular Per-Anders Svensson for advice while choosing the topic for this thesis and the assistance throughout; Karl-Olof Lindahl for the useful lectures on the thesis process; and finally my family for supporting me during my studies. 1 Contents 1 Introduction 4 1.1 Motivation and aim .
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