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# Novel methods for primality testing and factoring

Hammad, Yousef Bani (2005) Novel methods for primality testing and factoring. PhD thesis, Queensland University of Technology.

 Yousef Bani Hammad Thesis (PDF 3128Kb)

## Abstract

From the time of the Greeks, primality testing and factoring have fascinated mathematicians, and for centuries following the Greeks primality testing and factorization were pursued by enthusiasts and professional mathematicians for their intrisic value. There was little practical application. One example application was to determine whether or not the Fermat numbers, that is, numbers of the form F;, = 2'" + 1 were prime. Fermat conjectured that for all n they were prime. For n = 1,2,3,4, the Fermat numbers are prime, but Euler showed that F; was not prime and to date no F,, n 2 5 has been found to be prime. Thus, for nearly 2000 years primality testing and factorization was largely pure mathematics. This all changed in the mid 1970's with the advent of public key cryptography. Large prime numbers are used in generating keys in many public key cryptosystems and the security of many of these cryptosystems depends on the difficulty of factoring numbers with large prime factors. Thus, the race was on to develop new algorithms to determine the primality or otherwise of a given large integer and to determine the factors of given large integers. The development of such algorithms continues today. This thesis develops both of these themes. The first part of this thesis deals with primality testing and after a brief introduction to primality testing a new probabilistic primality algorithm, ALI, is introduced. It is analysed in detail and compared to Fermat and Miller-Rabin primality tests. It is shown that the ALI algorithm is more efficient than the Miller-Rabin algorithm in some aspects. The second part of the thesis deals with factoring and after looking closely at various types of algorithms a new algorithm, RAK, is presented. It is analysed in detail and compared with Fermat factorization. The RAK algorithm is shown to be significantly more efficient than the Fermat factoring algorithm. A number of enhancements is made to the basic RAK algorithm in order to improve its performance. The RAK algorithm with its enhancements is known as IMPROVEDRAK. In conjunction with this work on factorization an improvement to Shor's factoring algorithm is presented. For many integers Shor's algorithm uses a quantum computer multiple times to factor a composite number into its prime factors. It is shown that Shor's alorithm can be modified in a way such that the use of a quantum computer is required just once. The common thread throughout this thesis is the application of factoring and primality testing techniques to integer types which commonly occur in public key cryptosystems. Thus, this thesis contributes not only in the area of pure mathematics but also in the very contemporary area of cryptology.

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