| Study of the modular arithmetic; the properties of integers numbers; the prime number distribution and the way to identify them; the most usual cryptographic systems, specially the RSA public key one; the random and pseudorandom bit sequence generators; the factoring algorithms; and finally, study of the computer architecture features most directly related to the execution time optimization.; Analysis of the different available approaches to implement operations with very long integers; the several existing tests for prime condition of numbers; the various pseudorandom bit sequence generators, being selected the BBS one as the most secure for cryptography; the alternative implementations and improvements carried out in the factoring algorithm, based in the continued fractions technique, the optimum parameter value used for better yield and the comparisons of the implemented algorithm with those of Pomerance (QS) and Lenstra (NFS); the different conditions that the RSA cryptosystem must fulfill to avoid attacks and tricks; an analysis process regarding the code and computer interaction to achieve a significant reduction of execution time.; Design of a new model of long integer, defining its domain, that is, its rank of the allowed values and their operators; a bit sequence random generator; an orderly and systematic code optimization protocol.; Implementation of every tool needed for optimum calculus regarding either the long integer model or the designed random bit sequence generator or the BBS pseudorandom one; the prime condition tests algorithms; the needed processes to factorize long integers, as a product of two prime numbers, such as function libraries required for the CFRAC algorithm and others programs to factorize millions of integers; macros for friendly interfaces and speedy programs; and also the implementation of any tool needed to analyze software and hardware interaction.; Finally, optimization of the code that has been studied, analyzed, designed and implemented to factorize integers obtained as the product of two large prime numbers. Following an innovated protocol in order to minimize execution time, number of instructions, cache misses and branch instructions. Our objective has always been to simplify computer algorithms and to maximize the performance of current computers. |
