A Modified Non-Sieving Quadratic Sieve For Factoring Simple Blur Integers

This paper addresses the problem-how to factor big simple blur integers quickly. We definite simple blur integer at first, and then give some theorems about it. Based on these theorems and the primitive non-sieving quadratic sieve (PNQS), we proposed a modified non-sieving quadratic sieve(MNQS). In MNQS, we not only reduce the times of squares and modulo n, but also imply another important conclusion, that is, we don´t need to choose a factor base and find the greatest common divisor of two integers as we do in PNQS. We argue that each algorithm has its virtue when factoring some special form of n. We find that when factoring n whose two factors´s difference is little, MNQS is fast than elliptic curve algorithm. We verify our conclusion by some examples.