Abstract
The algorithm for the online matrix solving is proposed. The rate of acceleration of the basic quadratic sieve method based on the online matrix solving is investigated. Acceleration of the quadratic sieve method will reduce the runtime, the complexity of the algorithm and expand the set of numbers, where this algorithm is the best. It is shown that the modified algorithm has increased the number of successful decompositions. That is, the number of cases where the basic quadratic sieve (standard sieving interval and size of the factor base) failed to form a matrix to obtain a solution was reduced. This became possible due to the fact that in the modified algorithm there is no need to obtain all L a +2 B-smooth numbers prior to diagonalization of the matrix, as in the case of the basic method. Among other important characteristics of this method, it should be noted that when used, the same operations as in the basic quadratic sieve method are performed, only their order is changed. The computing complexity decreases if the set of B-smooth numbers, for which the power matrix vectors form a linearly dependent system, are found quickly. According to the data obtained, the modified QS method, based on the online matrix solving, provides an acceleration of about 5.45 percent for numbers of 10 130 in size. It is shown that improvements associated with solving the matrix cannot lead to a significant increase in the sieving interval. After all, the rate of acceleration decreases with increasing number N. Further improvement to the quadratic sieve method should be related to methods aimed at a significant reduction of the sieving interval and the size of the factor base, which in relative terms should be the greater, the higher N
Highlights
Integer factorization is one of the oldest problems in mathematics
In 1994, factorization of the RSA-129 number was performed by means of the quadratic sieve algorithm (QS) [4]
In the modified algorithm presented in [17], it is possible to achieve a decrease in the size of the sieving interval and the matrix only when in the set of Lmax+2 B-smooth numbers all odd powers of factors are assigned to the sequence numbers of the factor base elements, which do not exceed Lmax≤La
Summary
Integer factorization is one of the oldest problems in mathematics. major breakthroughs have occurred over the past 30 years, especially after the introduction of public-key cryptography, and in particular, the RSA cryptosystem. We can say that if factorization is solved effectively, the RSA cryptosystem will be extremely vulnerable. It is interesting that latest introductions of factorization algorithms are closely related to the RSA challenge. The authors carried out the studies [1, 2] of methods of cryptographic analysis of the RSA algorithm. [3] shows that known examples of compromise of the RSA algorithm work only for specific implementations, and usually are not more effective than the factorization problem. In 1994, factorization of the RSA-129 number was performed by means of the quadratic sieve algorithm (QS) [4]. Modification of the quadratic sieve algorithm will allow reducing the running time of the algorithm and increasing the limit value of the factorized number for which the algorithm of the quadratic sieve method is the best. The study of new ways to reduce its computing complexity is relevant
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