Improvement of the quadratic sieve method on the basis of the extended factor base and using available quantity of B – smooth numbers
Keywords:Quadratic sieve, extended factor base, available quantity of B-smooth, sieving interval, prime numbers.
In information and telecommunication systems, RSA algorithms are often used to solve information security problems. At the core of the cryptostability of the most popular today asymmetric cryptographic algorithm RSA is the complexity of the factorization of large integers. Quadratic sieve method is the best for factorization of integers under 110 decimal digits or so. The most time consuming part of the algorithm of a quadratic sieve is the sieving process. The size of the factor base is one of the key parameters that determine the effectiveness of the sieving algorithm. Too large factor base requires the search for a large number of B–smooth numbers, which increases the total execution time of the algorithm. When the size is less than necessary, it will not be possible to find a sufficient number of B–smooth numbers. In this paper, the method for determining and applying a sufficient size of B–smooth numbers with doubling the factor base size in comparison with the basic algorithm of a quadratic sieve is proposed. With the expansion of the factor base, the number of N numbers increases, which can be decomposed into factors by the quadratic sieve method. It is also noted that its increase leads to an increase in computational complexity, since it is advisable to find a greater number of B–smooth numbers. However, when conducting numerical experiments, where the size of the factor base increased twice, it turned out that using the proposed algorithm, the time necessary to find a sufficient number of B-smooth numbers, on the contrary, decreased.
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