Conference Paper

A Parallel GNFS Algorithm Based on a Reliable Look-Ahead Block Lanczos Method for Integer Factorization.

DOI: 10.1007/11802167_13 Conference: Embedded and Ubiquitous Computing, International Conference, EUC 2006, Seoul, Korea, August 1-4, 2006, Proceedings
Source: DBLP

ABSTRACT The Rivest-Shamir-Adleman (RSA) algorithm is a very popular and secure public key cryptosystem, but its security relies on
the difficulty of factoring large integers. The General Number Field Sieve (GNFS) algorithm is currently the best known method
for factoring large integers over 110 digits. Our previous work on the parallel GNFS algorithm, which integrated the Montgomery’s
block Lanczos method to solve large and sparse linear systems over GF(2), is less reliable. In this paper, we have successfully
implemented and integrated the parallel General Number Field Sieve (GNFS) algorithm with the new look-ahead block Lanczos
method for solving large and sparse linear systems generated by the GNFS algorithm. This new look-ahead block Lanczos method
is based on the look-ahead technique, which is more reliable, avoiding the break-down of the algorithm due to the domain of
GF(2). The algorithm can find more dependencies than Montgomery’s block Lanczos method with less iterations. The detailed
experimental results on a SUN cluster will be presented in this paper as well.

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    ABSTRACT: RSA algorithm is a very popular public key cryptosystem which has been widely used in industries. Its security relies on the difficulty of factoring large integers. The general number field sieve (GNFS) is so far the best known algorithm for factoring large integers over 110 digits. The Montgomery's block Lanczos method from Linbox is for solving large and sparse linear systems over finite fields and it can be integrated into GNFS algorithm. This paper introduces an improved Montgomery block Lanczos method, based on the version developed in Linbox, integrated with our previously developed parallel GNFS algorithm. This method has a better performance comparing with the original one, can find more solutions or dependencies than the original one with less time complexities. Implementation details and experimental results will be provided as well in the paper as well.
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    ABSTRACT: Integer factorization is known to be one of the most important and useful methods in number theory and arithmetic. It also has a very close relationship to some algorithms in cryptography such as RSA algorithm. The RSA cryptosystem is one of the most popular and attractive public-key cryptosystems in the world today. Its security is based on the difficulty of integer factorization. Solving a large and sparse linear system over GF(2) is one of the most time consuming steps in most modern integer factorization algorithms including the fastest one, GNFS algorithm.The Montgomery block Lanczos method from Linbox [13] is for solving large and sparse linear systems over finite fields and it can be integrated into the general number field sieve (GNFS) algorithm which is the best known algorithm for factoring large integers over 110 digits. This paper will present an improved Montgomery block Lanczos method integrated with parallel GNFS algorithm. The experimental results show that the improved Montgomery block Lanczos method has a better performance compared with the original method. It can find more solutions or dependencies than the original method with less time complexities. Implementation details and experimental results are provided in this paper as well.
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