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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References
Briggs, M.E.: An introduction to the general number field sieve. Master’s thesis, Virginia Polytechnic Institute and State University (1998)
Case, M.: A beginner’s guide to the general number field sieve. Oregon State University, ECE575 Data Security and Cryptography Project (2003)
Dreibellbis, J.: Implementing the general number field sieve, 5–14 (June 2003)
Granlund, T.: The GNU Multiple Precision Arithmetic Library. TMG Datakonsult, Boston, MA, USA, 2.0.2 edition (June 1996)
Gropp, W., Lusk, E., Skjellum, A.: Using MPI: Portable Parallel Programming with the Message-Passing Interface. MIT Press, Cambridge (1994)
Gutknecht, M.H.: Block krylov space methods for linear systems with multiple right-hand sides. In: The Joint Workshop on Computational Chemistry and Numerical Analysis (CCNA 2005), Tokyo (December 2005)
Gutknecht, M.H., Schmelzer, T.: A QR-decomposition of block tridiagonal matrices generated by the block lanczos process. In: Proceedings IMACS World Congress, Paris (July 2005)
Hovinen, B.: Blocked lanczos-style algorithms over small finite fields. Master Thesis of Mathematics, University of Waterloo, Canada (2004)
Lambert, R.: Computational Aspects of Discrete Logarithms. PhD thesis, University of Waterloo (1996)
Lanczos, C.: An iteration method for the solution of the eigenvalue problem of linear differential and integral operators. Journal of Research of the National Bureau of Standards 45, 255–282 (1950)
Lanczos, C.: Solutions of linread equations by minimized iterations. Journal of Research of the National Bureau of Standards 49, 33–53 (1952)
Lenstra, A.K.: Integer factoring. Designs, Codes and Cryptography 19(2-3), 101–128 (2000)
Lenstra, H.W.: Factoring integers with elliptic curves. Annals of Mathematics (2) 126, 649–673 (1987)
Lenstra, H.W., Pomerance, C., Buhler, J.P.: Factoring integers with the number field sieve. In: The Development of the Number Field Sieve, New York. Lecture Notes in Mathematics, vol. 1554, pp. 50–94. Springer, Heidelberg (1993)
Monico, C.: General number field sieve documentation. GGNFS Documentation (November 2004)
Montgomery, P.L.: A block lanczos algorithm for finding dependencies over gf(2). In: Guillou, L.C., Quisquater, J.-J. (eds.) EUROCRYPT 1995. LNCS, vol. 921, pp. 106–120. Springer, Heidelberg (1995)
Parlett, B.N., Taylor, D.R., Liu, Z.A.: A look-ahead lanczos algorithm for unsymetric matrics. Mathematics of Computation 44, 105–124 (1985)
Pollard, J.M.: Theorems on factorization and primality testing. In: Proceedings of the Cambridge Philosophical Society, pp. 521–528 (1974)
Pomerance, C.: The quadratic sieve factoring algorithm. In: Beth, T., Cot, N., Ingemarsson, I. (eds.) EUROCRYPT 1984. LNCS, vol. 209, pp. 169–182. Springer, Heidelberg (1985)
Rivest, R.L., Shamir, A., Adelman, L.M.: A method for obtaining digital signatures and public-key cryptosystems. Technical Report MIT/LCS/TM-82 (1977)
Wunderlich, M.C., Selfridge, J.L.: A design for a number theory package with an optimized trial division routine. Communications of ACM 17(5), 272–276 (1974)
Xu, L., Yang, L.T., Lin, M.: Parallel general number field sieve method for integer factorization. In: Proceedings of the 2005 International Conference on Parallel and Distributed Processing Techniques and Applications (PDPTA 2005), Las Vegas, USA, June 2005, pp. 1017–1023 (2005)
Yang, L.T., Xu, L., Lin, M.: Integer factorization by a parallel gnfs algorithm for public key cryptosystem. In: Yang, L.T., Zhou, X.-s., Zhao, W., Wu, Z., Zhu, Y., Lin, M. (eds.) ICESS 2005. LNCS, vol. 3820, pp. 683–695. Springer, Heidelberg (2005)
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Yang, L.T., Xu, L., Lin, M., Quinn, J. (2006). A Parallel GNFS Algorithm Based on a Reliable Look-Ahead Block Lanczos Method for Integer Factorization. In: Sha, E., Han, SK., Xu, CZ., Kim, MH., Yang, L.T., Xiao, B. (eds) Embedded and Ubiquitous Computing. EUC 2006. Lecture Notes in Computer Science, vol 4096. Springer, Berlin, Heidelberg. https://doi.org/10.1007/11802167_13
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DOI: https://doi.org/10.1007/11802167_13
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