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Issue No.09 - September (2010 vol.59)

pp: 1264-1280

Kris Gaj , George Mason University, Fairfax

Soonhak Kwon , Sungkyunkwan University, Suwon

Patrick Baier , Siemens PLM Software

Paul Kohlbrenner , George Mason University, Fairfax

Hoang Le , University of Southern California, Los Angeles

Mohammed Khaleeluddin , Hughes Network Systems, Germantown

Ramakrishna Bachimanchi , Thomas Jefferson National Accelerator Facility

Marcin Rogawski , George Mason University, Fairfax

DOI Bookmark: http://doi.ieeecomputersociety.org/10.1109/TC.2009.191

ABSTRACT

A novel portable hardware architecture of the Elliptic Curve Method of factoring, designed and optimized for application in the relation collection step of the Number Field Sieve, is described and analyzed. A comparison with an earlier proof-of-concept design by Pelzl et al. has been performed, and a substantial improvement has been demonstrated in terms of both the execution time and the area-time product. The ECM architecture has been ported across five different families of FPGA devices in order to select the family with the best performance to cost ratio. A timing comparison with the highly optimized software implementation, GMP-ECM, has been performed. Our results indicate that low-cost families of FPGAs, such as Spartan-3 and Spartan-3E, offer at least an order of magnitude improvement over the same generation of microprocessors in terms of the performance to cost ratio, without the use of embedded FPGA resources, such as embedded multipliers.

INDEX TERMS

Cipher-breaking, factoring, ECM, FPGA, NFS.

CITATION

Kris Gaj, Soonhak Kwon, Patrick Baier, Paul Kohlbrenner, Hoang Le, Mohammed Khaleeluddin, Ramakrishna Bachimanchi, Marcin Rogawski, "Area-Time Efficient Implementation of the Elliptic Curve Method of Factoring in Reconfigurable Hardware for Application in the Number Field Sieve",

*IEEE Transactions on Computers*, vol.59, no. 9, pp. 1264-1280, September 2010, doi:10.1109/TC.2009.191REFERENCES

- [1] L. Batina and G. Muurling, "Montgomery in Practice: How to Do It More Efficiently in Hardware,"
Topics in Cryptology—CT-RSA—The Cryptographers' Track at the RSA Conf., pp. 40-52, 2002.- [2] D.J. Bernstein, "Circuits for Integer Factorization: A Proposal," http://cr.yp.topapers.html#nfscircuit, 2010.
- [3] D.J. Bernstein, http://cr.yp.to/talks/2004.07.29slides.pdf , 2010.
- [4] R.P. Brent, "Some Integer Factorization Algorithms Using Elliptic Curves,"
Australian Computer Science Comm., vol. 8, pp. 149-163, 1986.- [5] H. Cohen,
A Course in Computational Algebraic Number Theory, second ed. Springer Graduate Texts in Math., 1993.- [6] D. Coppersmith, "Modifications to the Number Field Sieve,"
J. Cryptology, vol. 6, pp. 169-180, 1993.- [7] R. Crandall and C. Pomerance,
Prime Numbers—A Computational Perspective, 2001.- [8] F. Bahr, M. Boehm, J. Franke, and T. Kleinjung, "Factorization of RSA-200," http://crypto-world.com/announcementsrsa200.txt , 2010.
- [9] J. Fougeron, L. Fousse, A. Kruppa, D. Newman, and P. Zimmermann, "GMP-ECM," http://www.komite.net/laurent/soft/ecmecm-6.0.1.html , 2005.
- [10] J. Franke, T. Kleinjung, C. Paar, J. Pelzl, C. Priplata, and C. Stahlke, "SHARK—A Realizable Hardware Architecture for Factoring 1024-Bit Composites with the GNFS,"
Cryptographic Hardware and Embedded Systems (CHES '05), Springer-Verlag, 2005.- [11] K. Gaj, S. Kwon, P. Baier, P. Kohlbrenner, H. Le, M. Khaleeluddin, and R. Bachimanchi, "Implementing the Elliptic Curve Method of Factoring in Reconfigurable Hardware,"
Proc. Special Purpose Hardware for Attacking Cryptographic Systems (SHARCS '06), Apr. 2006.- [12] T. Güneysu, C. Paar, G. Pfeiffer, and M. Schimmler, "Enhancing COPACOBANA for Advanced Applications in Cryptography and Cryptanalysis,"
Int'l Conf. Field Programmable Logic and Applications (FPL '08), pp. 675-678, Sept. 2008.- [13] W. Geiselmann, F. Januszewski, H. Koepfer, J. Pelzl, and R. Steinwandt, "A Simpler Sieving Device: Combining ECM and TWIRL," Proc. Int'l Conf. Information Security and Cryptology (ICISC), 2006.
- [14] R. Golliver, A.K. Lenstra, and K. McCurley, "Lattice Sieving and Trial Division,"
Proc. First Int'l Symp. Algorithmic Number Theory (ANTS I Conf.), pp. 18-27, 1994.- [15] I. Kuon and J. Rose, "Measuring the Gap between FPGAs and ASICs,"
IEEE Trans. Computer-Aided Design of Integrated Circuits and Systems, vol. 62, no. 2, pp. 203-215, Feb. 2007.- [16] H.W. Lenstra, "Factoring Integers with Elliptic Curves,"
Ann. of Math., vol. 126, pp. 649-673, 1987.- [17] A.K. Lenstra and H.W. Lenstra,
The Development of the Number Field Sieve. Springer, 1993.- [18] A.K. Lenstra and H.W. LenstraJr., "Algorithms in Number Theory,"
Handbook of Theoretical Computer Science, Volume A, Algorithms and Complexity, J. van Leeuwen, ed., chapter 12, pp. 673-715, Elsevier, 1990.- [19] C. McIvor, M. McLoone, J. McCanny, A. Daly, and W. Marnane, "Fast Montgomery Modular Multiplication and RSA Cryptographic Processor Architectures,"
Proc. 37th IEEE CS Asilomar Conf. Signals, Systems and Computers, pp. 379-384, Nov. 2003.- [20] P.L. Montgomery, "Modular Multiplication without Trivial Division,"
Math. of Computation, vol. 44, pp. 519-521, 1985.- [21] P.L. Montgomery, "Speeding the Pollard and Elliptic Curve Methods of Factorization,"
Math. of Computation, vol. 48, pp. 243-264, 1987.- [22] P.L. Montgomery, "An FFT Extension of the Elliptic Curve Method of Factorization," PhD thesis, UCLA, 1992.
- [23] G. de Meulenaer, F. Gosset, G.M. de Dormale, and J.-J. Quisquater, "Integer Factorization Based on Elliptic Curve Method: Towards Better Exploitation of Reconfigurable Hardware,"
15th Ann. IEEE Symp. Field-Programmable Custom Computing Machines (FCCM '07), pp. 197-206, Apr. 2007.- [24] J. Pelzl, J. Šimka, T. Kleinjung, J. Franke, C. Priplata, C. Stahlke, M. Drutarovsky, V. Fischer, and C. Paar, "Area-Time Efficient Hardware Architecture for Factoring Integers with the Elliptic Curve Method,"
IEE Proc. Information Security, vol . 152, no. 1, pp. 67-78, 2005.- [25] J.M. Pollard, "Factoring with Cubic Integers,"
The Development of the Number Field Sieve, pp. 4-10, Springer, 1993.- [26] R.D. Silverman and S.S. Wagstaff, "A Practical Analysis of the Elliptic Curve Factoring Algorithm,"
Math. of Computation, vol. 61, no. 203, pp. 465-462, 1993.- [27] M. Šimka, J. Pelzl, T. Kleinjung, J. Franke, C. Priplata, C. Stahlke, M. Drutarovsky, V. Fischer, and C. Paar, "Hardware Factorization Based Elliptic Curve Method,"
IEEE Symp. Field-Programmable Custom Computing Machines (FCCM '05), 2005.- [28] SRC Computers, Inc., http:/www.srccomp.com, 2010.
- [29] Synopsys 90 nm Generic Library for Teaching IC Design, http://www.synopsys.com/Community/UniversityProgram/ Pages Library.aspx, 2010.
- [30] C.D. Walter, "Precise Bounds for Montgomery Modular Multiplication and Some Potentially Insecure RSA Moduli,"
Topics in Cryptology—CT-RSA—The Cryptographers' Track at the RSA Conf., pp. 30-39, 2002.- [31] P. Zimmermann and B. Dodson, "20 years of ECM,"
Proc. Seventh Algorithmic Number Theory Symp. (ANTS VII), pp. 525-542, 2006. |