Integer Factorization of Semi-Primes Based on Analysis of a Sequence of Modular Elliptic Equations

References

[1] R. Crandall and C. Pomerance, “Prime Numbers: A Computational Perspective,” Springer, New York, 2001.

[2] H. Cohen, “A Course in Computational Algebraic Number Theory,” Springer-Verlag, New York, 1996.

[3] D. Shanks, “Class Number, a Theory of Factorization and Genera,” Proceedings of Symposium of Pure Mathematics, Vol. 20, 1969, pp. 415-440.

[4] S. Lehman, “Factoring Large Integers,” Mathematics of Computation, Vol. 28, 1974, pp. 637-646.
doi:10.1090/S0025-5718-1974-0340163-2

[5] J. Pollard, “Theorems on Factorization and Primality Testing,” Mathematical Proceedings of the Cambridge Philosophical Society, Vol. 76, 1974, pp. 521-528.
doi:10.1017/S0305004100049252

[6] J. Pollard, “Factoring with Cubic Integers,” The Development of the Number Field Sieve, Lecture Notes in Mathematics, Vol. 1554, 1993, pp. 4-10.
doi:10.1007/BFb0091536

[7] C. Pomerance, “Analysis and Comparison of Some Integer Factoring Algorithms,” In: H. W. Lenstra and R. Tijdeman, Eds., Computational Methods in Number Theory, Math Centre Tracts—Part 1, Math Centrum, Amsterdam, 1982, pp. 89-139.

[8] C. Pomerance, “The Quadratic Sieve Factoring Algorithm,” Advances in Cryptology, Proceedings of Eurocrypt’84, LNCS, Springer-Verlag, Berlin, 1985, 169-182.

[9] R. D. Silverman, “The Multiple Polynomial Quadratic Sieve,” Mathematics of Computation, Vol. 48, 1987, pp. 329-339. doi:10.1090/S0025-5718-1987-0866119-8

[10] J. Buhler, H. W. Lenstra and C. Pomerance, “Factoring Integers with the Number Field Sieve,” In: A. K. Lenstra and H. W. Lenstra, Eds., The Development of the Number Field Sieve, Lecture Notes in Mathematics, Springer-Verlag, Berlin, Vol. 1554, 1993, pp. 50-94.
doi:10.1007/BFb0091539

[11] A. K. Lenstra and A. Shamir, “Analysis and Optimization of the TWINKLE Factoring Device,” Advances in Cryptology—EUROCRYPT 2000, Lecture Notes in Computer Science, Springer-Verlag, New York, Vol. 1807, 2000, pp. 35-52.

[12] A. Shamir and E. Tromer, “Factoring Large Numbers with the TWIRL Device,” Advances in Cryptology— CRYPTO 2003, Lecture Notes in Computer Science, Springer-Verlag, New York, Vol. 2729, 2003, pp. 1-26.

[13] P. W. Shor, “Polynomial-Time Algorithms for Prime Fac- torization and Discrete Logarithms on a Quantum Com- puter,” SIAM Journal on Computing, Vol. 26, No. 5, 1997, pp. 1484-1509. doi:10.1137/S0097539795293172

[14] R. P. Brent, “Some Integer Factorization Algorithms Using Elliptic Curves,” Proceedings of 9th Australian Computer Science Conference, Canberra, January 1985.

[15] H. W. Lenstra Jr., “Factoring Integers with Elliptic Cur- ves,” Annals of Mathematics, Vol. 126, No. 2, 1987, pp. 649-673. doi:10.2307/1971363

[16] P. L. Montgomery, “A FFT Extension of the Elliptic Curve Method of Factorization,” PhD Thesis, University of California, Los Angeles, 1992.

[17] R. Schoof, “Counting Points of Elliptic Curves over Finite Fields,” Journal de Théorie des Nombres de Bor- deaux, Vol. 7, No. 1, 1995, pp. 219-254.
doi:10.5802/jtnb.142

[18] R. Lencier, D. Lubicz and F. Vercauteren, “Point Counting on Elliptic and Hyperelliptic Curves,” In: H. Cohen and G. Frey, Eds., Handbook of Elliptic and Hyperelliptic Curve Cryptography, Chapman & Hall/CRC, Boca Raton, 2006, pp. 407-453.

[19] A. G. B. Lauder and D. Wan, “Counting Points on Varieties over Finite Fields of Small Characteristics,” In: J. P. Buhler and P. Stevenhagen, Eds., Algorithmic Number Theory, Cambridge University Press, Cambridge, 2008, pp. 579-612.

[20] A. Weil, “Number of Solutions of Equations in Finite Fields,” Bulletin of American Mathematical Society, Vol. 55, 1949, pp. 497-508.
doi:10.1090/S0002-9904-1949-09219-4

[21] A. G. B. Lauder, “Counting Solutions to Equations in Many Variables over Finite Fields,” Foundation of Com- putational Mathematics, Vol. 4, No. 3, 2004, pp. 221-267.
doi:10.1007/s10208-003-0093-y

[22] Boris S. Verkhovsky, “Integer Factorization: Solution via Algorithm for Constrained Discrete Logarithm Problem,” Journal of Computer Science, Vol. 5, No. 9, 2009, pp. 674-679. doi:10.3844/jcssp.2009.674.679

[23] “RSA Factoring Challenge,”
http://en.wikipedia.org/wiki/RSA_Factoring_Challenge