The special function field sieve

SIAM Journal on Discrete Mathematics

Volumn 16, Issue 1, 2003, Pages 81-98

  Schirokauer, Oliver ^a  

^a OBERLIN COLLEGE   (United States)

Author keywords

Discrete logarithm; Finite field; Function field sieve

Indexed keywords

ALGORITHMS; ASYMPTOTIC STABILITY; COMPUTATIONAL COMPLEXITY; CONFORMAL MAPPING; CONVERGENCE OF NUMERICAL METHODS; CRYPTOGRAPHY; MATRIX ALGEBRA; MAXIMUM LIKELIHOOD ESTIMATION; POLYNOMIALS; SECURITY OF DATA; VECTORS;

DISCRETE LOGARITHM; FINITE FIELD; FUNCTION FIELD SIEVE;

FUNCTION EVALUATION;

EID: 0037285628     PISSN: 08954801     EISSN: None     Source Type: Journal    
DOI: 10.1137/S0895480100372668     Document Type: Article

References (31)

1. L.M. Adleman, The function field sieve, in Algorithmic Number Theory, ANTS-I, Lecture Notes in Comput. Sci. 877, L.M. Adleman and M.-D. Huang, eds., Springer-Verlag, Berlin, 1994, pp. 108-121.
2. L.M. Adleman and M-D. Huang, Function field method for discrete logarithms over finite fields, Inform. and Comput., 151 (1999), pp. 5-16.
3. R.L. Bender and C. Pomerance, Rigorous Discrete Logarithm Computations in Finite Fields via Smooth Polynomials, AMS/IP Stud. Adv. Math., 7, AMS, Providence, RI, 1998, pp. 221-232.
4. E. Berlekamp, Factoring polynomials over large finite fields, Math. Comp., 25 (1970), pp. 713-735.
5. J.P. Buhler, H.W. Lenstra, Jr., and C. Pomerance, Factoring integers with the number field sieve, in The Development of the Number Field Sieve, Lecture Notes in Math. 1554, Springer-Verlag, Berlin, 1993, pp. 50-94.
6. D. Coppersmith, Fast evaluation of discrete logarithms in fields of characteristic two, IEEE Trans. Inform. Theory, 30 (1984), pp. 587-594
7. D.M. Gordon, Discrete logarithms in GF(P) using the number field sieve, SIAM J. Discrete Math. 6 (1993), pp. 124-138.
8. A. Joux and R. Lercier, Improvements on the General Number Field Sieve for Discrete Logarithms in Prime Fields, preprint.
9. A. Joux and R. Lercier, The function field sieve is quite special, in Algorithmic Number Theory, ANTS-V, Lecture Notes in Comput. Sci. 2369, C. Fieker and D.R. Kohel, eds., Springer-Verlag, Berlin, 2002, pp. 131-445.
10. A.K. Lenstra and H. W. Lenstra, Jr., eds., The Development of the Number Field Sieve, Lecture Notes in Math. 1554, Springer-Verlag, Berlin, 1993.
11. A.K. Lenstra, H.W. Lenstra, Jr., M.S. Manasse, and J.M. Pollard, The number field sieve, in The Development of the Number Field Sieve, Lecture Notes in Math. 1554, Springer-Verlag, Berlin, 1993, pp. 11-42.
12. A.K. Lenstra, H.W. Lenstra, Jr., M.S. Manasse, and J.M. Pollard, The factorization of the ninth Fermat number, Math Comp., 61 (1993), pp. 319-349.
13. D. Lorenzini, An Invitation to Arithmetic Geometry, Grad. Stud. Math. 9, AMS, Providence, RI, 1996.
14. E. Manstavičius, Semigroup elements free of large prime factors, in New Trends in Probability and Statistics, 2: Analytic and Probabilistic Methods in Number Theory, F. Schweiger and E. Manstavicius, eds, VSP, Utrecht, The Netherlands, 1992, pp. 135-153.
15. E. Manstavičius, Remarks on elements in semigroups that are free of large prime factors, Lithuanian Math. J., 132 (1993), pp. 400-409.
16. K. McCurley, The discrete logarithm problem, in Cryptology and Computational Number Theory, Proc. Sympos. Appl. Math. 42, C. Pomerance, ed., AMS, Providence, RI, 1990, pp. 49-74.
17. R.J. McEliece, The Theory of Information and Coding: A Mathematical Framework for Communications, Encyclopedia Math. Appl. 3, Addison-Wesley, Reading, MA, 1977.
18. A.M. Odlyzko, Discrete logarithms: The past and the future, Des. Codes Cryptogr., 19 (2000), pp. 129-145.
19. D. Panario, X. Gourdon, and P. Flajolet, An analytic approach to smooth polynomials over finite fields, in Algorithmic Number Theory, ANTS-III, Lecture Notes in Comput. Sci. 1424, J. Buhler, ed., Springer-Verlag, Berlin, 1998, pp. 237-246.
20. O. Schirokauer, Discrete logarithms and local units, in Theory and Applications of Numbers Without Large Prime Factors, Philos. Trans. Roy. Soc. London Ser. A 345, R.C. Vaughan, ed., Royal Society, London, 1993, pp. 409-424.
21. O. Schirokauer, Using number fields to compute logarithms in finite fields, Math. Comp., 69 (2000), pp. 1267-1283.
22. O. Schirokauer, The impact of the number field sieve on the discrete logarithm problem, in Algorithmic Number Theory: Lattices, Number Fields, Curves, and Cryptography, Cambridge University Press, Cambridge, UK, to appear.
23. O. Schirokauer, D. Weber, and T. Denny, Discrete logarithms: The effectiveness of the index calculus method, in Algorithmic Number Theory, ANTS-II, Lecture Notes in Comput. Sci. 1123, H. Cohen, ed., Springer-Verlag, Berlin, 1996, pp. 337-361.
24. I.A. Semaev, Special prime numbers and discrete logs in prime finite fields, Math. Comp., 71 (2002), pp. 363-377.
25. V. Shoup, Searching for primitive roots in finite fields, Math. Comp., 58 (1992), pp. 369-380.
26. K. Soundararajan, Smooth Polynomials: Analogies and Asymptotics, preprint.
27. H. Stichtenoth, Algebraic Function Fields and Codes, Springer-Verlag, Berlin, 1993.
28. 607), in Advances in Cryptology - Asiacrypt 2001, Lecture Notes in Comput. Sci. 2248, C. Boyd, ed., Springer-Verlag, Berlin, 2001, pp. 107-124.
29. D. Weber, Computing discrete logarithms with the number field sieve, in Algorithmic Number Theory, ANTS-II, Lecture Notes in Comput. Sci. 1123, H. Cohen, ed., Springer-Verlag, Berlin, 1996, pp. 391-403.
30. D. Weber and T. Denny, The solution of McCurley's discrete log challenge, in Advances in Cryptology - Crypto '98, Lecture Notes in Comput. Sci. 1462, H. Krawczyk, ed., Springer-Verlag, Berlin, 1998, pp. 458-471.
31. D.H. Wiedemann, Solving sparse linear equations over finite fields, IEEE Trans. Inform. Theory, 32 (1986), pp. 54-62.