Abstract
The fastest method known for factoring integers is the ‘number field sieve’. An analogous method over function fields is developed, the ‘function field sieve’, and applied to calculating discrete logarithms over GF(p n). An heuristic analysis shows that there exists a cε ℜ>0 such that the function field sieve computes discrete logarithms within random time: L p n[1/3, c] when log(p) ≤ n 9(n) where g is any function such that g: N → ℜ <.98>0 approaches zero as n → ∞.
Preview
Unable to display preview. Download preview PDF.
Similar content being viewed by others
References
Adleman L.M., Factoring numbers using singular integers, Proc. 23rd Annual ACM Symposium on Theory of Computing, 1991, pp. 64–71.
Berlekamp E., Factoring polynomials over large finite fields. Math. Comp. 24, 1970. pp. 713–735.
Cassels J.W.S. and Fröhlich A., Algebraic Number Theory, Thompson Book Company, Washington, D.C. 1967.
Coppersmith D., Modifications to the Number Field Sieve. IBM Research Report #RC 16264, November, 1990.
Coppersmith D., Fast Evaluation of Logarithms in Fields of Characteristic Two. IEEE Trans on Information Theory, vol IT-30, No 4, July 1984, pp. 587–594.
Fulton W., Algebraic Curves, The Benjamin/Cummings Publishing Company, Menlo Park, 1969.
Gordon D.M., Discrete logarithms in GF(p) using the number field sieve, manuscript, April 4, 1990.
Gordon D.M., Discrete logarithms in GF(p n) using the number field sieve (preliminary version), manuscript, November 29, 1990.
Hasse H., Number Theory. English Translation by H. Zimmer. Springer-Verlag, Berlin. 1980.
Kalofen E., Fast parallel absolute irreducibility testing. J. Symbolic Computation 1, 1985, pp. 57–67.
A.K. Lenstra and H.W. Lenstra Jr. (Eds.), The development of the number field sieve, Lecture Notes in Mathematics 1554, Springer-Verlag, Berlin. 1993.
Lovorn R., Rigorous Subexponential Algorithms For Discrete Logarithms Overr Finite Fields. Ph.D. Thesis. Universiy of Georgia, Athens, Georgia. 1992.
Wiedermann D. Solving sparse linear equations over finite fields. IEEE Trans. Inform. Theory. IT-32, pp. 54–62
Author information
Authors and Affiliations
Editor information
Rights and permissions
Copyright information
© 1994 Springer-Verlag Berlin Heidelberg
About this paper
Cite this paper
Adleman, L.M. (1994). The function field sieve. In: Adleman, L.M., Huang, MD. (eds) Algorithmic Number Theory. ANTS 1994. Lecture Notes in Computer Science, vol 877. Springer, Berlin, Heidelberg. https://doi.org/10.1007/3-540-58691-1_48
Download citation
DOI: https://doi.org/10.1007/3-540-58691-1_48
Published:
Publisher Name: Springer, Berlin, Heidelberg
Print ISBN: 978-3-540-58691-3
Online ISBN: 978-3-540-49044-9
eBook Packages: Springer Book Archive