The function field sieve
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(pn). 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) ≤ n9(n) where g is any function such that g: N → ℜ >0 <.98 approaches zero as n → ∞.
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- [Ad1]Adleman L.M., Factoring numbers using singular integers, Proc. 23rd Annual ACM Symposium on Theory of Computing, 1991, pp. 64–71.Google Scholar
- [Be1]Berlekamp E., Factoring polynomials over large finite fields. Math. Comp. 24, 1970. pp. 713–735.Google Scholar
- [CF1]Cassels J.W.S. and Fröhlich A., Algebraic Number Theory, Thompson Book Company, Washington, D.C. 1967.Google Scholar
- [Co1]Coppersmith D., Modifications to the Number Field Sieve. IBM Research Report #RC 16264, November, 1990.Google Scholar
- [Co2]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.Google Scholar
- [Fu1]Fulton W., Algebraic Curves, The Benjamin/Cummings Publishing Company, Menlo Park, 1969.Google Scholar
- [Go1]Gordon D.M., Discrete logarithms in GF(p) using the number field sieve, manuscript, April 4, 1990.Google Scholar
- [Go2]Gordon D.M., Discrete logarithms in GF(p n) using the number field sieve (preliminary version), manuscript, November 29, 1990.Google Scholar
- [Ha1]Hasse H., Number Theory. English Translation by H. Zimmer. Springer-Verlag, Berlin. 1980.Google Scholar
- [Ka1]Kalofen E., Fast parallel absolute irreducibility testing. J. Symbolic Computation 1, 1985, pp. 57–67.Google Scholar
- [LL1]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.Google Scholar
- [Lo1]Lovorn R., Rigorous Subexponential Algorithms For Discrete Logarithms Overr Finite Fields. Ph.D. Thesis. Universiy of Georgia, Athens, Georgia. 1992.Google Scholar
- [Wi1]Wiedermann D. Solving sparse linear equations over finite fields. IEEE Trans. Inform. Theory. IT-32, pp. 54–62Google Scholar