TOPICS
Search

Number Field Sieve

An extremely fast factorization method developed by Pollard which was used to factor the RSA-130 number. This method is the most powerful known for factoring general numbers, and has complexity

O{exp[c(logn)^(1/3)(loglogn)^(2/3)]},
(1)

reducing the exponent over the continued fraction factorization algorithm and quadratic sieve. There are three values of c relevant to different flavors of the method (Pomerance 1996). For the "special" case of the algorithm applied to numbers near a large power,

c=((32)/9)^(1/3)=1.526285...,
(2)

for the "general" case applicable to any odd positive number which is not a power,

c=((64)/9)^(1/3)=1.922999...,
(3)

and for a version using many polynomials (Coppersmith 1993),

c=1/3(92+26sqrt(13))^(1/3)=1.901883....
(4)

See also

General Number Field Sieve, Quadratic Sieve, RSA Number

Explore with Wolfram|Alpha

References

Coppersmith, D. "Modifications to the Number Field Sieve." J. Cryptology 6, 169-180, 1993.Coppersmith, D.; Odlyzko, A. M.; and Schroeppel, R. "Discrete Logarithms in GF(p)." Algorithmics 1, 1-15, 1986.Cowie, J.; Dodson, B.; Elkenbracht-Huizing, R. M.; Lenstra, A. K.; Montgomery, P. L.; Zayer, J. A. "World Wide Number Field Sieve Factoring Record: On to 512 Bits." In Advances in Cryptology--ASIACRYPT '96 (Kyongju) (Ed. K. Kim and T. Matsumoto.) New York: Springer-Verlag, pp. 382-394, 1996.Elkenbracht-Huizing, R.-M. "A Multiple Polynomial General Number Field Sieve." Algorithmic Number Theory (Talence, 1996). New York: Springer-Verlag, pp. 99-114, 1996.Elkenbracht-Huizing, R.-M. "An Implementation of the Number Field Sieve." Experiment. Math. 5, 231-253, 1996.Elkenbracht-Huizing, R.-M. "Historical Background of the Number Field Sieve Factoring Method." Nieuw Arch. Wisk. 14, 375-389, 1996.Elkenbracht-Huizing, R.-M. Factoring Integers with the Number Field Sieve. Doctor's Thesis, Leiden University, 1997.Lenstra, A. K. and Lenstra, H. W. Jr. "Algorithms in Number Theory." In Handbook of Theoretical Computer Science, Volume A: Algorithms and Complexity (Ed. J. van Leeuwen). New York: Elsevier, pp. 673-715, 1990.Lenstra, A. K. and Lenstra, H. W. Jr. The Development of the Number Field Sieve. Berlin: Springer-Verlag, 1993.Pomerance, C. "A Tale of Two Sieves." Not. Amer. Math. Soc. 43, 1473-1485, 1996.

Referenced on Wolfram|Alpha

Number Field Sieve

Cite this as:

Weisstein, Eric W. "Number Field Sieve." From MathWorld--A Wolfram Web Resource. https://mathworld.wolfram.com/NumberFieldSieve.html

Subject classifications