Limitations of Algorithm
This algorithm, as mentioned above, is very efficient for numbers of the form re±s, for r and s relatively small. It is also efficient for any integers which can be represented as a polynomial with small coefficients. This includes integers of the more general form a're±b'sf, and also for many integers whose binary representation has low Hamming weight. The reason for this is as follows: The Number Field Sieve performs sieving in two different fields. The first field is usually the rationals. The second is a higher degree field. The efficiency of the algorithm strongly depends on the norms of certain elements in these fields. When an integer can be represented as a polynomial with small coefficients, the norms that arise are much smaller than those that arise when an integer is represented by a general polynomial. The reason is that a general polynomial will have much larger coefficients, and the norms will be correspondingly larger. The algorithm attempts to factor these norms over a fixed set of prime numbers. When the norms are smaller, these numbers are more likely to factor.
Read more about this topic: Special Number Field Sieve
Famous quotes containing the words limitations of and/or limitations:
“The motion picture made in Hollywood, if it is to create art at all, must do so within such strangling limitations of subject and treatment that it is a blind wonder it ever achieves any distinction beyond the purely mechanical slickness of a glass and chromium bathroom.”
—Raymond Chandler (18881959)
“That all may be so, but when I begin to exercise that power I am not conscious of the power, but only of the limitations imposed on me.”
—William Howard Taft (18571930)