The Number Field Sieve Factoring Algorithm
نویسنده
چکیده
The idea of the NFS is rooted in work by Coppersmith, Odlyzko, and Schroeppel [9], who used the Gaussian integers, Z[i], to compute discrete logarithms in GF (p) in subexponential time. They looked for coprime integers, a and b such that the integer a+bx is smooth for some fixed integer value of x and over some factor base, and such that a+ bi is smooth over some factor base of primes in Z[i]. In 1988, Pollard [15] generalized this idea to factor integers of the form x − k, where |k| is small, using the cubic number field Q( 3 √ k) and (a subset of) its ring of integers Z[ 3 √ −k]. This method was initially tested by factoring the 7th Fermat number, F7 = 2 128 + 1. F7 does not fit the desired form, so he instead factored 2F7 = x 3 + 2, where x = 2, into nontrivial factors. Note that this was not the first time F7 had been factored, since it was first factored in 1970 by Morrison and Brillhart using the continued fraction method.
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