Choice of Parameters
Not every number is an appropriate choice for the SNFS: you need to know in advance a polynomial f of appropriate degree (the optimal degree is conjectured to be, which is 4, 5, or 6 for the sizes of N currently feasible to factorise) with small coefficients, and a value x such that where N is the number to factorise. There is an extra condition: x must satisfy for a and b no bigger than .
One set of numbers for which such polynomials exist are the numbers from the Cunningham tables; for example, when NFSNET factored 3^479+1, they used the polynomial x^6+3 with x=3^80, since (3^80)^6+3 = 3^480+3, and .
Numbers defined by linear recurrences, such as the Fibonacci and Lucas numbers, also have SNFS polynomials, but these are a little more difficult to construct. For example, has polynomial, and the value of x satisfies .
If you already know some factors of a large SNFS-number, you can do the SNFS calculation modulo the remaining part; for the NFSNET example above, 3^479+1 = (4*158071*7167757*7759574882776161031) times a 197-digit composite number (the small factors were removed by ECM), and the SNFS was performed modulo the 197-digit number. The number of relations required by SNFS still depends on the size of the large number, but the individual calculations are quicker modulo the smaller number.
Read more about this topic: Special Number Field Sieve
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