Special Number Field Sieve - Choice of Parameters

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.