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by jedwards1211 521 days ago
The wiki article on GNFS says:

> The size of the input to the algorithm is log2 n or the number of bits in the binary representation of n. Any element of the order nc for a constant c is exponential in log n. The running time of the number field sieve is super-polynomial but sub-exponential in the size of the input.

"Super-polynomial" means that the running time increases by more than the square.

In any case, even if the algorithm were just polynomial, the argument about squaring costs doesn't work out.