If one should attempt to factorize a number like the RSA-2048, or in general any number with $n$ decimal digits, using the best algorithm available and a modern desktop PC, what is the approximate length in time it would take (as a function of $n$) ? I'd like a general formula (possible parameterized with CPU speed and/or # of cpu's) so I can apply it to other numbers and PCs.
Thanks
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As Brandon Carter said, the Number Field Sieve is the fastest known algorithm for factoring numbers that are at the limits of our capability. But it won't run on a modern desktop PC, because the matrix step at the end requires too much RAM. More precisely, for numbers that are small enough to factor on a PC using the Number Field Sieve, the Multiple Polynomial Quadratic Sieve is faster.
But in any case RSA-2048 is beyond reach. It's like interstellar travel: any program that you launch today to factor RSA-2048 will inevitably be beaten by a program using a better algorithm, discovered some time in the next 100 years.
The number field sieve algorithm is generally the most efficient factoring algorithm and has a running time of $$O(\exp(c \sqrt[3]{(\log N)(\log \log N)^2}))$$ but I don't think it feasible or meaningful to derive a formula for the absolute time of computation.