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What is the time to complexity to solve Discrete log problem now in $Z_p$?

Initially for $n$ bit prime $p$, it was $Exp(n^{1/3})$.

user15864
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These days the fastest general method to solve discrete logarithms modulo primes is the number field sieve, which has the asymptotic complexity

$$ e^{(1.92+o(1)) (\log p)^{1/3} (\log\log p)^{2/3}} $$

Samuel Neves
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  • Note that this is a heuristic asymptotic complexity, not a proven one. – Reid Jan 13 '14 at 20:42
  • You're right, of course. Given it's little-oh, that factor can actually be any function, as long as it becomes insignificant at infinity input sizes. – Samuel Neves Jan 13 '14 at 20:45
  • Thanks for fixing the formula (save for the dots after 1.92). Notice that the little-oh in the exponent makes it impossible to give a big-Oh expression for the complexity. If that $o(1)$ is taken to be $0.01$, the effort is raised by $37%$ for $1024$-bit $p$, and $52%$ for $2048$-bit $p$. – fgrieu Jan 14 '14 at 15:40