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For example, log 10 10000 = 4, and log 10 0.001 = −3. These are instances of the discrete logarithm problem. Other base-10 logarithms in the real numbers are not instances of the discrete logarithm problem, because they involve non-integer exponents. For example, the equation log 10 53 = 1.724276… means that 10 1.724276… = 53.
The discrete logarithm algorithm and the factoring algorithm are instances of the period-finding algorithm, and all three are instances of the hidden subgroup problem. On a quantum computer, to factor an integer , Shor's algorithm runs in polynomial time, meaning the time taken is polynomial in . [6]
For arbitrary groups, it is known that the hidden subgroup problem is solvable using a polynomial number of evaluations of the oracle. [3] However, the circuits that implement this may be exponential in log | G | {\displaystyle \log |G|} , making the algorithm not efficient overall; efficient algorithms must be polynomial in the number of ...
In computer science, polylogarithmic functions occur as the order of time for some data structure operations. Additionally, the exponential function of a polylogarithmic function produces a function with quasi-polynomial growth, and algorithms with this as their time complexity are said to take quasi-polynomial time. [2]
Here, "quickly" means an algorithm that solves the task and runs in polynomial time (as opposed to, say, exponential time) exists, meaning the task completion time is bounded above by a polynomial function on the size of the input to the algorithm. The general class of questions that some algorithm can answer in polynomial time is "P" or "class ...
The third stage searches for a power s of the generator g which, when multiplied by the argument h, may be factored in terms of the factor base g s h = (−1) f 0 2 f 1 3 f 2 ···p r f r. Finally, in an operation too simple to really be called a fourth stage, the results of the second and third stages can be rearranged by simple algebraic ...
[2] [3] Some problems that are considered good candidates for being NP-intermediate are the graph isomorphism problem, and decision versions of factoring and the discrete logarithm. Under the exponential time hypothesis, there exist natural problems that require quasi-polynomial time, and can be solved in that time, including finding a large ...
Inverting this function requires computing the discrete logarithm. Currently there are several popular groups for which no algorithm to calculate the underlying discrete logarithm in polynomial time is known. These groups are all finite abelian groups and the general discrete logarithm problem can be described as thus.