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The discrete logarithm problem is considered to be computationally intractable. That is, no efficient classical algorithm is known for computing discrete logarithms in general. A general algorithm for computing log b a in finite groups G is to raise b to larger and larger powers k until the desired a is found.
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]
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 hidden subgroup problem is especially important in the theory of quantum computing for the following reasons.. Shor's algorithm for factoring and for finding discrete logarithms (as well as several of its extensions) relies on the ability of quantum computers to solve the HSP for finite abelian groups.
It is a major open problem to find an algorithm for integer factorization that runs in time polynomial in the size of representation (). The security of many cryptographic protocols rely on the assumption that integer factorization is hard (i.e. cannot be solved in polynomial time).
Polynomial-time algorithms are closed under composition. Intuitively, this says that if one writes a function that is polynomial-time assuming that function calls are constant-time, and if those called functions themselves require polynomial time, then the entire algorithm takes polynomial time. One consequence of this is that P is low for itself.
That is, assuming a solution for H takes 1 unit time, H ' s solution can be used to solve L in polynomial time. [1] [2] As a consequence, finding a polynomial time algorithm to solve a single NP-hard problem would give polynomial time algorithms for all the problems in the complexity class NP.
This team was able to compute discrete logarithms in GF(2 30750) using 25,481,219 core hours on clusters based on the Intel Xeon architecture. This computation was the first large-scale example using the elimination step of the quasi-polynomial algorithm. [9] Previous records in a finite field of characteristic 2 were announced by: