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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 N {\displaystyle N} , Shor's algorithm runs in polynomial time , meaning the time taken is polynomial in log N {\displaystyle \log ...
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:
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.
If a deterministic polynomial time algorithm A computes the discrete logarithm for a 1/poly(n) fraction of all inputs (where n = log |G| is the input size), then there is a randomized polynomial time algorithm for discrete logarithm for all inputs.
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]
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.
The set S 0 of initial states can be computed in time polynomial in n and log(X). Let V j be the set of all values that can appear in coordinate j in a state. Then, the ln of every value in V j is at most a polynomial P 1 (n,log(X)). If d j =0, the cardinality of V j is at most a polynomial P 2 (n,log(X)).