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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]
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.
Pollard gives the time complexity of the algorithm as (), using a probabilistic argument based on the assumption that acts pseudorandomly. Since a , b {\displaystyle a,b} can be represented using O ( log b ) {\displaystyle O(\log b)} bits, this is exponential in the problem size (though still a significant improvement over the trivial brute ...
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]
Dedicated to the discrete logarithm in (/) where is a prime, index calculus leads to a family of algorithms adapted to finite fields and to some families of elliptic curves. The algorithm collects relations among the discrete logarithms of small primes, computes them by a linear algebra procedure and finally expresses the desired discrete ...
ECC2K-108, involving taking a discrete logarithm on a Koblitz curve over a field of 2 108 elements. The prize was awarded on 4 April 2000 to a group of about 1300 people represented by Robert Harley. They used a parallelized Pollard rho method with speedup. ECC2-109, involving taking a discrete logarithm on a curve over a field of 2 109 ...
There is no known deterministic algorithm that runs in polynomial time for finding such a . However, if the generalized Riemann hypothesis is true, there exists a quadratic nonresidue z < 2 ln 2 p {\displaystyle z<2\ln ^{2}{p}} , [ 7 ] making it possible to check every z {\displaystyle z} up to that limit and find a suitable z ...