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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 N {\displaystyle N} , Shor's algorithm runs in polynomial time , meaning the time taken is polynomial in log N {\displaystyle \log ...
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.
[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 ...
The empirical average-case complexity (time vs. problem size) of such algorithms can be surprisingly low. An example is the simplex algorithm in linear programming, which works surprisingly well in practice; despite having exponential worst-case time complexity, it runs on par with the best known polynomial-time algorithms. [27]
Graphs of functions commonly used in the analysis of algorithms, showing the number of operations versus input size for each function. The following tables list the computational complexity of various algorithms for common mathematical operations.
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 ...
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 ...