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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 ...
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]
It runs in polynomial time on inputs that are in SUBSET-SUM if and only if P = NP: // Algorithm that accepts the NP-complete language SUBSET-SUM. // // this is a polynomial-time algorithm if and only if P = NP. // // "Polynomial-time" means it returns "yes" in polynomial time when // the answer should be "yes", and runs forever when it is "no".
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)).
Shor's algorithm solves the discrete logarithm problem and the integer factorization problem in polynomial time, [9] whereas the best known classical algorithms take super-polynomial time. It is unknown whether these problems are in P or NP-complete. It is also one of the few quantum algorithms that solves a non-black-box problem in polynomial ...
Using these proofs, the prover can not only prove the knowledge of the discrete logarithm, but also that the discrete logarithm is of a specific form. For instance, it is possible to prove that two logarithms of y 1 {\displaystyle y_{1}} and y 2 {\displaystyle y_{2}} with respect to bases g 1 {\displaystyle g_{1}} and g 2 {\displaystyle g_{2 ...
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: