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For the case n = 2, an extension of the Euclidean algorithm can find any integer relation that exists between any two real numbers x 1 and x 2.The algorithm generates successive terms of the continued fraction expansion of x 1 /x 2; if there is an integer relation between the numbers, then their ratio is rational and the algorithm eventually terminates.
An early successful application of the LLL algorithm was its use by Andrew Odlyzko and Herman te Riele in disproving Mertens conjecture. [5]The LLL algorithm has found numerous other applications in MIMO detection algorithms [6] and cryptanalysis of public-key encryption schemes: knapsack cryptosystems, RSA with particular settings, NTRUEncrypt, and so forth.
Pollard's rho algorithm is an algorithm for integer factorization. It was invented by John Pollard in 1975. [ 1 ] It uses only a small amount of space, and its expected running time is proportional to the square root of the smallest prime factor of the composite number being factorized.
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.
Using repeated squaring, the running time of this algorithm is O(k n 3), for an n-digit number, and k is the number of rounds performed; thus this is an efficient, polynomial-time algorithm. FFT -based multiplication, for example the Schönhage–Strassen algorithm , can decrease the running time to O( k n 2 log n log log n ) = Õ ( k n 2 ) .
These algorithms can also be used for mixed integer linear programs (MILP) - programs in which some variables are integer and some variables are real. [23] The original algorithm of Lenstra [ 14 ] : Sec.5 has run-time 2 O ( n 3 ) ⋅ p o l y ( d , L ) {\displaystyle 2^{O(n^{3})}\cdot poly(d,L)} , where n is the number of integer variables, d is ...
To find the needed , , , and the algorithm uses Floyd's cycle-finding algorithm to find a cycle in the sequence =, where the function: + is assumed to be random-looking and thus is likely to enter into a loop of approximate length after steps.
Pollard's p − 1 algorithm is a number theoretic integer factorization algorithm, invented by John Pollard in 1974. It is a special-purpose algorithm, meaning that it is only suitable for integers with specific types of factors; it is the simplest example of an algebraic-group factorisation algorithm .