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2. Integer relation algorithm - Wikipedia

   en.wikipedia.org/wiki/Integer_relation_algorithm

   An integer relation algorithm is an algorithm for finding integer relations. Specifically, given a set of real numbers known to a given precision, an integer relation algorithm will either find an integer relation between them, or will determine that no integer relation exists with coefficients whose magnitudes are less than a certain upper bound .

3. Lattice reduction - Wikipedia

   en.wikipedia.org/wiki/Lattice_reduction

   LLL is widely used in the cryptanalysis of public key cryptosystems. When used to find integer relations, a typical input to the algorithm consists of an augmented identity matrix with the entries in the last column consisting of the elements (multiplied by a large positive constant to penalize vectors that do not sum to zero) between which the ...

4. Lenstra–Lenstra–Lovász lattice basis reduction algorithm

   en.wikipedia.org/wiki/Lenstra–Lenstra–Lovász...

   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.

5. General number field sieve - Wikipedia

   en.wikipedia.org/wiki/General_number_field_sieve

   This results in many rather complicated aspects of the algorithm, as compared to the simpler rational sieve. The size of the input to the algorithm is log 2 n or the number of bits in the binary representation of n. Any element of the order n c for a constant c is exponential in log n.

6. Euclidean algorithm - Wikipedia

   en.wikipedia.org/wiki/Euclidean_algorithm

   The Euclidean algorithm was the first integer relation algorithm, which is a method for finding integer relations between commensurate real numbers. Several novel integer relation algorithms have been developed, such as the algorithm of Helaman Ferguson and R.W. Forcade (1979) [ 49 ] and the LLL algorithm .

7. Mersenne Twister - Wikipedia

   en.wikipedia.org/wiki/Mersenne_Twister

   The 'Extract number' section shows an example where integer 0 has already been output and the index is at integer 1. 'Generate numbers' is run when all integers have been output. For a w -bit word length, the Mersenne Twister generates integers in the range [ 0 , 2 w − 1 ] {\displaystyle [0,2^{w}-1]} .