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- Algebraic-group_factorisation_algorithm abstract "Algebraic-group factorisation algorithms are algorithms for factoring an integer N by working in an algebraic group defined modulo N whose group structure is the direct sum of the 'reduced groups' obtained by performing the equations defining the group arithmetic modulo the unknown prime factors p1, p2, ... By the Chinese remainder theorem, arithmetic modulo N corresponds to arithmetic in all the reduced groups simultaneously.The aim is to find an element which is not the identity of the group modulo N, but is the identity modulo one of the factors, so a method for recognising such one-sided identities is required. In general, one finds them by performing operations that move elements around and leave the identities in the reduced groups unchanged. Once the algorithm finds a one-sided identity all future terms will also be one-sided identities, so checking periodically suffices.Computation proceeds by picking an arbitrary element x of the group modulo N and computing a large and smooth multiple Ax of it; if the order of at least one but not all of the reduced groups is a divisor of A, this yields a factorisation. It need not be a prime factorisation, as the element might be an identity in more than one of the reduced groups.Generally, A is taken as a product of the primes below some limit K, and Ax is computed by successive multiplication of x by these primes; after each multiplication, or every few multiplications, the check is made for a one-sided identity.".
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- Algebraic-group_factorisation_algorithm wikiPageRevisionID "667551874".
- Algebraic-group_factorisation_algorithm wikiPageWikiLink Algebraic_group.
- Algebraic-group_factorisation_algorithm wikiPageWikiLink Binary_exponentiation.
- Algebraic-group_factorisation_algorithm wikiPageWikiLink Category:Integer_factorization_algorithms.
- Algebraic-group_factorisation_algorithm wikiPageWikiLink Chinese_remainder_theorem.
- Algebraic-group_factorisation_algorithm wikiPageWikiLink Elliptic_curve.
- Algebraic-group_factorisation_algorithm wikiPageWikiLink Elliptic_curve_method.
- Algebraic-group_factorisation_algorithm wikiPageWikiLink Exponentiation_by_squaring.
- Algebraic-group_factorisation_algorithm wikiPageWikiLink Greatest_common_divisor.
- Algebraic-group_factorisation_algorithm wikiPageWikiLink Hasses_theorem_on_elliptic_curves.
- Algebraic-group_factorisation_algorithm wikiPageWikiLink Integer_factorization.
- Algebraic-group_factorisation_algorithm wikiPageWikiLink Inverse_function.
- Algebraic-group_factorisation_algorithm wikiPageWikiLink Lenstra_elliptic_curve_factorization.
- Algebraic-group_factorisation_algorithm wikiPageWikiLink Modular_arithmetic.
- Algebraic-group_factorisation_algorithm wikiPageWikiLink Multiplicative_group.
- Algebraic-group_factorisation_algorithm wikiPageWikiLink Pollards_p-1_algorithm.
- Algebraic-group_factorisation_algorithm wikiPageWikiLink Pollards_p_xe2x88x92_1_algorithm.
- Algebraic-group_factorisation_algorithm wikiPageWikiLink Quadratic_residue.
- Algebraic-group_factorisation_algorithm wikiPageWikiLink Smooth_number.
- Algebraic-group_factorisation_algorithm wikiPageWikiLink Williams_p_+_1_algorithm.
- Algebraic-group_factorisation_algorithm wikiPageWikiLinkText "Algebraic-group factorisation algorithm".
- Algebraic-group_factorisation_algorithm wikiPageWikiLinkText "algebraic-group factorisation algorithm".
- Algebraic-group_factorisation_algorithm hasPhotoCollection Algebraic-group_factorisation_algorithm.
- Algebraic-group_factorisation_algorithm wikiPageUsesTemplate Template:Unreferenced.
- Algebraic-group_factorisation_algorithm subject Category:Integer_factorization_algorithms.
- Algebraic-group_factorisation_algorithm hypernym Algorithms.
- Algebraic-group_factorisation_algorithm type Algorithm.
- Algebraic-group_factorisation_algorithm comment "Algebraic-group factorisation algorithms are algorithms for factoring an integer N by working in an algebraic group defined modulo N whose group structure is the direct sum of the 'reduced groups' obtained by performing the equations defining the group arithmetic modulo the unknown prime factors p1, p2, ...".
- Algebraic-group_factorisation_algorithm label "Algebraic-group factorisation algorithm".
- Algebraic-group_factorisation_algorithm sameAs m.03d8bcr.
- Algebraic-group_factorisation_algorithm sameAs Q4723979.
- Algebraic-group_factorisation_algorithm sameAs Q4723979.
- Algebraic-group_factorisation_algorithm wasDerivedFrom Algebraic-group_factorisation_algorithm?oldid=667551874.
- Algebraic-group_factorisation_algorithm isPrimaryTopicOf Algebraic-group_factorisation_algorithm.