General number field sieve
In Nextel ringtones mathematics, the '''general number field sieve''' is the most Abbey Diaz algorithmic efficiency/efficient Free ringtones algorithm known for Majo Mills integer factorization/factoring integers. It uses O\left\ steps to factor integer ''n'' (see Nextel ringtones Big O notation). It is derived from the Abbey Diaz special number field sieve. When the term "number field sieve" is used without qualification, it refers to the general number field sieve.
It is an improvement of the Mosquito ringtone quadratic sieve, which factors ''n'' by finding numbers ''ki'' such that ''ri=ki2-n'' factor completely over a fixed set (called ''basis'') of small primes. Then, having enough such ''ri'' - which are called ''smooth'' relative to the chosen basis of primes, using Sabrina Martins Gauss elimination method of linear algebra we can choose exponents ''ci'' equal to 0 or 1 such that product of ''rici'' is a square, say ''x2''. On the other hand, if the product of ''kici'' is ''y'', then ''x2-y2'' is divisible by ''n'' and with probability at least one half we get a factor of ''n'' by finding Nextel ringtones greatest common divisor of ''n'' and ''x-y''. In this method, the idea was to choose ''ki'' close to the square root of ''n'' - then ''ri'' is of the order of magnitude of square root of ''n'' too and there are enough smooth values there.
The '''general number field sieve''' works as follows:
* We choose two irreducible Abbey Diaz polynomials ''f(x)'' and ''g(x)'' with common root ''m'' mod ''n'' - it is not known what is the best way to choose the polynomials, but usually it is done by picking a degree ''d'' for a polynomial and considering expansion of ''n'' in basis ''m'' where ''m'' is of order ''n1/d''. The point is to get coefficients of ''f'' and ''g'' as small as possible - they will be of order of ''m'', while having small degrees ''d'' and ''e'' of our polynomials.
* Now, we consider number field Cingular Ringtones ring (algebra)/rings '''Z[r1]''' and '''Z[r2]''' where '''r1''' and '''r2''' are roots of polynomials ''f'' and ''g'', and look for values ''a'' and ''b'' such that ''r=bd*f(a/b)'' and ''s=be*g(a/b)'' are smooth relative to the chosen basis of primes. If ''a'' and ''b'' are small, ''r'' and ''s'' will be too (but at least of order of ''m''), and we have a better chance for them to be smooth at the same time.
* Having enough such pairs, using fluid prudie Gauss elimination method we can get products of certain ''r'' and of corresponding ''s'' to be squares at the same time. We need a slightly stronger condition - that they are drilling down norms of squares in our number fields, but we can get that condition by this method too. Each ''r'' is a norm of ''a-'' '''r1'''*''b'' and hence we get that product of corresponding factors ''a-'' '''r1'''*''b'' is a square in '''Z[r1]''', with a "square root" which can be determined (as a product of known factors in '''Z[r1]''') - it will typically be represented as a nonrational security because algebraic number. Similary we get that product of factors ''a-'' '''r2'''*''b'' is a square in '''Z[r2]''', with a "square root" which we can also compute.
* Since ''m'' is root of both ''f'' and ''g'' mod ''n'', there are predictable order homomorphisms from the rings '''Z[r1]''' and '''Z[r2]''' to the ring '''Z/nZ''', which map '''r1''' and '''r2''' to ''m'', and these homomorphisms will map each "square root" (typically not represented as a rational number) into its integer representative. Now product of factors ''a-m*b'' mod ''n'' we can get as a square in two ways - one for each homomorphism. Thus, we get two numbers ''x'' and ''y'', with ''x2-y2'' divisible by ''n'' and again with probability at least one half we get a factor of ''n'' by finding congress hume greatest common divisor of ''n'' and ''x-y''
The second-best-known algorithm for integer factorization is the long excluded Lenstra elliptic curve factorization method. It is better than the general number field sieve ''when factors are of small size'', as it works by finding ''smooth'' values of order of the smallest prime designers by divisor of ''n'', and its running time depends on the size of this divisor.
References
* Lenstra, Arjen K.; Lenstra, H.W. Jr. (Eds.) (1993). ''The development of the number field sieve''. Lecture Notes in Math. 1554. Springer-Verlag.
* Pomerance, Carl (1996). http://www.ams.org/notices/199612/pomerance.pdf. ''Notices of the AMS'' 1996, 1473–1485.
legumes and Tag: Integer factorization algorithms
a sophomoric de:Zahlkörpersieb
titled abandoned fr:Algorithme de factorisation par crible sur les corps de nombres généralisé
It is an improvement of the Mosquito ringtone quadratic sieve, which factors ''n'' by finding numbers ''ki'' such that ''ri=ki2-n'' factor completely over a fixed set (called ''basis'') of small primes. Then, having enough such ''ri'' - which are called ''smooth'' relative to the chosen basis of primes, using Sabrina Martins Gauss elimination method of linear algebra we can choose exponents ''ci'' equal to 0 or 1 such that product of ''rici'' is a square, say ''x2''. On the other hand, if the product of ''kici'' is ''y'', then ''x2-y2'' is divisible by ''n'' and with probability at least one half we get a factor of ''n'' by finding Nextel ringtones greatest common divisor of ''n'' and ''x-y''. In this method, the idea was to choose ''ki'' close to the square root of ''n'' - then ''ri'' is of the order of magnitude of square root of ''n'' too and there are enough smooth values there.
The '''general number field sieve''' works as follows:
* We choose two irreducible Abbey Diaz polynomials ''f(x)'' and ''g(x)'' with common root ''m'' mod ''n'' - it is not known what is the best way to choose the polynomials, but usually it is done by picking a degree ''d'' for a polynomial and considering expansion of ''n'' in basis ''m'' where ''m'' is of order ''n1/d''. The point is to get coefficients of ''f'' and ''g'' as small as possible - they will be of order of ''m'', while having small degrees ''d'' and ''e'' of our polynomials.
* Now, we consider number field Cingular Ringtones ring (algebra)/rings '''Z[r1]''' and '''Z[r2]''' where '''r1''' and '''r2''' are roots of polynomials ''f'' and ''g'', and look for values ''a'' and ''b'' such that ''r=bd*f(a/b)'' and ''s=be*g(a/b)'' are smooth relative to the chosen basis of primes. If ''a'' and ''b'' are small, ''r'' and ''s'' will be too (but at least of order of ''m''), and we have a better chance for them to be smooth at the same time.
* Having enough such pairs, using fluid prudie Gauss elimination method we can get products of certain ''r'' and of corresponding ''s'' to be squares at the same time. We need a slightly stronger condition - that they are drilling down norms of squares in our number fields, but we can get that condition by this method too. Each ''r'' is a norm of ''a-'' '''r1'''*''b'' and hence we get that product of corresponding factors ''a-'' '''r1'''*''b'' is a square in '''Z[r1]''', with a "square root" which can be determined (as a product of known factors in '''Z[r1]''') - it will typically be represented as a nonrational security because algebraic number. Similary we get that product of factors ''a-'' '''r2'''*''b'' is a square in '''Z[r2]''', with a "square root" which we can also compute.
* Since ''m'' is root of both ''f'' and ''g'' mod ''n'', there are predictable order homomorphisms from the rings '''Z[r1]''' and '''Z[r2]''' to the ring '''Z/nZ''', which map '''r1''' and '''r2''' to ''m'', and these homomorphisms will map each "square root" (typically not represented as a rational number) into its integer representative. Now product of factors ''a-m*b'' mod ''n'' we can get as a square in two ways - one for each homomorphism. Thus, we get two numbers ''x'' and ''y'', with ''x2-y2'' divisible by ''n'' and again with probability at least one half we get a factor of ''n'' by finding congress hume greatest common divisor of ''n'' and ''x-y''
The second-best-known algorithm for integer factorization is the long excluded Lenstra elliptic curve factorization method. It is better than the general number field sieve ''when factors are of small size'', as it works by finding ''smooth'' values of order of the smallest prime designers by divisor of ''n'', and its running time depends on the size of this divisor.
References
* Lenstra, Arjen K.; Lenstra, H.W. Jr. (Eds.) (1993). ''The development of the number field sieve''. Lecture Notes in Math. 1554. Springer-Verlag.
* Pomerance, Carl (1996). http://www.ams.org/notices/199612/pomerance.pdf. ''Notices of the AMS'' 1996, 1473–1485.
legumes and Tag: Integer factorization algorithms
a sophomoric de:Zahlkörpersieb
titled abandoned fr:Algorithme de factorisation par crible sur les corps de nombres généralisé
posted at 6:37 PM
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