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To factorize a small integer {{mvar|n}} using mental or pen-and-paper arithmetic, the simplest method is [[trial division]]: checking if the number is divisible by prime numbers {{math|2}}, {{math|3}}, {{math|5}}, and so on, up to the [[square root]] of {{mvar|n}}. For larger numbers, especially when using a computer, various more sophisticated factorization algorithms are more efficient. A prime factorization algorithm typically involves [[primality test|testing whether each factor is prime]] each time a factor is found.
When the numbers are sufficiently large, no efficient non-[[quantum computer|quantum]] integer factorization [[algorithm]] is known. However, it has not been proven that such an algorithm does not exist. The presumed [[Computational hardness assumption|difficulty]] of this problem is important for the algorithms used in [[cryptography]] such as [[RSA (cryptosystem)|RSA public-key encryption]] and the [[Digital Signature Algorithm|RSA digital signature]].<ref>{{Citation |last=Lenstra |first=Arjen K. |title=Integer Factoring |date=2011
Not all numbers of a given length are equally hard to factor. The hardest instances of these problems (for currently known techniques) are [[semiprime]]s, the product of two prime numbers. When they are both large, for instance more than two thousand [[bit]]s long, randomly chosen, and about the same size (but not too close, for example, to avoid efficient factorization by [[Fermat's factorization method]]), even the fastest prime factorization algorithms on the fastest classical computers can take enough time to make the search impractical; that is, as the number of digits of the integer being factored increases, the number of operations required to perform the factorization on any classical computer increases drastically.
Many cryptographic protocols are based on the presumed difficulty of factoring large composite integers or a related
== Prime decomposition ==
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Among the {{math|''b''}}-bit numbers, the most difficult to factor in practice using existing algorithms are those [[semiprimes]] whose factors are of similar size. For this reason, these are the integers used in cryptographic applications.
In 2019,
| last1 = Kleinjung | first1 = Thorsten
| last2 = Aoki | first2 = Kazumaro
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| title = Advances in Cryptology - CRYPTO 2010, 30th Annual Cryptology Conference, Santa Barbara, CA, USA, August 15-19, 2010. Proceedings
| volume = 6223
| year = 2010
}}</ref>
The largest such semiprime yet factored was [[RSA numbers#RSA-250|RSA-250]], an 829-bit number with 250 decimal digits, in February 2020. The total computation time was roughly 2700 core-years of computing using Intel [[Skylake (microarchitecture)#Xeon Gold (quad processor)|Xeon Gold]] 6130 at 2.1 GHz. Like all recent factorization records, this factorization was completed with a highly optimized implementation of the [[general number field sieve]] run on hundreds of machines.
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== External links ==
* [
* Richard P. Brent, "Recent Progress and Prospects for Integer Factorisation Algorithms", ''Computing and Combinatorics"'', 2000, pp. 3–22. [http://citeseer.ist.psu.edu/327036.html download]
* [[Manindra Agrawal]], Neeraj Kayal, Nitin Saxena, "PRIMES is in P." Annals of Mathematics 160(2): 781–793 (2004). [http://www.cse.iitk.ac.in/users/manindra/algebra/primality_v6.pdf August 2005 version PDF]
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