Integer Factorization Task
An Integer Factorization Task is a decomposition task of a composite number into smaller non-trivial divisors
- AKA: Prime Factorization.
- Context:
- It is an NP Decision Task, given integers n and m, and an integer f with 1 < f < m, and a claim that f is a factor of n (the certificate) one can check the answer in polynomial time by performing the division n / f.
- It can be solved by an Integer Factorization System (that implements an integer factorization algorithm).
- Example(s):
- given integers n and m, is there an integer f with 1 < f < m such that f divides n (f is a small factor of n)?
- See: RSA-768, Cryptography, RSA (Algorithm), Elliptic Curve, Algebraic Number Theory.
References
2014
- (Wikipedia, 2014) ⇒ http://en.wikipedia.org/wiki/integer_factorization Retrieved:2014-7-26.
- In number theory, integer factorization or prime factorization is the decomposition of a composite number into smaller non-trivial divisors, which when multiplied together equal the original integer.
When the numbers are very large, no efficient, non-quantum integer factorization algorithm is known; an effort by several researchers concluded in 2009, factoring a 232-digit number (RSA-768), utilizing hundreds of machines over a span of two years. The presumed difficulty of this problem is at the heart of widely used algorithms in cryptography such as RSA. Many areas of mathematics and computer science have been brought to bear on the problem, including elliptic curves, algebraic number theory, and quantum computing.
Not all numbers of a given length are equally hard to factor. The hardest instances of these problems (for currently known techniques) are semiprimes, the product of two prime numbers. When they are both large, for instance more than two thousand bits long, randomly chosen, and about the same size (but not too close, e.g. to avoid efficient factorization by Fermat's factorization method), even the fastest prime factorization algorithms on the fastest computers can take enough time to make the search impractical; that is, as the number of digits of the primes being factored increases, the number of operations required to perform the factorization on any computer increases drastically.
Many cryptographic protocols are based on the difficulty of factoring large composite integers or a related problem — for example, the RSA problem. An algorithm that efficiently factors an arbitrary integer would render RSA-based public-key cryptography insecure.
- In number theory, integer factorization or prime factorization is the decomposition of a composite number into smaller non-trivial divisors, which when multiplied together equal the original integer.