Integer factorization
In number theory, integer factorization is the decomposition of a composite number into a product of smaller integers. whether these factors are further restricted to prime numbers, the process is called prime factorization.
When the numbers are sufficiently large, no efficient, non-quantum integer factorization algorithm is known. However, it has non been proven that no able algorithm exists. The presumed difficulty of this problem is at the heart of widely used algorithms in cryptography such(a) as RSA. many areas of mathematics as well as computer science realize been brought to bear on the problem, including elliptic curves, algebraic number theory, together with quantum computing.
In 2019, Fabrice Boudot, Pierrick Gaudry, Aurore Guillevic, Nadia Heninger, Emmanuel Thomé and Paul Zimmermann factored a 240-digit 795-bit number RSA-240 utilizing about 900 core-years of computing power. The researchers estimated that a 1024-bit RSA modulus would realize about 500 times as long.
Not all numbers of a given length are equally tough to factor. The hardest instances of these problems for currently so-called techniques are 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 asked to perform the factorization on all 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 administer RSA-based public-key cryptography insecure.