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A general-purpose factoring algorithm, also known as a Category 2, Second Category, or Kraitchik family algorithm, [10] has a running time which depends solely on the size of the integer to be factored. This is the type of algorithm used to factor RSA numbers. Most general-purpose factoring algorithms are based on the congruence of squares method.
Integer factorization is the process of determining which prime numbers divide a given positive integer.Doing this quickly has applications in cryptography.The difficulty depends on both the size and form of the number and its prime factors; it is currently very difficult to factorize large semiprimes (and, indeed, most numbers that have no small factors).
The RSA Factoring Challenge was a challenge put forward by RSA Laboratories on March 18, 1991 [1] to encourage research into computational number theory and the practical difficulty of factoring large integers and cracking RSA keys used in cryptography.
The challenge was to find the prime factors of each number. It was created by RSA Laboratories in March 1991 to encourage research into computational number theory and the practical difficulty of factoring large integers. The challenge was ended in 2007. [1]
As far as is known, this is not possible using classical (non-quantum) computers; no classical algorithm is known that can factor integers in polynomial time. However, Shor's algorithm shows that factoring integers is efficient on an ideal quantum computer, so it may be feasible to defeat RSA by constructing a large quantum computer.
To make factoring harder, p and q should be chosen at random, be both large and have a large difference. [1] For choosing them the standard method is to choose random integers and use a primality test until two primes are found. p and q are kept secret. Compute n = pq. n is used as the modulus for both the public and private keys.
The Conjecture is that this is true for all natural numbers (positive integers from 1 through infinity). Down the Rabbit Hole: The Math That Helps the James Webb Space Telescope Sit Steady in Space
The most efficient method known to solve the RSA problem is by first factoring the modulus N, a task believed to be impractical if N is sufficiently large (see integer factorization). The RSA key setup routine already turns the public exponent e , with this prime factorization, into the private exponent d , and so exactly the same algorithm ...