Automatic Generation of Prime Factorization Algorithms Using Genetic Programming

This paper describes the application of the principles of genetic programming to the field of prime factorization. Any prime factorization algorithm is given one integer and must generate a complete list of primes such that, when multiplied together in varying degrees, produces the original integer. Constructing even a limited factoring algorithm in GP turns out to be extremely challenging and potentially impossible. Introduction and Overview Deriving an algorithm which scales better than O(n) to numerically factor integers into its constituent prime factors has been a crucial problem in recent times. Most encryption schemes, including RSA, depend on the fact that it is compute-intensive to factor large numbers in order to obscure sensitive data. Solutions to this problem include resorting to highly parallelized compute environments such as quantum computing, DNA-based computations, or massivelyparallelized computing farms. On the surface, attempting to derive an algorithm to factor integers into prime constituents seems to be a problem well suited for genetic programming. The goal is well defined: the genetic programming process must evolve an algorithm which takes an integer i and outputs n prime integer factors, such that when the ∏ product of these factors is calculated, it returns the original integer i. The fitness cases are similarly well defined: an individual is considered to be fit when its outputs are all prime integers that, when multiplied together, produces the original integer i. Traditional primality tests, such as the Rabin-Miller Strong Pseudoprime Test, are well understood and can be efficiently calculated for numbers up to 3.4 x 1014 (Rabin 1980). Additional fitness cases to evaluate compute-expense may be added in order to evolve higher performance algorithms; however, first, it is important to determine whether GP is capable of evolving a simple factoring algorithm. Prime Factorization The Fundamental Theorem of Arithmetic (unique factorization theorem) states that any positive integer can be represented in exactly one way as a product of primes. Mathematicians such as Euler and Fermat used to factor primes without computers at an astounding rate. Ever since these feats, there has been speculation that there was once a secret method of factoring primes that has since been lost.