A Beginner’s Guide To The General Number Field Sieve

RSA is a very popular public key cryptosystem. This algorithm is known to be secure, but this fact relies on the difficulty of factoring large numbers. Because of the popularity of the algorithm, much research has gone into this problem of factoring a large number. The size of the number that we are able to factor increases exponentially year by year. This fact is partly due to advancements in computing hardware, but it is largely due to advancements in factoring algorithms. The General Number Field Sieve is an example of just such an advanced factoring algorithm. This is currently the best known method for factoring large numbers. This paper is a presentation of the General Number Field Sieve. It begins with a discussion of the algorithm in general and covers the theory that is responsible for its success. Because often the best way to learn an algorithm is by applying it, an extensive numerical example is included as well.