RSA Key size selection is the first important decision when selecting RSA for a cryptosystem. The size of the key actually refers to the size (in bits) of the modulus, N, not the size of any of the public or private keys. Two randomly selected primes, p and q, should be chosen such that they are approximately the same length to ensure that any attempts to factor the modulus are much more difficult. Key sizes for public-key cryptosystems should be much larger than private-key cryptosystems and so comparisons between the two should not be made. The decision of the key size to be used should be based on a thorough assessment of the security solution requirements for the cryptosystem. This entails an evaluation of the value of the data to be protected as well as the length of time for which it needs to be protected. A corresponding factor is also an appraisal of who might wish to devise such an attack as well as what resources they have available. A best guess can then be made based upon the extrapolation of hardware advances, to hypothesize the computational time possible to break the cryptosystem as well as the cost such a design would involve. Increasing the key size will also cause a corresponding increase in the computational load. A rough approximation is that doubling the length of the key size will increase public key operations by a factor of four and private key operations by a factor of eight. Public key operations are less sensitive to key size increase because the public exponent can be fixed, while in private key operations the length of the private exponent increases proportionately. Doubling the length of the key size will also result in a ~16 factor increase in key generation operations. At first consideration, there might be a concern that for a fixed key length that the number of primes available is finite and with a bound on the number of possible primes, the set of primes may be so limited that it would be vulnerable to exploitation. However, from the Prime Number Theorem it is known that the number of primes less than or equal to N is asymptotic to N / ln N. Therefore, the number of prime numbers of length 512 bits or less is roughly 3.77 · 10151. The set of available primes is large enough to be effectively considered infinite. Current implementations of attacks attempting to break RSA cryptosystems must consider two factors. The first is obviously the number of operations required to factor large numbers. The second factor is the actual cost, which is based upon the cost of the hardware times the running time. Several enhancements have been proposed to reduce the cost of implementing the General Number Field Sieve Method, by reducing the amount of memory required. It has been estimated that by using these enhancements, a 1024-bit key could be broken at a cost of roughly $1 billion. However, the running time for such a design would still be measured in decades. Predicting the security of a given key size is difficult; some assumptions regarding the rate of processor performance and cost of hardware must be made. In 2000, several papers were published predicting that within 10 years it would be possible build a system for $250 million which would be able to break a 1024-bit key in a day. This assumed that the rate of processor performance would double every 18 months as well as the rate of improvement in factoring algorithms would also continue. However, much higher estimates of cost and time have also been proposed. At this point it is generally accepted that data encrypted using a key size of 1024-bits should be safe until 2015, although for more sensitive data, a key size of at least 2048-bits might be best.