Faster Methods of Generating Certain Cryptographic Keys
Siguna Mueller
One of the main tasks to be faced in modern information security is the selection of
cryptographic keys. In fact, the security of the most commonly used cryptographic
schemes such as RSA, DSA, etc. depends on choosing the correct keys. In all these
contexts, these keys have to consist of suitable prime numbers.
The most challenging criterium in the generation of primes is the size of the keys that are
required. To be on the safe side of designing cryptographic schemes, one needs to take
into account both the rapid growth in algorithmic developments and the rapid increase of
available computing power. Consequently, the parameters underlying the cryptosystems
need to be chosen sufficiently large. As of today, the factorization of a 155 digit RSA
modulus can be obtained. In order to achieve the desired security, the underlying primes
must be very large. The problem is how such large primes can be obtained efficiently.
Since August 2002 it has been known that primality testing can be done in deterministic
polynomial time. Nonetheless, more than one year later, this striking new result still
remains of only theoretical interest. Indeed, both the original Agrawal-Kayal-Saxena test
as well as the improvements made since then are far from being practical.
Essentially all practical applications for finding large prime numbers rely on the MillerRabin test. While MR is extremely fast, the problem is that it may wrongly declare
composite numbers as prime. This is why the U.S. Department (Federal Information
Processing Standards of the National Institute of Standards and Technology) proposed
iterating MR for random input parameters. In order to achieve an "acceptable probability
of error" it is proposed to iterate MR at least 50 times (FIPS PUB 186-2, 01.2000).
We propose an enhanced version of a MR type test which is much more efficient than the
original MR. Our methods can easily be implemented and achieve the same level of
security by requiring less running time. For example, a probability measure of 2^(-100)
can be achieved in approximately 19/50 of the time required when using MR.