18.104.22.168 How does one find random numbers for keys?
Whether using a secret-key cryptosystem or a public-key cryptosystem, one needs a good source of random numbers for key generation. The main features of a good source are that it produces numbers that are unknown and unpredictable by potential adversaries. Random numbers obtained from a physical process are in principle the best, since many physical processes appear truly random. One could use a hardware device, such as a noisy diode; some are sold commercially on computer add-in boards for this purpose. Another idea is to use physical movements of the computer user, such as inter-key stroke timings measured in microseconds. Techniques using the spinning of disks to generate random data are not truly random, as the movement of the disk platter cannot be considered truly random. A negligible-cost alternative is available; Davis et al. designed a random number generator based on the variation of a disk drive motor's speed [DIP94]. This variation is caused by air turbulence, which has been shown to be unpredictable. By whichever method they are generated, the random numbers may still contain some correlation, thus preventing sufficient statistical randomness. Therefore, it is best to run them through a good hash function (see Question 2.1.6) before actually using them [ECS94].
Another approach is to use a pseudo-random number generator fed by a random seed. The primary difference between random and pseudo-random numbers is that pseudo-random numbers are necessarily periodic whereas truly random numbers are not. Since pseudo-random number generators are deterministic algorithms, it is important to find one that is cryptographically secure and also to use a good random seed; the generator effectively acts as an "expander" from the seed to a larger amount of pseudo-random data. The seed must be sufficiently variable to deter attacks based on trying all possible seeds.
It is not sufficient for a pseudo-random number generator just to pass a variety of statistical tests, as described in Knuth [Knu81] and elsewhere, because the output of such generators may still be predictable. Rather, it must be computationally infeasible for an attacker to determine any bit of the output sequence, even if all the others are known, with probability better than 1/2. Blum and Micali's generator based on the discrete logarithm problem [BM84] satisfies this stronger definition, assuming that computing discrete logarithm is difficult (see Question 2.3.7). Other generators perhaps based on DES (see Section 3.2) or a hash function (see Question 2.1.6) can also be considered to satisfy this definition, under reasonable assumptions.
A summary of methods for generating random numbers in software can be found in [Mat96].
Note that one does not need random numbers to determine the public and private exponents in RSA. After generating the primes, and hence the modulus (see Question 3.1.1), one can simply choose an arbitrary value (subject to the standard constraints) for the public exponent, which then determines the private exponent.