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The two main criticisms of electronic draws are the problem of security and the question of randomness. Computerised draws are no harder to secure than mechanical ones; the draw computer is physically isolated in the same way as a mechanical draw machine; more than one official observes and records the draw result. The second objection - that of randomness - is perhaps more complex to explain, but no harder to justify.
One of the first RNGs was proposed by John von Neumann in 1946. His middle-square method involved taking a number, squaring it, then taking the middle digit of the result and repeating. It was simple to carry out, but its simplicity was its weakness: it had a very short period (how many numbers are generated before the sequence begins to repeat, in this case only 142). The generated numbers were not truly random but pseudo-random, and it is a problem which has dogged RNGs ever since.
As computational means improved and number theory research advanced, RNG periods have gradually increased, until now they are so long that for all practical purposes the generated sequences can be said to be random, and actually pass the most rigorous statistical tests for randomness. Modern RNG algorithms are triumphs of computation and the mathematician's craft. They have truly astonishing periods. The current best RNG is MT19937, developed by two Japanese mathematicians, Makoto Matsumoto and Takuji Nishimura in 1998, and revised in 2002 to remove undesirable results obtained from certain seed values (a seed is an initial number fed into an RNG to start it off).
MT19937 is a variant of the twisted generalized feedback shift-register algorithm, and has a Mersenne prime period of 2^19937 - 1 (about 10^6000). Let's put this gargantuan number into perspective: at the rate of 1,000,000,000 numbers PER SECOND it would take 10^5983 years before MT19937 began to repeat. Our universe is estimated to be only about 1.2 x 10^12 years old. Naturally such a powerful RNG is no mere academic plaything. It is used in quantum chromodynamic calculations, lattice field theory simulations, cosmological computer models, simulations of nuclear weapon explosions, and numerical tests of high-end supercomputers. MT19937 has even been adapted for use in cryptography. MT19937 has passed the most rigorous statistical tests for randomness yet devised, the toughest of which is the ominously-named Diehard suite of tests, devised by American mathematician George Marsaglia.
MT19937 represents the pinnacle of RNG science, but more modest algorithms would serve just as well for a lottery draw. One known as gsl_rng_rand has a period of 2^31, more than 2,000,000,000 numbers. At the rate of six numbers per week this would run for the best part of seven million years before it began to repeat.
Today, many countries use computer RNGs in their national lotteries (one of the largest being in China). The UK National Lottery has an online generator for players, and its shop terminals can also deliver computer-generated numbers. Now that the limitations of early RNGs have been overcome - and more people are becoming familiar with computers - objections to computerised draws can be viewed as increasingly irrational.
Checking the Randomness of the UK National Lottery (PDF from 2004)
Matsumoto and Nishimura's original 1998 paper on MT19937
Download Matsumoto and Nishimura's MT19937 PDF
George Marsaglia's Diehard tests
On the Reliability of Random Number Generators PDF site
Download the Reliability of Random Number Generators PDF
Perfectly Random Sampling with Markov Chains
The RNG Random Number Library PDF site
Download the RNG Random Number Library PDF
Random number article from MathWorld
Wikipedia entry about Diehard tests
Wikipedia entry about pseudo-random numbers