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The Spectral Test for Randomness
Lign 17, A. Kehler
Random values are the result of an arbitrary selection from among a set of alternatives.
For our purposes, we’ll assume that the alternatives are the values 0 and 1, i.e., bits. A
random sequence is simply a sequence of randomlygenerated values. How can we tell if
particular series of bits is likely to have been generated at random?
First impressions can be misleading. Suppose that we are looking at a series of 10000
bits, and we see a series of 12 ones in a row: 111111111111. That many ones in a row might
not look very random, but it turns out that the odds are less than 1 in 11 that we would
not
see a series of ones that long
somewhere
in the sequence. The question is whether this
pattern – indeed, any pattern – shows up either much more or much less frequently than the
laws of statistics would suggest.
We would thus like a more mechanistic test that could indicate whether a sequence is
randomlygenerated, keeping in mind that the answer will necessarily be
probabilistic
, since
there is always a chance that
any
given sequence of values was drawn randomly – it is just
that the odds might be very small. One such method is the
Spectral Test
for randomness.
The basic idea is that we can do frequency analysis with respect to various patterns, and
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 Winter '08
 Kehler

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