Cryptography Reference
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of the emitted message and random, is added to the useful value (spurious
effects of attenuation can also be added, like on the Rayleigh channel). Thermal
noise is well represented by a Gaussian random process. When demodulation
is performed in an optimal way, it results in a random Gaussian variable whose
sign represents the best hypothesis concerning the binary symbol emitted. The
channel is then characterized by its signal to noise ratio , defined as the ratio
of the power of the useful signal to that of the perturbing noise. For a given
signal to noise ratio, the decisions taken on the binary symbols emitted are
assigned a constant error probability, which leads to the simple model of the
binary symmetric channel.
3.1.2 An example: the binary symmetric channel
This is the simplest channel model, and it has already been mentioned in Sec-
tion 1.3. A channel can generally be described by the probabilities of the tran-
sition of the symbols that are input, towards the symbols that are output. A
binary symmetric channel is thus represented in Figure 3.1. This channel is
memoryless , in the sense that it operates separately on the successive input
bits. Its input symbol X and its output symbol Y are both binary. If X =0
(respectively X =1 ), there exists a probability p that Y =1 (resp. Y =0 ). p
is called the error probability of the channel.
Figure 3.1 - Binary symmetric channel with error probability p . The transition prob-
abilities of an input symbol towards an output symbol are equal two by two.
Another description of the same channel can be given in the following way:
let E be a binary random variable taking value 1 with a probability p< 1 / 2 and
value 0 with the probability 1
p . The hypothesis that p< 1 / 2 does not restrict
the generality of the model because changing the arbitrary signs 0 and 1 leads
to replacing an initial error probability p> 1 / 2 by 1
p< 1 / 2 . The behaviour
of the channel can be described by the algebraic expression Y = X
E ,where
X and Y are the binary variables at the input and at the output of the channel
respectively, E a binary error variable, and
represents the modulo 2 addition.
Configurations of errors on the binary symmetric channel
Let us now suppose that we no longer consider a particular single symbol,
but a set of n symbols (consecutive or not) making up a word , denoted
 
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