You are given a black box f : Z10 → Z10 that contains either a random permutation or a random function. Your distinguisher is allowed to invoke f twice. What is the best advantage you can achieve? Express your answer as a reduced fraction without any spaces (eg, 1/3 and not 12/36), or as 0 or 1 if appropriate.

Solution :

The function :       be a random permutation.

f is a permutation on ,  i.e. f is permutation 10.

Now we know that the total number of distinct permutation on to symbolize 10!.

Each of these 10! permutation to a permutation function    

Therefore, total number of permutation functions     are 10!.

Now we want the total number of permutation functioning :

  such that  f(0) = 0 and f(1)= 1

Now we notice that when f(0)=0 and f(1)=1, then two symbol '0' and '1 are fixed under permutation f.

So essentially when f(0) = 0 and f(1) = 1, f becomes permutation on 8 symbol.

Total number of permutation functioning    , f(0)=0 and f(1)=1 are 8!

Now we want the probability that a random permutation     satisfies f(0) = 0 and f(1) = 1.

The number of permutation function    , i.e.

The probability that a random permutation     satisfies f(0) = 0 and f(1) = 1 is

Therefore, the probability that a random permutation     satisfies f(0)= 0 and f(1)=1 is  

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16.9 times 2.85

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