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  

the system of inequity is

step-by-step explanation:

from the given graph it is noticed that we have two related lines. one is solid and another is dashed.

if a line passing through two points, then the equation of line is

the solid line passing through the points (0,4) and (4,0).

the related equation of solid line is .

similarly, find the related equation of dashed line. the dashed liune passing through the points (0,0) and (3,1).

the related equation of dashed line is

from the given graph it is noticed that the point (0,1) lies on the shaded region, therefore the point (0,1) must be satisfied by the related equations.

check the related equations by the point (0,1).

this statement is true if the sign of inequality is < instead of equality. since it is an solid line therefore the sign of inequality will be . the required inequality is

similarly, check the second related equation.

it is a dashed line, so this statement is true if the sign of inequality is > . the required inequality is

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step-by-step explanation:

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