Respuesta :
Answer:
for a) the time required is 228488.5 seconds= 63.469 hours
for b) the time required is 54 seconds
Explanation:
for a) since each combination is equally probable , then the number of possible combinations is
CT=Combinations = number of characters^length of password = 26⁴
then the number of combinations at time t will be the total , less the ones already tried:
Ct = CT - (n-1) , since n=α*t → Ct=CT-α*t
since each combination is equally probable , then the probability to succeed
pt = 1/Ct = 1/ (CT- α*t +1)
but the probability of having a success in time t , means also not succeeding in the previous trials , then
Pt = pt*П(1-pk), for k=1 to t-1
Pt = 1/ (CT- α*t +1) П[1-1/ (CT- α*k +1)] = 1/ (CT- α*t +1) П[(CT- α*k )] /(CT- α*k +1)]
since α=1 ,
Pt = 1/ (CT- t +1) П[(CT- k )] /(CT- k +1)] = 1/ (CT- t +1) * [CT- (t-1) ]/CT = 1/CT
then the expected value of the random variable t= time to discover the correct password is
E(t) = ∑ t* Pt = ∑ t *1/CT , for t=1 until t=CT/α =CT
E(t) = ∑ t *(1/CT) = (1/CT) ∑ t = (1/CT) * CT*(CT+1)/2 = (CT+1)/2
therefore
E(t) = (CT+1)/2 = (26⁴ +1)/2 = 228488.5 seconds = 63.469 hours
for b)
time required = time to find character 1 + time to find character 2 +time to find character 3 +time to find character 4 = 4*time to find character
since the time to find a character is the same case as before but with CT2=Combinations = 26 ,then
t= 4*tc
E(t) = 4*E(tc) = 4*(CT2+1)/2 = 4*(26+1)/2 = 54 seconds
The expected time to discover the correct password is 456,976 seconds.
How to calculate the expected time
It should be noted that because there is only one attempt per possible combination, then this follows a uniform distribution.
Then assuming no feedback to the adversary until each attempt has been completed, the expected time to discover the correct password will be:
= 26⁴ = 456,976 sec.
Assuming feedback to the adversary flagging an error as each incorrect character is entered, the expected time to discover the correct password will be:
= (26 × 4)/2
= 52 seconds
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