Question

What is the maximum amount of time that it would take a computer that can test 1000 billion (10¹²) keys per second to carry out a brute-force attack on a file encrypted using 128-bit AES? Give your answer in years, using scientific notation, correct to 2 significant figures. 

Answer 1

No of possible keys for 128-bit AES =  $\huge 2^{128}$

This is because each bit can have 2 values (0 or 1) and therefore, number of different 128-bit keys = $\huge 2 \times 2 \times2 \times .....\text{(total 128 times)} = 2^{128}$ . This is our search space and these many keys we would have to examine.

Our computer can test $\huge 10^{12}$ keys per second. In other words,

Time taken to test  $\huge 10^{12}$ keys = 1 second

$\huge \text{Time taken to test 1 key }=\frac{1}{10^{12}} \text{ second}$

$\huge \text{Time taken to test }2^{128} \text{ keys }=\frac{2^{128}}{10^{12}} \text{ second}$

As 1 year is approximately equal to $\huge 3.2 \times 10^7$ second (given in the question), we have,

$\huge \text{Time taken to test }2^{128} \text{ keys }=\frac{2^{128}}{10^{12}\times 3.2 \times 10^7} \text{ years}$

$\huge =\frac{2^{128}}{10^{18}\times 32} \text{ years}$

  $\huge =\frac{2^{128}}{10^{18}\times 2^5} \text{ years}$

  

2123 1018 years

$\huge =1.063 \times 10^{19} \text{ years}$

$\huge =\mathbf{1.1 \times 10^{19}} \text{ years, correct to two significant figures}$.

So. the machine will take $\huge 1.1 \times 10^{19}$ years to crack a 128-bit AES.