## 4 Discussion of the Results

Table 4: Mean number of samples required to retrieve the whole 128 bits key with the access approach (first column), our approach (second column) and the improvement our approach means. (V) means virtualized scenario. \begin{table}
\begin{tabular}{c c c c c} \hline \hline  & _Flush+Reload_ & _Flush+Flush_ & _Prime+Probe_ & _Prime+Abort_ \\ \hline
**AES 128** & 3000 & 15,000 & 14,000 & 3500 \\
**AES 256** & 8000 & 35,000 & 38,000 & 8000 \\ \hline \hline \end{tabular}
\end{table} 
Table 2: Mean number of samples required to retrieve the whole key in non-virtualized environments. \begin{table}
\begin{tabular}{c c c c} \hline \hline  & _Flush+Reload_ & _Flush+Flush_ & _Prime+Probe_ \\ \hline
**AES 128** & 10,000 & 40,000 & 45,000 \\
**AES 256** & 28,000 & 100,000 & 110,000 \\ \hline \hline \end{tabular}
\end{table} 
Table 3: Mean number of samples required to retrieve the whole key in virtualized environments (cross-VM). We could completely recover the full AES 256 key using the EBD attack. Comparing the number of traces required to obtain the 128 bit key and the number of traces required to obtain the 256 bit key, AES-256 happened to be only three times harder to attack with our approach. The different probabilities of non-accessing a line explained our results: 2.69% for AES-256 and 7.5% for AES-128, whithin theory indicates 2.8 \(\times\) more encryptions being needed to get the same information. In practice, we measured 5.29%, 3.87%, 4.55% and 4.31% of cache misses within our samples for AES-128 and each of the lines, respectively, and 1.52%, 0.93%, 1.31% and 1.02% for AES-256 for _Flush+Reload_. These results allow us to discuss the results in [49]. They state that, although larger key lengths translate into an exponential increase in the complexity of a brute force approach, this cannot be applied to side-channel attacks. In addition, using the two last rounds of the encryption algorithm (a version that uses a different table for computing the last round of the encryption), they arrive at the conclusion that AES-256 only provides an increase in complexity of 6-7 compared to cache-based attacks on AES-128. We, in contrast, demonstrate that this factor is equal to 3. In the case of the EBD attack, we did not find similar results or experimental setups to compare with. All studied attacks can be described as eminently theoretical. In the literature, [PERSON] et al. [50] stated that their attack can be applied to the second round of the encryption, but they did not provide experimental results. In 2009, [PERSON] et al. [51] presented a distinguisher and related key attack on AES-256 with 2\({}^{131}\) data complexity. Later, they slightly improved their results [52], achieving 2\({}^{119}\) data complexity. In [53], they described an attack using two related keys to recover the 256-bit key, but their target cipher was the nine-round version of AES-256 and used chosen-plaintexts. Another attack [54] runs in a related-subkey scenario with four related keys targeting the 13-round AES-256; it has 2\({}^{76}\) time and data complexity. [PERSON] et al. [55] presented a technique of cryptanalysis with bicliques, also leading to a non-practical complexity recovering the key. [PERSON] [56] explained also theoretically how to apply differential fault analysis to reduce the required number of faults in AES-192 and AES-256. As stated above, each retrieved sample that carries information about non-accesses to the T-Tables reduces the key search space. This means that, if the attack is interrupted for some reason before it finishes and the attacker decides to apply brute force to retrieve the key, the number of keys he has to check is much smaller. In Figure 6, we present the base 2 logarithm of the key search space vs. the total number of samples retrieved, i.e., the number of bits that remain unknown after each performed encryption.