International Association
for Cryptologic Research

Impossible differential attacks, which are taking advantage of differentials that cannot occur, are powerful attacks for block cipher primitives. The power of such attacks is often measured in terms of the advantage — number of key-bits found during the key sieving phase — which determines the time complexity of the exhaustive key search phase. The statistical model used to compute this advantage has been introduced in the seminal work about the resistance of the DEAL cipher to impossible differential attacks. This model, which has not been modified since the end of the 1990s, is implicitly based on the Poisson approximation of the binomial distribution. In this paper, we investigate this commonly used model and experimentally illustrate that random permutations do not follow it. Based on this observation, we propose more accurate estimates of the advantage of an impossible differential attack. The experiments illustrate the accuracy of the estimate derived from the multivariate hypergeometric distribution. The maximal advantage –using the full codebook– of an impossible differential attack is also derived.

Statistical attacks form an important class of attacks against block ciphers. By analyzing the distribution of the statistics involved in the attack, cryptanalysts aim at providing a good estimate of the data complexity of the attack. Recently multiple papers have drawn attention to how to improve the accuracy of the estimated success probability of linear key-recovery attacks. In particular, the effect of the key on the distribution of the sample correlation and capacity has been investigated and new statistical models developed. The major problem that remains open is how to obtain accurate estimates of the mean and variance of the correlation and capacity. In this paper, we start by presenting a solution for a linear approximation which has a linear hull comprising a number of strong linear characteristics. Then we generalize this approach to multiple and multidimensional linear cryptanalysis and derive estimates of the variance of the test statistic. Our simplest estimate can be computed given the number of the strong linear approximations involved in the offline analysis and the resulting estimate of the capacity. The results tested experimentally on SMALLPRESENT-[4] show the accuracy of the estimated variance is significantly improved. As an application we give more realistic estimates of the success probability of the multidimensional linear attack of Cho on 26 rounds of PRESENT.