Another Look at Normal Approximations in Cryptanalysis

Statistical analysis of attacks on symmetric ciphers often require assuming the normal behaviour of a test statistic. Typically such an assumption is made in an asymptotic sense. In this work, we consider concrete versions of some important normal approximations that have been made in the literature. To do this, we use the Berry-Esséen theorem to derive explicit bounds on the approximation errors. Analysing these error bounds in the cryptanalytic context throws up several surprising results. One important implication is that this puts in doubt the applicability of the order statistics based approach for analysing key recovery attacks on block ciphers. This approach has been earlier used to obtain several results on the data complexities of (multiple) linear and differential cryptanalysis. The non-applicability of the order statistics based approach puts a question mark on the data complexities obtained using this approach. Fortunately, we are able to recover all of these results by utilising the hypothesis testing framework. Detailed consideration of the error in normal approximation also has implications for $\chi^2$ and the log-likelihood ratio (LLR) based test statistics. The normal approximation of the $\chi^2$ test statistics has some serious and counter-intuitive restrictions. One such restriction is that for multiple linear cryptanalysis as the number of linear approximations grows so does the requirement on the number of plaintext-ciphertext pairs for the approximation to be proper. The issue of satisfactorily addressing the problems with the application of the $\chi^2$ test statistics remains open. For the LLR test statistics, previous work used a normal approximation followed by another approximation to simplify the parameters of the normal approximation. We derive the error bound for the normal approximation which turns out to be difficult to interpret. We show that the approximation required for simplifying the parameters restricts the applicability of the result. Further, we argue that this approximation is actually not required. More generally, the message of our work is that all cryptanalytic attacks should properly derive and interpret the error bounds for any normal approximation that is made

Another Look at Key Randomisation Hypotheses

In the context of linear cryptanalysis of block ciphers, let $p_0$ (resp. $p_1$) be the probability that a particular linear approximation holds for the right (resp. a wrong) key choice. The standard right key randomisation hypothesis states that $p_0$ is a constant $p\neq 1/2$ and the standard wrong key randomisation hypothesis states that $p_1=1/2$. Using these hypotheses, the success probability $P_S$ of the attack can be expressed in terms of the data complexity $N$. The resulting expression for $P_S$ is a monotone increasing function of $N$. Building on earlier work by Daemen and Rijmen (2007), Bogdanov and Tischhauser (2014) argued that $p_1$ should be considered to be a random variable. They postulated the adjusted wrong key randomisation hypothesis which states that $p_1$ follows a normal distribution. A non-intuitive consequence was that the resulting expression for $P_S$ is no longer a monotone increasing function of $N$. A later work by Blondeau and Nyberg (2017) argued that $p_0$ should also be considered to be a random variable and they postulated the adjusted right key randomisation hypothesis which states that $p_0$ follows a normal distribution. In this work, we revisit the key randomisation hypotheses. While the argument that $p_0$ and $p_1$ should be considered to be random variables is indeed valid, we consider the modelling of their distributions by normal to be inappropriate. Being probabilities, the support of the distributions of $p_0$ and $p_1$ should be subsets of $[0,1]$ which does not hold for normal distributions. We show that if $p_0$ and $p_1$ follow any distributions with supports which are subsets of $[0,1]$, and $E[p_0]=p$ and $E[p_1]=1/2$, then the expression for $P_S$ that is obtained is exactly the same as the one obtained using the standard key randomisation hypotheses. Consequently, $P_S$ is a monotone increasing function of $N$ even when $p_0$ and $p_1$ are considered to be random variables

A New Test Statistic for Key Recovery Attacks Using Multiple Linear Approximations

The log-likelihood ratio (LLR) and the chi-squared distribution based test statistics have been proposed in the literature for performing statistical analysis of key recovery attacks on block ciphers. A limitation of the LLR test statistic is that its application requires the full knowledge of the corresponding distribution. Previous work using the chi-squared approach required {\em approximating} the distribution of the relevant test statistic by chi-squared and normal distributions. Problematic issues regarding such approximations have been reported in the literature. Perhaps more importantly, both the LLR and the chi-squared based methods are applicable only if the success probability $P_S$ is greater than 0.5. On the other hand, an attack with success probability less than $0.5$ is also of considerable interest. This work proposes a new test statistic for key recovery attacks which has the following features. Its application does not require the full knowledge of the underlying distribution; it is possible to carry out an analysis using this test statistic without using any approximations; the method applies for all values of the success probability. The statistical analysis of the new test statistic follows the hypothesis testing framework and uses Hoeffding\u27s inequalities to bound the probabilities of Type-I and Type-II errors

Multiple Differential Cryptanalysis: A Rigorous Analysis

Statistical analyses of multiple differential attacks are considered in this paper. Following the work of Blondeau and Gérard, the most general situation of multiple differential attack where there are no restrictions on the set of differentials is studied. We obtain closed form bounds on the data complexity in terms of the success probability and the advantage of an attack. This is done under two scenarios -- one, where an independence assumption used by Blondeau and Gérard is assumed to hold and second, where no such assumption is made. The first case employs the Chernoff bounds while the second case uses the Hoeffding bounds from the theory of concentration inequalities. In both cases, we do not make use of any approximations in our analysis. As a consequence, the results are more generally applicable compared to previous works. The analysis without the independence assumption is the first of its kind in the literature. We believe that the current work places the statistical analysis of multiple differential attack on a more rigorous foundation than what was previously known

Rigorous Upper Bounds on Data Complexities of Block Cipher Cryptanalysis

Statistical analysis of symmetric key attacks aims to obtain an expression for the data complexity which is the number of plaintext-ciphertext pairs needed to achieve the parameters of the attack. Existing statistical analyses invariably use some kind of approximation, the most common being the approximation of the distribution of a sum of random variables by a normal distribution. Such an approach leads to expressions for data complexities which are {\em inherently approximate}. Prior works do not provide any analysis of the error involved in such approximations. In contrast, this paper takes a rigorous approach to analysing attacks on block ciphers. In particular, no approximations are used. Expressions for upper bounds on the data complexities of several basic and advanced attacks are obtained. The analysis is based on the hypothesis testing framework. Probabilities of Type-I and Type-II errors are upper bounded using standard tail inequalities. In the cases of single linear and differential cryptanalysis, we use the Chernoff bound. For the cases of multiple linear and multiple differential cryptanalysis, Hoeffding bounds are used. This allows bounding the error probabilities and obtaining expressions for data complexities. We believe that our method provides important results for the attacks considered here and more generally, the techniques that we develop should have much wider applicability

Success Probability of Multiple/Multidimensional Linear Cryptanalysis Under General Key Randomisation Hypotheses

This work considers statistical analysis of attacks on block ciphers using several linear approximations. A general and unified approach is adopted. To this end, the general key randomisation hypotheses for multidimensional and multiple linear cryptanalysis are introduced. Expressions for the success probability in terms of the data complexity and the advantage are obtained using the general key randomisation hypotheses for both multidimensional and multiple linear cryptanalysis and under the settings where the plaintexts are sampled with or without replacement. Particularising to standard/adjusted key randomisation hypotheses gives rise to success probabilities in 16 different cases out of which in only five cases expressions for success probabilities have been previously reported. Even in these five cases, the expressions for success probabilities that we obtain are more general than what was previously obtained. A crucial step in the analysis is the derivation of the distributions of the underlying test statistics. While we carry out the analysis formally to the extent possible, there are certain inherently heuristic assumptions that need to be made. In contrast to previous works which have implicitly made such assumptions, we carefully highlight these and discuss why they are unavoidable. Finally, we provide a complete characterisation of the dependence of the success probability on the data complexity

Another Look at Success Probability in Linear Cryptanalysis

This work studies the success probability of key recovery attacks based on using a single linear approximation. Previous works had analysed success probability under different hypotheses on the distributions of correlations for the right and wrong key choices. This work puts forward a unifying framework of general key randomisation hypotheses. All previously used key randomisation hypotheses as also zero correlation attacks can be seen to special cases of the general framework. Derivations of expressions for the success probability are carried out under both the settings of the plaintexts being sampled with and without replacements. Compared to previous analysis, we uncover several new cases which have not been considered in the literature. For most of the cases which have been considered earlier, we provide complete expressions for the respective success probabilities. Finally, the complete picture of the dependence of the success probability on the data complexity is revealed. Compared to the extant literature, our work provides a deeper and more thorough understanding of the success probability of single linear cryptanalysis