Cryptology ePrint Archive: Report 2015/679

Another Look at Normal Approximations in Cryptanalysis

Subhabrata Samajder and Palash Sarkar

Abstract: 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.

Category / Keywords: secret-key cryptography / block cipher, linear cryptanalysis, differential cryptanalysis, $\chi^2$ test, log-likelihood test, order statistics, normal distribution

Date: received 6 Jul 2015, last revised 21 Sep 2015

Contact author: subhabrata samajder at gmail com; palash sarkar@gmail com;

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Version: 20150921:080619 (All versions of this report)

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