Statistical Model of Correlation Difference and Related-Key Linear Cryptanalysis

The goal of this work is to propose a related-key model for linear cryptanalysis. We start by giving the mean and variance of the difference of sampled correlations of two Boolean functions when using the same sample of inputs to compute both correlations. This result is further extended to determine the mean and variance of the difference of correlations of a pair of Boolean functions taken over a random data sample of fixed size and over a random pair of Boolean functions. We use the properties of the multinomial distribution to achieve these results without independence assumptions. Using multivariate normal approximation of the multinomial distribution we obtain that the distribution of the difference of related-key correlations is approximately normal. This result is then applied to existing related-key cryptanalyses. We obtain more accurate right-key and wrong-key distributions and remove artificial assumptions about independence of sampled correlations. We extend this study to using multiple linear approximations and propose a Χ2-type statistic, which is proven to be Χ2 distributed if the linear approximations are independent. We further examine this statistic for multidimensional linear approximation and discuss why removing the assumption about independence of linear approximations does not work in the related-key setting the same way as in the single-key setting.

Multivariate Normal Approximation on the Wiener Space: New Bounds in the Convex Distance

We establish explicit bounds on the convex distance between the distribution of a vector of smooth functionals of a Gaussian field and that of a normal vector with a positive-definite covariance matrix. Our bounds are commensurate to the ones obtained by Nourdin et al. (Ann Inst Henri Poincaré Probab Stat 46(1):45–58, 2010) for the (smoother) 1-Wasserstein distance, and do not involve any additional logarithmic factor. One of the main tools exploited in our work is a recursive estimate on the convex distance recently obtained by Schulte and Yukich (Electron J Probab 24(130):1–42, 2019). We illustrate our abstract results in two different situations: (i) we prove a quantitative multivariate fourth moment theorem for vectors of multiple Wiener–Itô integrals, and (ii) we characterize the rate of convergence for the finite-dimensional distributions in the functional Breuer–Major theorem.

Marker-Trait Complete Analysis

1AbstractA recurring problem in genomics involves testing association of one or more traits of interest to multiple genomic features. Feature-trait squared correlations r2 are commonly-used statistics, sensitive to trend associations. It is often of interest to perform testing across collections {r2} over markers and/or traits using both maxima and sums. However, both trait-trait correlations and marker-marker correlations may be strong and must be considered. The primary tools for multiple testing suffer from various shortcomings, including p-value inaccuracies due to asymptotic methods that may not be applicable. Moreover, there is a lack of general tools for fast screening and follow-up of regions of interest.To address these difficulties, we propose the MTCA approach, for Marker-Trait Complete Analysis. MTCA encompasses a large number of existing approaches, and provides accurate p-values over markers and traits for maxima and sums of r2 statistics. MTCA uses the conditional inference implicit in permutation as a motivational frame-work, but provides an option for fast screening with two novel tools: (i) a multivariate-normal approximation for the max statistic, and (ii) the concept of eigenvalue-conditional moments for the sum statistic. We provide examples for gene-based association testing of a continuous phenotype and cis-eQTL analysis, but MTCA can be applied in a much wider variety of settings and platforms.