New Links Between Differential and Linear Cryptanalysis

Recently, a number of relations have been established among previously known statistical attacks on block ciphers. Leander showed in 2011 that statistical saturation distinguishers are on average equivalent to multidimensional linear distinguishers. Further relations between these two types of distinguishers and the integral and zero-correlation distinguishers were established by Bogdanov et al.. Knowledge about such relations is useful for classification of statistical attacks in order to determine those that give essentially complementary information about the security of block ciphers. The purpose of the work presented in this paper is to explore relations between differential and linear attacks. The mathematical link between linear and differential attacks was discovered by Chabaud and Vaudenay already in 1994, but it has never been used in practice. We will show how to use it for computing accurate estimatesof truncated differential probabilities from accurate estimates of correlations of linear approximations. We demonstrate this method in practice and give the first instantiation of multiple differential cryptanalysis using the LLR statistical test on PRESENT. On a more theoretical side,we establish equivalence between a multidimensional linear distinguisher and a truncated differential distinguisher, and show that certain zero-correlation linear distinguishers exist if and only if certain impossible differentials exist

Nonlinarity of Boolean functions and hyperelliptic curves

We study the nonlinearity of functions defined on a finite field with 2^m elements which are the trace of a polynomial of degree 7 or more general polynomials. We show that for m odd such functions have rather good nonlinearity properties. We use for that recent results of Maisner and Nart about zeta functions of supersingular curves of genus 2. We give some criterion for a vectorial function not to be almost perfect nonlinear

A Generalization of APN Functions for Odd Characteristic

Almost perfect nonlinear (APN) functions on finite fields of characteristic two have been studied by many researchers. Such functions have useful properties and applications in cryptography, finite geometries and so on. However APN functions on finite fields of odd characteristic do not satisfy desired properties. In this paper, we modify the definition of APN function in the case of odd characteristic, and study its properties

Non-Boolean almost perfect nonlinear functions on non-Abelian groups

The purpose of this paper is to present the extended definitions and characterizations of the classical notions of APN and maximum nonlinear Boolean functions to deal with the case of mappings from a finite group K to another one N with the possibility that one or both groups are non-Abelian.Comment: 17 page

Классы отображений с тривиальной линейной структурой над конечным полем

Получены допускающие простую практическую проверку условия, при которых отображение над конечным полем обладает свойством тривиальности линейной структуры, важным в криптографических приложениях.Отримано умови, що допускають просту практичну перевірку, за якими відображення над скінченим полем володіє властивістю тривіальності лінійної структури, що є важливим для криптографічних застосувань.Important for cryptographic applications conditions that allow simple examination and at which mapping over a finite field has a trivial linear structure, are obtained

Показатели и оценки стойкости блочных шифров относительно статистических атак первого порядка

Получены аналитические верхние оценки надежности различающей и, соответственно, «вскрывающей» статистической атаки первого порядка на блочные шифры. Указанные оценки позволяют ввести теоретически обоснованные показатели стойкости блочных шифров относительно обобщенного линейного, билинейного и ряда других методов криптоанализа. В случае линейной различающей атаки полученная оценка стойкости блочных шифров является более точной по сравнению с ранее известной.Отримано аналітичні верхні оцінки надійності розрізнювальної та, відповідно, «вскриваючої» статистичної атаки першого порядку на блокові шифри. Зазначені оцінки дозволяють ввести теоретично обґрунтовані показники стійкості блокових шифрів відносно узагальненого лінійного, білінійного і низки інших методів криптоаналізу. У випадку лінійної розрізнювальної атаки отримана оцінка стійкості блокових шифрів є більш точною у порівнянні з раніше відомою.Analytical upper estimations of the success probability of a distinguishing and, consequently, a «breaking» first order statistical attack on block ciphers are obtained. These estimations form a foundament for the definition of measures that characterize provable security of block ciphers against generalized linear, bilinear and some other cryptanalysis techniques. For the case of linear distinguishing attack, the obtained estimation of block ciphers security is more accurate that the previous well-known estimation

Partially APN Boolean functions and classes of functions that are not APN infinitely often

In this paper we define a notion of partial APNness and find various characterizations and constructions of classes of functions satisfying this condition. We connect this notion to the known conjecture that APN functions modified at a point cannot remain APN. In the second part of the paper, we find conditions for some transformations not to be partially APN, and in the process, we find classes of functions that are never APN for infinitely many extensions of the prime field \F_2, extending some earlier results of Leander and Rodier.Comment: 24 pages; to appear in Cryptography and Communication