Constructions of Almost Optimal Resilient Boolean Functions on Large Even Number of Variables

In this paper, a technique on constructing nonlinear resilient Boolean functions is described. By using several sets of disjoint spectra functions on a small number of variables, an almost optimal resilient function on a large even number of variables can be constructed. It is shown that given any $m$, one can construct infinitely many $n$-variable ($n$ even), $m$-resilient functions with nonlinearity $>2^{n-1}-2^{n/2}$. A large class of highly nonlinear resilient functions which were not known are obtained. Then one method to optimize the degree of the constructed functions is proposed. Last, an improved version of the main construction is given.Comment: 14 pages, 2 table

Implementing Symmetric Cryptography Using Sequence of Semi-Bent Functions

Symmetric cryptography is a cornerstone of everyday digital security, where two parties must share a common key to communicate. The most common primitives in symmetric cryptography are stream ciphers and block ciphers that guarantee confidentiality of communications and hash functions for integrity. Thus, for securing our everyday life communication, it is necessary to be convinced by the security level provided by all the symmetric-key cryptographic primitives. The most important part of a stream cipher is the key stream generator, which provides the overall security for stream ciphers. Nonlinear Boolean functions were preferred for a long time to construct the key stream generator. In order to resist several known attacks, many requirements have been proposed on the Boolean functions. Attacks against the cryptosystems have forced deep research on Boolean function to allow us a more secure encryption. In this work we describe all main requirements for constructing of cryptographically significant Boolean functions. Moreover, we provide a construction of Boolean functions (semi-bent Boolean functions) which can be used in the construction of orthogonal variable spreading factor codes used in code division multiple access (CDMA) systems as well as in certain cryptographic applications

A Construction of Bent Functions of n + 2 Variables from a Bent Function of n Variables and Its Cyclic Shifts

We present a method to iteratively construct new bent functions of n + 2 variables from a bent function of n variables and its cyclic shift permutations using minterms of n variables and minterms of 2 variables. In addition, we provide the number of bent functions of n + 2 variables that we can obtain by applying the method here presented, and finally we compare this method with a previous one introduced by us in 2008 and with the Rothaus and Maiorana-McFarland constructions.The work of the first author was partially supported by Spanish Grant MTM2011-24858 of the Ministerio de Economía y Competitividad of the Gobierno de España

The complexity of Boolean functions from cryptographic viewpoint

Cryptographic Boolean functions must be complex to satisfy Shannon\u27s principle of confusion. But the cryptographic viewpoint on complexity is not the same as in circuit complexity. The two main criteria evaluating the cryptographic complexity of Boolean functions on $F_2^n$ are the nonlinearity (and more generally the $r$-th order nonlinearity, for every positive $r< n$) and the algebraic degree. Two other criteria have also been considered: the algebraic thickness and the non-normality. After recalling the definitions of these criteria and why, asymptotically, almost all Boolean functions are deeply non-normal and have high algebraic degrees, high ($r$-th order) nonlinearities and high algebraic thicknesses, we study the relationship between the $r$-th order nonlinearity and a recent cryptographic criterion called the algebraic immunity. This relationship strengthens the reasons why the algebraic immunity can be considered as a further cryptographic complexity criterion

Algorithm 959: VBF: A Library of C plus plus Classes for Vector Boolean Functions in Cryptography

VBF is a collection of C++ classes designed for analyzing vector Boolean functions (functions that map a Boolean vector to another Boolean vector) from a cryptographic perspective. This implementation uses the NTL library from Victor Shoup, adding new modules that call NTL functions and complement the existing ones, making it better suited to cryptography. The class representing a vector Boolean function can be initialized by several alternative types of data structures such as Truth Table, Trace Representation, and Algebraic Normal Form (ANF), among others. The most relevant cryptographic criteria for both block and stream ciphers as well as for hash functions can be evaluated with VBF: it obtains the nonlinearity, linearity distance, algebraic degree, linear structures, and frequency distribution of the absolute values of the Walsh Spectrum or the Autocorrelation Spectrum, among others. In addition, operations such as equality testing, composition, inversion, sum, direct sum, bricklayering (parallel application of vector Boolean functions as employed in Rijndael cipher), and adding coordinate functions of two vector Boolean functions are presented. Finally, three real applications of the library are described: the first one analyzes the KASUMI block cipher, the second one analyzes the Mini-AES cipher, and the third one finds Boolean functions with very high nonlinearity, a key property for robustness against linear attacks