A NOVEL ALGORITHM ENUMERATING BENT FUNCTIONS

By the relationship between the Walsh spectra at partial points and the Walsh spectra of its sub-functions, by the action of general linear group on the set of Boolean functions, and by the Reed-Muller transform, a novel method is developed, which can theoretically construct all bent functions. With this method, we enumerate all bent functions in 6 variables; in 8-variable case, our method is more efficient than the method presented by Clark though we still can not enumerate all bent functions; enumeration of all homogeneous bent functions of degree 3 in eight variables can be done in one minute by a P4 1.7G HZ computer; construction of homogenous bent function of degree 3 in 10 variables is efficient too; the nonexistence of homogeneous bent functions in 10 variables of degree 4 is prove

Rotation symmetric Boolean functions---count and cryptographic properties

The article of record as published may be located at http://dx.doi.org/10.1.1.137.6388Rotation symmetric (RotS) Boolean functions have been used as components of different cryptosystems. This class of Boolean functions are invariant under circular translation of indices. Using Burnsideﾒs lemma it can be seen that the number of n-variable rotation symmetric Boolean functions is 2gn, where gn = 1 nPt|n (t) 2n t , and (.) is the Euler phi-function. In this paper, we find the number of short and long cycles of elements in Fn2 having fixed weight, under the RotS action. As a consequence we obtain the number of homogeneous RotS functions having algebraic degree w. Our results make the search space of RotS functions much reduced and we successfully analyzed important cryptographic properties of such functions by executing computer programs. We study RotS bent functions up to 10 variables and observe (experimentally) that there is no homogeneous rotation symmetric bent function having degree > 2. Further, we studied the RotS functions on 5, 6, 7 variables by computer search for correlation immunity and propagation characteristics and found some functions with very good cryptographic properties which were not known earlier

Analysis and design of some cryptographic Boolean functions

Boolean functions are vital components of symmetric-key ciphers such as block ciphers, stream ciphers and hash functions. When used in cipher systems, Boolean functions should satisfy several cryptographic properties such as balance, high nonlinearity, resiliency and high algebraic degree. Bent functions achieve the maximum possible nonlinearity and hence they provide optimal resistance to several cryptographic attacks such as linear and differential cryptanalysis. We present some simple constructions for binary bent functions of length 2 2 k using a known bent function of length 2 2 k -2 . Adams and Tavares introduced two classes of bent functions: bent-based bent functions and linear-based bent functions. In this thesis we explore different bent-based constructions. In particular, we show that all nonlinear resilient functions with maximum order resiliency are either bent-based or linear-based. We provide an explicit count for the number of such resilient functions that belong to both classes. We also provide a simple proof that all symmetric functions that achieve the maximum possible nonlinearity are bent-based. In particular, for n even, we have 4 bent-based bent functions. For n odd, we also have 4 bent-based functions. We also prove that there is no bent-based homogeneous functions with algebraic degree >2. Almost all cryptographic properties of Boolean functions can be determined efficiently from its Walsh transform. In this thesis, we present some restrictions on the partial sum of the Walsh transform of binary functions. In several parts of the thesis, we extend the obtained results to functions defined over GF(p

On the Systematic Constructions of Rotation Symmetric Bent Functions with Any Possible Algebraic Degrees

In the literature, few constructions of $n$-variable rotation symmetric bent functions have been presented, which either have restriction on $n$ or have algebraic degree no more than $4$. In this paper, for any even integer $n=2m\ge2$, a first systemic construction of $n$-variable rotation symmetric bent functions, with any possible algebraic degrees ranging from $2$ to $m$, is proposed