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

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

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