On applications of simulated annealing to cryptology

Boolean functions are critical building blocks of symmetric-key ciphers. In most cases, the security of a cipher against a particular kind of attacks can be explained by the existence of certain properties of its underpinning Boolean functions. Therefore, the design of appropriate functions has received significant attention from researchers for several decades. Heuristic methods have become very powerful tools for designing such functions. In this thesis, we apply simulated annealing methods to construct Boolean functions with particular properties. Our results meet or exceed the best results of available theoretical constructions and/or heuristic searches in the literature, including a 10-variable balanced Boolean function with resiliency degree 2, algebraic degree 7, and nonlinearity 488 for the first time. This construction affirmatively answers the open problem about the existence of such functions. This thesis also includes results of cryptanalysis for symmetric ciphers, such as Geffe cipher and TREYFER cipher

Systematic Constructions of Bent-Negabent Functions, 2-Rotation Symmetric Bent-Negabent Functions and Their Duals

Bent-negabent functions have many important properties for their application in cryptography since they have the flat absolute spectrum under the both Walsh-Hadamard transform and nega-Hadamard transform. In this paper, we present four new systematic constructions of bent-negabent functions on $4k, 8k, 4k+2$ and $8k+2$ variables, respectively, by modifying the truth tables of two classes of quadratic bent-negabent functions with simple form. The algebraic normal forms and duals of these constructed functions are also determined. We further identify necessary and sufficient conditions for those bent-negabent functions which have the maximum algebraic degree. At last, by modifying the truth tables of a class of quadratic 2-rotation symmetric bent-negabent functions, we present a construction of 2-rotation symmetric bent-negabent functions with any possible algebraic degrees. Considering that there are probably no bent-negabent functions in the rotation symmetric class, it is the first significant attempt to construct bent-negabent functions in the generalized rotation symmetric class

On lower bounds of second-order nonlinearities of cubic bent functions constructed by concatenating Gold functions

In this paper we consider cubic bent functions obtained by Leander and McGuire (J. Comb. Th. Series A, 116 (2009) 960-970) which are concatenations of quadratic Gold functions. A lower bound of second-order nonlinearities of these functions is obtained. This bound is compared with the lower bounds of second-order nonlinearities obtained for functions belonging to some other classes of functions which are recently studied

On the Lower Bounds of the Second Order Nonlinearity of some Boolean Functions

The $r$-th order nonlinearity of a Boolean function is an important cryptographic criterion in analyzing the security of stream as well as block ciphers. It is also important in coding theory as it is related to the covering radius of the Reed-Muller code $\mathcal{R}(r, n)$. In this paper we deduce the lower bounds of the second order nonlinearity of the two classes of Boolean functions of the form \begin{enumerate} \item $f_{\lambda}(x) = Tr_1^n(\lambda x^{d})$ with $d=2^{2r}+2^{r}+1$ and $\lambda \in \mathbb{F}_{2^{n}}$ where $n = 6r$. \item $f(x,y)=Tr_1^t(xy^{2^{i}+1})$ where $x,y \in \mathbb{F}_{2^{t}}, n = 2t, n \ge 6$ and $i$ is an integer such that $1\le i < t$, $\gcd(2^t-1, 2^i+1) = 1$. \end{enumerate} For some $\lambda$, the first class gives bent functions whereas Boolean functions of the second class are all bent, i.e., they achieve optimum first order nonlinearity

Generalized rotation symmetric and dihedral symmetric boolean functions - 9 variable boolean functions with nonlinearity 242

Recently, 9-variable Boolean functions having nonlinearity 241, which is strictly greater than the bent concatenation bound of 240, have been discovered in the class of Rotation Symmetric Boolean Functions (RSBFs) by Kavut, Maitra and Yucel. In this paper, we present several 9-variable Boolean functions having nonlinearity of 242, which we obtain by suitably generalizing the classes of RSBFs and Dihedral Symmetric Boolean Functions (DSBFs). These functions do not have any zero in the Walsh spectrum values, hence they cannot be made balanced easily. This result also shows that the covering radius of the first order Reed-Muller code R(1, 9) is at least 242

Özilinti ve walsh spektrumlarında üstün kriptografik özelliklere sahip boole işlevleri.

We introduce a steepest-descent-like search algorithm for the design of Boolean functions, yielding multiple desirable cryptographic properties in their Walsh and autocorrelation spectra together. The algorithm finds some Boolean functions on 9, 10, 11, 13 variables with very good cryptographic properties unattained in the literature. More specifically, we have discovered 9-variable rotation symmetric Boolean functions (RSBFs) having nonlinearity of 241, which exceeds the bent concatenation bound and has remained as an open question in the literature for almost three decades. We have then shown that there is no RSBF having nonlinearity greater than 241, and that there are 8x189 many RSBFs having nonlinearity of 241, such that, among them there are only two that are different up to the affine equivalence. We also propose a generalization to RSBFs and dihedral symmetric Boolean functions (DSBFs), which improves the nonlinearity result of 9-variable Boolean functions to 242. Further, we classify all possible permutations (362, 880) on the input variables of 9-variable Boolean functions and find that there are only 30 classes, which are different with respect to the linear equivalence of invariant Boolean functions under some permutations. Some of these classes and their subsets yield new 9-variable Boolean functions having the nonlinearity of 242 with different autocorrelation spectra from those of the Boolean functions found in generalized RSBF and DSBF classes. Moreover, we have attained 13-variable balanced Boolean functions having nonlinearity of 4036 which is greater than the bent concatenation bound of 4032, and improves the recent result of 4034.Ph.D. - Doctoral Progra