A Conjecture on Binary String and Its Applications on Constructing Boolean Functions of Optimal Algebraic Immunity

In this paper, we propose a combinatoric conjecture on binary string, on the premise that our conjecture is correct we mainly obtain two classes of functions which are both algebraic immunity optimal: the first class of functions are also bent, moreover, from this fact we conclude that the algebraic immunity of bent functions can take all possible values except one. The second class are balanced functions, which have optimal algebraic degree and the best nonlinearity up to now

1-Resilient Boolean Functions on Even Variables with Almost Perfect Algebraic Immunity

Several factors (e.g., balancedness, good correlation immunity) are considered as important properties of Boolean functions for using in cryptographic primitives. A Boolean function is perfect algebraic immune if it is with perfect immunity against algebraic and fast algebraic attacks. There is an increasing interest in construction of Boolean function that is perfect algebraic immune combined with other characteristics, like resiliency. A resilient function is a balanced correlation-immune function. This paper uses bivariate representation of Boolean function and theory of finite field to construct a generalized and new class of Boolean functions on even variables by extending the Carlet-Feng functions. We show that the functions generated by this construction support cryptographic properties of 1-resiliency and (sub)optimal algebraic immunity and further propose the sufficient condition of achieving optimal algebraic immunity. Compared experimentally with Carlet-Feng functions and the functions constructed by the method of first-order concatenation existing in the literature on even (from 6 to 16) variables, these functions have better immunity against fast algebraic attacks. Implementation results also show that they are almost perfect algebraic immune functions

A general conjecture similar to T-D conjecture and its applications in constructing Boolean functions with optimal algebraic immunity

In this paper, we propose two classes of 2k-variable Boolean functions, which have optimal algebraic immunity under the assumption that a general combinatorial conjecture is correct. These functions also have high algebraic degree and high nonlinearity. One class contain more bent functions, and the other class are balanced

Balanced Boolean Functions with (Almost) Optimal Algebraic Immunity and Very High Nonlinearity

In this paper, we present a class of $2k$-variable balanced Boolean functions and a class of $2k$-variable $1$-resilient Boolean functions for an integer $k\ge 2$, which both have the maximal algebraic degree and very high nonlinearity. Based on a newly proposed conjecture by Tu and Deng, it is shown that the proposed balanced Boolean functions have optimal algebraic immunity and the $1$-resilient Boolean functions have almost optimal algebraic immunity. Among all the known results of balanced Boolean functions and $1$-resilient Boolean functions, our new functions possess the highest nonlinearity. Based on the fact that the conjecture has been verified for all $k\le 29$ by computer, at least we have constructed a class of balanced Boolean functions and a class of $1$-resilient Boolean functions with the even number of variables $\le 58$, which are cryptographically optimal or almost optimal in terms of balancedness, algebraic degree, nonlinearity, and algebraic immunity

Highly Nonlinear Boolean Functions with Optimal Algebraic Immunity and Good Behavior Against Fast Algebraic Attacks

In this paper, we present a new combinatorial conjecture about binary strings. Based on the new conjecture, two classes of Boolean functions of $2k$ variables with optimal algebraic immunity are proposed, where $k\ge 2$. The first class contains unbalanced functions having high algebraic degree and nonlinearity. The functions in the second one are balanced and have maximal algebraic degree and high nonlinearity. It is checked that, at least for small numbers of variables, both classes of functions have a good behavior against fast algebraic attacks. Compared with the known Boolean functions resisting algebraic attacks and fast algebraic attacks, the two classes of functions possess the highest lower bounds on nonlinearity. These bounds are however not enough for ensuring a sufficient nonlinearity for allowing resistance to the fast correlation attack. Nevertheless, as for previously found functions with the same features, there is a gap between the bound that we can prove and the actual values computed for small numbers of variables. Moreover, these values are very good and much better than for the previously found functions having all the necessary features for being used in the filter model of pseudo-random generators

About the Tu-Deng Conjecture for \w(t) Less Than or Equal to 10

Let $k \ge 2$ be an integer, define S_t^k:=\Bigg\{(a,b)\in \mathbb{Z}^2\ \Big| \ { 0 \le a,b \le 2^{k}-2,\ a+b\equiv t ~(\text{mod} \ 2^k-1),\ \w(a)+\w(b)\le{k-1}}\Bigg\}, where $t \in \mathbb{Z}, 1 \le t \le 2^k-2$. This paper gives the upper bound of cardinality of $S_t^k$ when \w(t)\le 10, proving that a conjecture proposed by Tu and Deng in the case

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