Bent functions of maximal degree

In this article a technique for constructing p-ary bent functions from plateaued functions is presented. This generalizes earlier techniques of constructing bent from near-bent functions. The Fourier spectrum of quadratic monomials is analysed, examples of quadratic functions with highest possible absolute values in their Fourier spectrum are given. Applying the construction of bent functions to the latter class of functions yields bent functions attaining upper bounds for the algebraic degree when $p=3,5$. Until now no construction of bent functions attaining these bounds was known

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

Improving the lower bound on the maximum nonlinearity of 1-resilient Boolean functions and designing functions satisfying all cryptographic criteria

In this paper, we improve the lower bound on the maximum nonlinearity of 1-resilient Boolean functions, for $n$ even, by proposing a method of constructing this class of functions attaining the best nonlinearity currently known. Thus for the first time, at least for small values of $n$, the upper bound on nonlinearity can be reached in a deterministic manner in difference to some heuristic search methods proposed previously. The nonlinearity of these functions is extremely close to the maximum nonlinearity attained by bent functions and it might be the case that this is the highest possible nonlinearity of 1-resilient functions. Apart from this theoretical contribution, it turns out that the cryptographic properties of these functions are overall good apart from their moderate resistance to fast algebraic attacks (FAA). This weakness is repaired by a suitable modification of the original functions giving a class of balanced functions with almost optimal resistance to FAA whose nonlinearity is better than the nonlinearity of other methods

Generating S-Boxes from Semi-fields Pseudo-extensions

Block ciphers, such as the AES, correspond to a very important family of secret-key cryptosystems. The security of such systems partly relies on what is called the S-box. This is a vectorial Boolean function f : F n 2 ֒→ F n 2 , where n is the size of the blocks. It is often the only non linear opera-tion in the algorithm. The most well-known attacks against block ciphers algorithms are the known-plaintext attacks called differential cryptanal-ysis [4, 10] and linear cryptanalysis [11]. To protect such cryptosystems against linear and differential attacks, S-boxes are designed to fulfill some cryptographic criteria (balancedness, high nonlinearity, high algebraic de-gree, avalanche, or transparency [2, 12]) and are usually defined on finite fields, like F2n [7, 3]. Unfortunately, it seems difficult to find good S-Boxes, at least for bijective ones: random generation does not work [8, 9] and the one used in the AES or Camellia are actually variations around a single function, the inverse function in F2n . Would the latter function have an unforeseen weakness (for instance if more practical algebraic attacks are developped), it would be desirable to have some replacement candidates. For that matter, we propose to weaken a little bit the algebraic part of the design of S-Boxes and use finite semi-fields instead of finite fields to build such S-Boxes. Finite semi-fields relax the associativity and com-mutativity of the multiplication law. While semi-fields of a given order are unique up to isomorphism, on the contrary semi-fields of a given order can be numerous: nowadays, on the one hand, it is for instance easy to generate all the 36 semi-fields of order 2 4 , but, on the other hand, it is not even known how many semi-fields are there of order 2 8 . Therefore, we propose to build S-Boxes via semi-fields pseudo extensions of the form S 2 2 4 , where S 2 4 is any semi-field of order 2 4 , and mimic in this structure the use of the inverse function in a finite field. We report here the construction of 10827 S-Boxes, 7052 non CCZ-equivalent, with maximal nonlinearity, differential invariants, degrees and bit interdependency. Among the latter 2963 had fix points, and among the ones without fix points, 3846 had the avalanche level of AES and 243 1 the better avalanche level of Camellia. Among the latter 232 have a better transparency level than the inverse function on a finite field

ON DILLON\u27S CLASS H OF BENT FUNCTIONS, NIHO BENT FUNCTIONS AND O-POLYNOMIALS

One of the classes of bent Boolean functions introduced by John Dillon in his thesis is family H. While this class corresponds to a nice original construction of bent functions in bivariate form, Dillon could exhibit in it only functions which already belonged to the well- known Maiorana-McFarland class. We first notice that H can be extended to a slightly larger class that we denote by H. We observe that the bent functions constructed via Niho power functions, which four examples are known, due to Dobbertin et al. and to Leander-Kholosha, are the univariate form of the functions of class H. Their restrictions to the vector spaces uF2n=2 , u 2 F? 2n, are linear. We also characterize the bent functions whose restrictions to the uF2n=2 \u27s are affine. We answer to the open question raised by Dobbertin et al. in JCT A 2006 on whether the duals of the Niho bent functions introduced in the paper are Niho bent as well, by explicitely calculating the dual of one of these functions. We observe that this Niho function also belongs to the Maiorana-McFarland class, which brings us back to the problem of knowing whether H (or H) is a subclass of the Maiorana-McFarland completed class. We then show that the condition for a function in bivariate form to belong to class H is equivalent to the fact that a polynomial directly related to its definition is an o-polynomial and we deduce eight new cases of bent functions in H which are potentially new bent functions and most probably not affine equivalent to Maiorana-McFarland functions