Constructions Of Boolean Functions And Multi-Output Boolean Functions In Cryptography

Cryptographic Boolean functions and Boolean permutations have wide applicationin stream ciphers and block ciphers. This dissertation investigates the constructions of Boolean functions and Boolean permutations. The author obtains main results as follows:(1). We first point out the Corollary5,6of the letter "A new class of bent functions" presented by Ma, et al. in2005and the Theory5,6of the paper "On bent and semi-bent quadratic boolean functions" presented by Charpin, et al. in2005are not very right. In addition, the corresponding correct conclusions are proposed. A new method for constructing bent functions in polynomial forms is presented by using both bent functions of two trace terms and permutations of polynomial.(2). Thirty years ago, Rothaus introduced the notion of bent function and presented a secondary construction (building new bent functions from already defined ones), which is now called the Rothaus’construction. This construction has a strict re-quirement for its initial functions. We first concentrate on the design of the ini-tial functions in the Rothaus construction. We show how to construct Maiorana-McFarland’s (M-M) bent functions, which can then be used as initial functions, from Boolean permutations and orthomorphic permutations. We present a lower bound of the number of bent functions that are constructed by using Rothaus’construc-tion. In addition, we present a new secondary construction of bent functions which generalizes the Rothaus construction. This construction requires initial functions with stronger conditions; we give examples of functions satisfying them. Further, we generalize the new secondary construction of bent functions and illustrate it with examples.(3). A novel secondary construction of1-resilient functions is obtained. Furthermore, we present the relationships between the properties of these constructed1-resilient functions and those of the initial functions. Based on the construction and a class of bent functions on n variables, we can obtain a class of (n+3)-variable1-resilient non-separable cryptographic functions with high algebraic immunity, whose nonlinearity is equal to the bent concatenation bound2n+2-2(n+2)/2. Furthermore, we propose a set of1-resilient non-separable functions on odd number of variables with optimal algebraic degree, high algebraic immunity, and high nonlinearity. On the basis of the construction of1-resilient functions, a method for constructing (n+3,[n/2])-resilient functions is presented. (4). A technique for constructing balanced Boolean functions on even number of vari-ables is presented. The main technique is to utilize a set of disjoint spectra functions and a special Boolean permutation on a small number of variables to derive a highly nonlinear balanced Boolean function of optimal algebraic degree. It is shown that the constructed functions are different from both Maiorana-McFarland’s Superclass functions introduced by Car let and modified Maiorana-McFarland’s Superclass func-tions presented by Zeng and Hu. In addition, we show that the newly constructed highly nonlinear balanced functions have no non-zero linear structures.(5). We first put forward a method to propose the inverse of a given Boolean permu-tation. It is shown that a Boolean permutation has an optimal algebraic degree if and only if its inverse has an optimal algebraic degree. Furthermore, A class of Boolean permutations is presented. In addition, we present the inverse of the constructed Boolean permutation, and show that the inverse permutation has the largest algebraic degree. Finally, we show that the constructed Boolean permuta-tions can achieve optimum algebraic degree by selecting an appropriate initial vector and illustrate it with examples.(6). A new recursive method on constructing an n-variable orthomorphic permutations from two (n-2)-variable ones is proposed. Furthermore, it is shown that the new constructed orthomorphic permutations are different from the known ones. In addition, we show that the constructed orthomorphic permutations are pair-wise different. Finally, combining the known orthomorphic permutations and a special class of orthomorphic permutations, we count the number of the new constructed orthomorphic permutations.