Several fuzzy connectives, including those proposed by Lotfi Zadeh, can be seen as linear extensions of the Boolean connectives from the scale ${0,1}$ into the scale $[0,1]$. We discuss these extensions, in particular, we focus on the dualities arising from the Boolean dualities. These dualities allow to transfer the results from some particular class of extended Boolean functions, e.g., from c...

the square map is one of the functions used in cryptography. for instance, the square map is used in rabin encryption scheme, block cipher rc6 and stream cipher rabbit, in different forms. in this paper, we study statistical properties of the output of the square map as a vectorial boolean function. we obtain the joint probability distribution of arbitrary number of the upper and the lower bits...

This paper discusses the design of binary image operators from training data and its relation to Boolean function learning. An extended version of the incremental splitting of intervals (ISI) algorithm for Boolean function learning and some improvements and heuristics to reduce its processing time are proposed. Some examples illustrate the application of the algorithm.

We use combinatorial methods and permutation groups to classify homogeneous boolean functions. The property of symmetry of a boolean function limits the size of the function’s class. We exhaustively searched for all boolean functions on V6. We found two interesting classes of degree 3 homogeneous boolean functions: the first class is degree 3 homogeneous bent boolean functions; and the second i...

In this paper, for the first time, we present quantum algorithms to check the order of resiliency of a Boolean function. We first show that the DeutschJozsa algorithm can be directly used for this purpose. We also point out how the quadratic improvement in query complexity over the Deutsch-Jozsa algorithm can be obtained using the well known Grover’s algorithm. While the worst case quantum quer...

All discrete function representations become exponential in size in the worst case. Binary decision diagrams have become a common method of representing discrete functions in computer-aided design applications. For many functions, binary decision diagrams do provide compact representations. This work presents a way to represent large decision diagrams as multiple smaller partial binary decision...