In this paper, we first recall the concept of 2-tuples distribution matrix, and further study its properties. Based on these properties, we find four special classes of 2-tuples distribution matrices. Then, we provide a new sufficient and necessary condition for n-variable rotation symmetric Boolean functions to be 2-correlation immune. Finally, we give a new method for constructing such functions when n=4t - 1 is prime, and we show an illustrative example.

Rotation symmetric Boolean functions which are invariant under the action of cyclic group have been used in many different cryptosystems. This paper presents a new construction of balanced odd-variable rotation symmetric Boolean functions with optimum algebraic immunity. It is checked that, at least for some small variables, such functions have very good behavior against fast algebraic attacks. Compared with some known rotation symmetric Boolean functions with optimum algebraic immunity, the new construction has really better nonlinearity. Further, the algebraic degree of the constructed functions is also high enough.

Recent research has shown that the class of rotation symmetric Boolean functions is beneficial to cryptographics. In this paper, for an odd prime p, two sufficient conditions for p-variable rotation symmetric Boolean functions to be 1-resilient are obtained, and then several concrete constructions satisfying the conditions are presented. This is the first time that resilient rotation symmetric Boolean functions have been systematically constructed. In particular, we construct a class of 2-resilient rotation symmetric Boolean functions when p=2m+1 for m ≥ 4. Moreover, several classes of 1-order correlation immune rotation symmetric Boolean functions are also got.

Rotation symmetric Boolean functions (RSBFs) that are invariant under circular translation of indices have been used as components of different cryptosystems. In this paper, odd-variable balanced RSBFs with maximum algebraic immunity (AI) are investigated. We provide a construction of n-variable (n=2k+1 odd and n ≥ 13) RSBFs with maximum AI and nonlinearity ≥ 2n-1-¥binom{n-1}{k}+2k+2k-2-k, which have nonlinearities significantly higher than the previous nonlinearity of RSBFs with maximum AI.

It is well known that Boolean functions used in stream and block ciphers should have high algebraic immunity to resist algebraic attacks. Up to now, there have been many constructions of Boolean functions achieving the maximum algebraic immunity. In this paper, we present several constructions of rotation symmetric Boolean functions with maximum algebraic immunity on an odd number of variables which are not symmetric, via a study of invertible cyclic matrices over the binary field. In particular, we generalize the existing results and introduce a new method to construct all the rotation symmetric Boolean functions that differ from the majority function on two orbits. Moreover, we prove that their nonlinearities are upper bounded by .