Summary: In designing cryptographic Boolean functions, it is challenging to achieve at the same time the desirable features of algebraic immunity, balancedness, nonlinearity, and algebraic degree for necessary resistance against algebraic attack, correlation attack, Berlekamp-Massey attack, etc. This paper constructs balanced rotation symmetric Boolean functions on \(n\) variables where \(n=2p\) and \(p\) is an odd prime. We prove the construction has an optimal algebraic immunity and is of high nonlinearity. We check that, at least for those primes \(p\) which are not of the form of a power of two plus one, the algebraic degree of the construction achieves in fact the upper bound \(n-1\).