The nonlinearity and Hamming weights of rotation symmetric Boolean functions of small degree

Let e, l and n be integers such that 1 <= e < n and 3 <= l <= n. Let < i > denote the least nonnegative residue of i mod n. In this paper, we investigate the following Boolean function F-l,e(n)(X-n) =( )Sigma(n-1)( i=0) x(i)x(< i+e >)x(< i+2e >)...x(< i+(l-1)e >), which plays an important role in cryptography and coding theory. We introduce some new subfunctions and provide some recursive formulas for the Fourier transform. Using these recursive formulas, we show that the nonlinearity of F-l,e(n)(x(n)) is the same as its weight for 5 <= l <= 7. Our result confirms partially a conjecture of Yang, Wu and Hong raised in 2013. It also gives a partial answer to a conjecture of Castro, Medina and Stanica proposed in 2018. Our result extends the result of Zhang, Guo, Feng and Li for the case l = 3 and that of Yang, Wu and Hong for the case l = 4.