Improved construction of nonlinear resilient S-boxes

We provide two new construction methods for nonlinear resilient functions. The first method is a simple modification of a construction due to Zhang and Zheng and constructs n-input, m-output resilient S-boxes with degree d>m. We prove by an application of the Griesmer bound for linear error-correcting codes that the modified Zhang-Zheng construction is superior to the previous method of Cheon in Crypto 2001. Our second construction uses a sharpened version of the Maiorana-McFarland technique to construct nonlinear resilient functions. The nonlinearity obtained by our second construction is better than previously known construction methods.