Variation on correlation immune Boolean and vectorial functions

Correlation immune functions were introduced to protect some shift register based stream ciphers against correlation attacks. For cryptographic applications, relaxing the concept of correlation immunity has been highlighted and proved to be more appropriate in several cryptographic situations. Various weakened notions of correlation immunity and resiliency have been widely introduced for cryptographic functions, but those notions are difficult to handle.   As a variation, we focus on the notion of $\varphi$-correlation immunity which is closely related to (fast) correlation attacks on stream ciphers based on nonlinear combiner model. In particular, we exhibit new connections between $\varphi$-correlation immunity and $\epsilon$-almost resiliency, which are two distinct approaches for characterizing relaxed resiliency. We also extend the concept of $\varphi$-correlation immunity introduced by Carlet et al. in 2006 for Boolean functions to vectorial functions and study the main cryptographic parameters of $\varphi$-correlation immune functions. Moreover, we provide new primary constructions of $\varphi$-resilient functions with good designed immunity profile. Specially, we propose a new recursive method to construct $\varphi$-resilient functions with high nonlinearity, high algebraic degree, and monotone increasing immunity profile.