New generalized almost perfect nonlinear functions

APN (almost perfect non-linear) functions over finite fields of even characteristic are widely studied due to their applications to the design of symmetric ciphers resistant to differential attacks. This notion was recently generalized to GAPN (generalized APN) functions by Kuroda and Tsujie to odd characteristic p. They presented some constructions of GAPN functions, and other constructions were given by Zha et al. We present new constructions of GAPN functions both in the case of monomial and multinomial functions. Our monomial GAPN functions can be viewed as a further generalization of the Gold APN functions. We show that a certain technique used by Hou to construct permutations over finite fields also yields monomial GAPN functions. We also present several new constructions of GAPN functions which are sums of monomial GAPN functions, as well as new GAPN functions of degree p which can be written as the product of two powers of linearized polynomials. For this latter construction we describe some interesting differences between even and odd characteristic and also obtain a classification in certain cases.