This paper introduces a new concept, distortion eigenvector space; it is a (higher dimensional) vector space in which bilinear pairings and distortion maps are available. A distortion eigenvector space can be efficiently realized on a supersingular hyperelliptic curve or a direct product of supersingular elliptic curves. We also introduce an intractable problem (with trapdoor) on distortion eigenvector spaces, the higher dimensional generalization of the vector decomposition problem (VDP). We define several computational and decisional problems regarding VDP, and clarify the relations among them. A trapdoor bijective function with algebraically rich properties can be obtained from the VDP on distortion eigenvector spaces. This paper presents two applications of this trapdoor bijective function; one is multivariate homomorphic encryption as well as a two-party protocol to securely evaluate 2DNF formulas in a higher dimensional manner, and the other is various types of signatures such as ordinary signatures, blind signatures, generically (selectively and universally) convertible undeniable signatures and their combination.