The class of bent functions contains a subclass of functions whose properties are still stronger and whose elements are still rarer. Youssef and Gong have proved the existence of such hyper-bent functions, for every even n. We prove that the hyper-bent functions they exhibit are exactly those elements of the well-known class, introduced by Dillon, up to the linear transformations xδx, . Hyper-bent functions seem still more difficult to generate at random than bent functions; however, by showing that they all can be obtained from some codewords of an extended cyclic code H_n with small dimension, we can enumerate them for up to 10 variables. We study the non-zeroes of H_n and we deduce that the algebraic degree of hyper-bent functions is n/2. We also prove that the functions of class are some codewords of weight 2^n-1-2^n/2-1 of a subcode of H_n and we deduce that for some n, depending on the factorization of 2ⁿ-1, the only hyper-bent functions on n variables are the elements of the class , obtained from by composing the functions by the transformations xδx, δ≠0, and by adding constant functions. We prove that non- hyper-bent functions exist for n=4, but it is not clear whether they exist for greater n. We also construct potentially new bent functions for n=12.