Charpin and Gong recently characterized a large class of hyperbent functions defined on fields of order 2n, which include the well known monomial functions with the Dillon exponent as a special case. We give a reformulation of the Charpin-Gong criterion in terms of the number of rational points on certain hyperelliptic curves. We present two applications of our result: The time needed to check the hyperbentness of a specific function is now polynomial in n, and hyperbent functions with subfield coefficients can be constructed.