Hashing into Hessian Curves

We describe a hashing function from the elements of the finite field Fq into points on a Hessian curve. Our function features the uniform and smaller size for the cardinalities of almost all fibers compared with the other known hashing functions for elliptic curves. Moreover, a point on the image set of the function is uniquely given by its abscissa. For ordinary Hessian curves, the cardinality of the image set of the function is exactly given by (q + i)/2 for some i = 1, 2, 3.