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. For ordinary Hessian curves, this function is 2 : 1 for almost all points. More precisely, for odd q, the cardinality of the image set of the function is exactly given by (q + i + 2)/2 for some i = -1, 1. Next, we present an injective hashing function from the elements of Zm into points on a Hessian curve over Fq with odd q and m = (q+i)/2 for some i = -1, 1, 3.

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