Hashing into Hessian Curves

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Abstract

We describe a hashing function from the elements of the finite field \(\mathbb{F}_q\) 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 ℤ m into points on a Hessian curve over \(\mathbb{F}_q\) with odd q and m = (q + i)/2 for some i = − 1,1,3.