A Computational Study of Icart\u27s Function

A hash function maps some elements of a larger, initial set to elements of a smaller, resultant set. By nature, this leads to collisions and, sometimes, not all elements in the smaller set will be mapped to as a result. The set in consideration here is all points on an elliptic curve. This is a special class of curve with two variables, which takes the form here as y2 = x3 + ax + b. A hash function is useful in offering a deterministic way to map an input to a pair of x and y values that satisfy such an equation. This paper experimentally verifies that an asymptotic result on the size of the image for Icart\u27s hash function provided by Fouque and Tibouchi is true for small primes less than 219 and for all curves of conductor less than or equal to 100. Combined with Fouque and Tibouchi\u27s asymptotic result, this proves that the coverage of Icart\u27s hash function is a 5/8 fraction of the points (with some error)