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The hash functions using elliptic curve cryptography are hash functions that are produced using both an elliptic curve and a twist of the elliptic curve. Hash points are assigned values that either correspond to points on the elliptic curve or to points on the twist, depending upon whether the scalar value of the corresponding message block produces a quadratic residue or a quadratic non-residue when substituted as the x-value into the elliptic curve equation. The corresponding hash point x-coordinates are concatenated to form the hash bit string. The hash points may be doubled, and the hash functions may be applied to multimedia data by applying a media compression method to the message data before computing the hash points.