Abstract: The hash functions with elliptic polynomial hopping are based upon an elliptic polynomial discrete logarithm problem. Security using hash functions is dependent upon the implementation of a computationally hard problem, and the elliptic polynomial discrete logarithm problem provides enough relative difficulty in computation to ensure that the produced hash functions, as applied to message bit strings, are optimally secure. The hash functions are produced as functions of both the elliptic polynomial as well as the twist of the elliptic polynomial, particularly using a method of polynomial hopping.

Abstract: 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.

Abstract: The message authentication code with elliptic polynomial hopping provides methods for the generation of message authentication codes (MACs) utilizing elliptic curves, which are based on the elliptic curve discrete logarithm problem. The elliptic curve discrete logarithm problem is well known to be a computationally “difficult” or “hard” problem, thus providing enhanced security for the MACs. Different elliptic polynomials are used for different blocks of the same plaintext, each elliptic polynomial for each message block being selected at random using an initial secret key and a random number generator.

Abstract: The cryptographic hash functions using of elliptic polynomial polynomials are based on the elliptic polynomial discrete logarithm problem, which is well known as a computationally hard problem. The hash functions are based on the elliptic polynomial equation in their generation, where different elliptic polynomials are used for different blocks of the same plain text. Particularly, the hash functions use an elliptic polynomial with more than one independent x-coordinate. More specifically, a set of elliptic polynomial points are used that satisfy an elliptic polynomial equation with more than one independent x-coordinate which is defined over a finite field F.

Abstract: The hash functions with elliptic polynomial hopping are based upon an elliptic polynomial discrete logarithm problem. Security using hash functions is dependent upon the implementation of a computationally hard problem, and the elliptic polynomial discrete logarithm problem provides enough relative difficulty in computation to ensure that the produced hash functions, as applied to message bit strings, are optimally secure. The hash functions are produced as functions of both the elliptic polynomial as well as the twist of the elliptic polynomial, particularly using a method of polynomial hopping.

Abstract: The message authentication code with elliptic polynomial hopping provides methods for the generation of message authentication codes (MACs) utilizing elliptic curves, which are based on the elliptic curve discrete logarithm problem. The elliptic curve discrete logarithm problem is well known to be a computationally “difficult” or “hard” problem, thus providing enhanced security for the MACs. Different elliptic polynomials are used for different blocks of the same plaintext, each elliptic polynomial for each message block being selected at random using an initial secret key and a random number generator.

Abstract: 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.

Abstract: The cryptographic hash functions using of elliptic polynomial polynomials are based on the elliptic polynomial discrete logarithm problem, which is well known as a computationally hard problem. The hash functions are based on the elliptic polynomial equation in their generation, where different elliptic polynomials are used for different blocks of the same plain text. Particularly, the hash functions use an elliptic polynomial with more than one independent x-coordinate. More specifically, a set of elliptic polynomial points are used that satisfy an elliptic polynomial equation with more than one independent x-coordinate which is defined over a finite field F.