Abstract: Methods and apparati are provided for use in determining “Squared Weil pairings” and/or “Squared Tate Pairing” based on an elliptic curve, for example, and which are then used to support cryptographic processing of selected information. Significant improvements are provided in computing efficiency over the conventional implementation of the Weil and Tate pairings. The resulting Squared Weil and/or Tate pairings can be substituted for conventional Weil or Tate pairings in a variety of applications.

Abstract: A cryptosystem based on a Jacobian of a hyperelliptic curve is being used. Various methods and apparatus are provided for generating a compressed data format that identifies one or more points on the Jacobian of the hyperelliptic curve, and for subsequently decompressing the compressed format data.

Abstract: An implementation of a technology, described herein, for facilitating cryptographic systems and techniques. At least one implementation, described herein, maximizes the speed and security of fast exponentiation while minimizing its expense. At least one implementation, described herein, employs elliptic curves with a fast exponentiation technique so that it maximizes speed and security while minimizing expense. At least one implementation, described herein, employs Koblitz exponentiation with “bucketing” techniques to maximize speed and security of cryptosystems while minimizing expense of such techniques. This abstract itself is not intended to limit the scope of this patent. The scope of the present invention is pointed out in the appending claims.

Abstract: A cryptosystem based on a Jacobian of a hyperelliptic curve is being used. Various methods and apparati are provided for generating a compressed data format that identifies one or more points—on the Jacobian of the hyperelliptic curve, and for subsequently decompressing the compressed format data.

Abstract: A cryptosystem has a secret based on an order of a group of points on a Jacobian of a curve. In certain embodiments, the cryptosystem is used to generate a product identifier corresponding to a particular product. The product identifier is generated by initially receiving a value associated with a copy (or copies) of a product. The received value is padded using a recognizable pattern, and the padded value is converted to a number represented by a particular number of bits. The number is then converted to an element of the Jacobian of the curve, and the element is then raised to a particular power. The result of raising the element to the particular power is then compressed and output as the product identifier. Subsequently, the encryption process can be reversed and the decrypted value used to indicate validity and/or authenticity of the product identifier.

Abstract: A cryptosystem based on a Jacobian of a hyperelliptic curve is being used. Various methods and apparati are provided for generating a compressed data format that identifies one or more points—on the Jacobian of the hyperelliptic curve, and for subsequently decompressing the compressed format data.

Abstract: An implementation of a technology, described herein, for facilitating cryptographic systems and techniques.. At least one implementation, described herein, maximizes the speed and security of fast exponentiation while minimizing its expense. At least one implementation, described herein, employs elliptic curves with a fast exponentiation technique so that it maximizes speed and security while minimizing expense. At least one implementation, described herein, employs Koblitz exponentiation with “bucketing” techniques to maximize speed and security of cryptosystems while minimizing expense of such techniques. This abstract itself is not intended to limit the scope of this patent. The scope of the present invention is pointed out in the appending claims.

Abstract: An implementation of a technology, described herein, for facilitating cryptography and other security processing. At least one implementation, described herein, maximizes the speed and security of fast exponentiation. At least one implementation, described herein, employs exponentiation with optimized elliptic curve “double-and-add” techniques to maximize speed and security of cryptosystems. This abstract itself is not intended to limit the scope of this patent. The scope of the present invention is pointed out in the appending claims.

Abstract: A cryptosystem has a secret based on an order of a group of points on a Jacobian of a curve. In certain embodiments, the cryptosystem is used to generate a product identifier corresponding to a particular product. The product identifier is generated by initially receiving a value associated with a copy (or copies) of a product. The received value is padded using a recognizable pattern, and the padded value is converted to a number represented by a particular number of bits. The number is then converted to an element of the Jacobian of the curve, and the element is then raised to a particular power. The result of raising the element to the particular power is then compressed and output as the product identifier. Subsequently, the encryption process can be reversed and the decrypted value used to indicate validity and/or authenticity of the product identifier.

Abstract: A technique for generating, for a given message to be signed, an authentic cryptographic signature that can be authenticated, by a recipient of the signed message, as having originated from a signor of the message; and appropriately authenticating such a signature. Specifically, this technique, given a message, such as, e.g., a numerical product copy identifier (83, 93), forms an authentic signature (87, 97), based on public-key cryptosystem, through use of generator value selected from points on an elliptic curve over a finite field. The authentic signature is generated using the generator value in conjunction with three keys; namely, a public key, a private key and a secret key, and thus substantially increases the security associated with cryptographic signatures generated through a conventional two-key public key cryptosystem. A unique product copy indicia can be formed by concatenating the identifier, for a given product copy, with its corresponding authentic signature.

Abstract: A technique for generating, for a given message to be signed, an authentic cryptographic signature that can be authenticated, by a recipient of the signed message, as having originated from a signor of the message; and appropriately authenticating such a signature. Specifically, this technique, given a message, such as, e.g., a numerical product copy identifier (83, 93), forms an authentic signature (87, 97), based on public-key cryptosystem, through use of generator value selected from points on an elliptic curve over a finite field. The authentic signature is generated using the generator value in conjunction with three keys; namely, a public key, a private key and a secret key, and thus substantially increases the security associated with cryptographic signatures generated through a conventional two-key public-key cryptosystem. A unique product copy indicia can be formed by concatenating the identifier, for a given product copy, with its corresponding authentic signature.