Hashing byte streams into elements of the Shafarevich-Tate group of an abelian variety
First Claim
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1. A computer-implemented method comprising:
- generating, via a computer, a Shafarevich-Tate group from an abelian variety;
determining, via the computer, a Cassels-Tate pairing based on elements of the Shafarevich-Tate group;
cryptographically processing, via the computer, data based on the Cassels-Tate pairing by hashing the data into an element of the Shafarevich-Tate group using Kolyvagin cohomology classes; and
communicating, by the computer, the cryptographically processed data that includes signed data and a calculated signature that is used to verify the signed data based on a calculated Cassels-Tate pairing.
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Abstract
Systems and methods for cryptographically processing data as a function of a Cassels-Tate pairing are described. In one aspect, a Shafarevich-Tate group is generated from an abelian variety. A Cassels-Tate pairing is determined as a function of elements of the Shafarevich-Tate group. Data is then cryptographically processed as a function of the Cassels-Tate pairing by using Kolyvagin cohomology classes to hash the data into an element of the Shafarevich-Tate group.
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Citations
20 Claims
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1. A computer-implemented method comprising:
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generating, via a computer, a Shafarevich-Tate group from an abelian variety; determining, via the computer, a Cassels-Tate pairing based on elements of the Shafarevich-Tate group; cryptographically processing, via the computer, data based on the Cassels-Tate pairing by hashing the data into an element of the Shafarevich-Tate group using Kolyvagin cohomology classes; and communicating, by the computer, the cryptographically processed data that includes signed data and a calculated signature that is used to verify the signed data based on a calculated Cassels-Tate pairing. - View Dependent Claims (2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12)
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13. A computer-implemented method comprising:
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generating, via a computer, a Shafarevich-Tate group from an abelian variety; determining, via the computer, a Cassels-Tate pairing based on elements of the Shafarevich-Tate group; cryptographically processing, via the computer, data based on the Cassels-Tate pairing by; selecting a public element x from the Shafarevich-Tate group of a dual of A; and hashing a messages M into the Shafarevich-Tate group of A using a Kolyvagin construction; and communicating, via the computer, the cryptographically processed data that includes signed data and a calculated signature that is used to verify the signed data based on a calculated Cassels-Tate pairing. - View Dependent Claims (14, 15, 16, 17, 18, 19)
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20. A computing device comprising:
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a processor; and a memory encoded with computer-executable instructions that, when executed, configure the computing device to perform operations comprising; generating a Shafarevich-Tate group from a cohomology group; determining a Cassels-Tate pairing based on elements of the Shafarevich-Tate group; cryptographically processing data based on the Cassels-Tate pairing by hashing the data into an element of the Shafarevich-Tate group using Kolyvagin cohomology classes; and communicating the cryptographically processed data that includes signed data and a calculated signature that is used to verify the signed data based on a calculated Cassels-Tate pairing.
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Specification