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.