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.