The BF Identity-Based Encryption System

We briefly review the notion of cyclic group equipped with a bilinear map, or pairing. In this note, we focus exclusively on symmetric pairings that support a cryptographic hashing operation directly into the bilinear group (these pairings are also called of type 1). These objects can be instantiated using the modified Weil or Tate pairing on certain algebraic curves over finite fields, that include families of supersingular elliptic curves of large characteristic. We refer the reader to the Boneh-Franklin paper [BF03] and some specialized treatises [BSS99] for details.