Mathematical Background Jean-Luc Beuchat, Nadia El Mrabet, Laura Fuentes-Castañeda, and Francisco Rodríguez-Henríquez
In this chapter we will review the essential mathematical concepts, definitions, and properties required for the formal description of bilinear pairings. The chapter is organized into three main parts. First, we give in Section 2.1 basic algebra definitions, which will be used in the remaining part of this book. Then, in Section 2.2, we state several facts on finite fields, extension finite fields, and their arithmetic. Finally, in Section 2.3, we give a basic introduction to elliptic curves and their properties, which are especially relevant for pairing-based cryptography.