Mathematical background

Abstract

In this chapter, the author reviews the essential mathematical concepts, definitions, and properties required for the formal description of bilinear pairings. Fields, and, more particularly, finite fields, together with elliptic curves, play a fundamental role in the construction of pairings. In pairing-based cryptography, one can consider finite fields of small characteristic or finite fields of a large prime characteristic. The computation of pairings relies on the arithmetic of finite fields. In particular, several important optimizations in a pairing execution are based on Fermat's little theorem. The choice of the multiplication algorithm is adapted to the bit size of the operands or the targeted device.