In this note we study an analogy between a positive definite quadratic form for elliptic curves over finite fields and a positive definite quadratic form for elliptic curves over the rational number field. A question is posed of which an affirmative answer would imply the analogue of the Riemann hypothesis for elliptic curves over the rational number field.

Since 2000 pairings became a very useful tool to design new protocols in cryptography. Short signaturesand identity-based encryption became also practical thanks to these pairings.This thesis contains two parts. One part is about optimized pairing implementation on different ellip-tic curves according to the targeted protocol. Pairings are implemented on supersingular elliptic curve...