Elliptic Curves over Finite Fields: Number Theoretic and Cryptographic Aspects

Abstract

We present a collection of several natural questions about elliptic curves, mostly over finite fields, that have led to some interesting number theoretic questions and whose solutions require rather involved techniques from various area of number theory. Some of these questions are of intrinsic value for the theory of elliptic curves; they stem from their application to cryptography.

The Sato–Tate distribution in families of elliptic curves with a rational parameter of bounded height

“…The properties of the specialisations E(t) modulo consecutive primes p ≤ x for a growing parameter x and for the parameter t that runs through some interesting sets T have recently being investigated quite intensively, see [10,23,27] and also Section 1.2. These sets T can be of integer or rational numbers of limited size, and sometimes also of certain arithmetic structure; for example T can be a set of primes in a given interval [1, T ], see [8].…”

Lang–Trotter and Sato–Tate distributions in single and double parametric families of elliptic curves

“…where EndQ(E) stands for the endomorphism ring of E; but if E is without complex multiplication, there are infinitely many distinct such Frobenius fields as prime p ∤ N E varies. Despite a series of interesting (conditional and unconditional) recent achievements, see [9,10,12,14,27,30] for surveys and some recent results, these conjectures are widely open.…”