Roughly four decades ago, Taher ElGamal put forward what is today one of the most widely known and best understood public key encryption schemes. ElGamal encryption has been used in many different contexts, chiefly among them by the OpenPGP standard. Despite its simplicity, or perhaps because of it, in reality there is a large degree of ambiguity on several key aspects of the cipher. Each library in the OpenPGP ecosystem seems to have implemented a slightly different "flavour" of ElGamal encryption. While --taken in isolation-- each implementation may be secure, we reveal that in the interoperable world of OpenPGP, unforeseen cross-configuration attacks become possible. Concretely, we propose different such attacks and show their practical efficacy by recovering plaintexts and even secret keys.

Lattices of compatibly embedded finite fields are useful in computer algebra systems for managing many extensions of a finite field \F_p at once. They can also be used to represent the algebraic closure \bar\F_p, and to represent all finite fields in a standard manner.

We present algorithms to construct and do arithmetic operations in the algebraic closure of the finite field Fp. Our approach is inspired by algorithms for constructing irreducible polynomials, which first reduce to prime power degrees, then use composita techniques. We use similar ideas to give efficient algorithms for embeddings and isomorphisms.

Inspired by previous work of Shoup, Lenstra-De Smit and Couveignes-Lercier, we give fast algorithms to compute in the first levels of) the l-adic closure of a finite field. In many cases, our algorithms have quasi-linear complexity.

An Artin-Schreier tower over the finite field Fp is a tower of field extensions generated by polynomials of the form Xp-X-α. Following Cantor and Couveignes, we give algorithms with quasi-linear time complexity for arithmetic operations in such towers. As an application, we present an implementation of Couveignes' algorithm for computing isogenies between elliptic curves using the p-torsion.