Algorithms for Finite Field Arithmetic

We review several algorithms to construct finite fields and perform operations such as field embedding. Following previous work by notably Shoup, de Smit and Lenstra or Couveignes and Lercier, as well as results obtained with De Feo and Doliskani, we distinguish between algorithms that build "towers" of finite fields, with degrees of the form l, l2, l3,... and algorithms for composita. We show in particular how techniques that originate from algorithms for computing with triangular sets can be useful in such a context.