McEliece public key cryptosystems using algebraic-geometric codes

McEliece (1978) proposed a public-key cryptosystem based on binary linear codes, in particular binary classical Goppa codes. We look at various aspects of McEliece´s scheme in the general setting of q-ary codes. In particular, we consider schemes based on much larger class of q-ary algebraic-geometric (AG) Goppa codes, subfield subcodes of AG codes, and concatenated codes. We give explicit constructions of several schemes which have very high work factor, excellent key-length/plain-text ratios, and relatively smaller size of the keys for given work factors. We also present its modifications and generalizations following Krouk (see Proceedings of The Sixth Swedish-Russian International Workshop on Information Theory, p.285, 1993) and others. Finally, we discuss some open problems