The multicovering radii of a code are recent generalizations of the covering radius of a code. For positive m, the m-covering radius of C is the least radius t such that every m-tuple of vectors is contained in at least one ball of radius t centered at some codeword. In this paper upper bounds are found for the multicovering radii of first order Reed-Muller codes. These bounds generalize the well-known Norse bounds for the classical covering radii of first order Reed-Muller codes. They are exact in some cases. These bounds are then used to prove the existence of secure families of keystreams against a general class of cryptanalytic attacks. This solves the open question that gave rise to the study of multicovering radii of codes.

Abstract. It is surveyed known (published) possible application of binary and n-ary quasigroups in cryptology.