Halving PGL (2, 2 f ), f odd: A series of cryptocodes

Abstract

A set S of permutations of k objects is μ-uniform, t-homogeneous if for every pair A, B of t-subsets of the ground set, there are exactly μ permutations in S mapping A onto B. Our main result (Theorem 1.2) is the construction of a (q − 1)-uniform, 2-homogeneous set of permutations of q + 1 objects contained in the projective group PGL(2, q), where q is a power of 2 with odd exponent. The main ingredient of the proof is a lemma concerning cubic equations in characteristic 2 (Lemma 2.6). The result is useful in the framework of theoretical secrecy and authentication. By a theorem of D.R. Stinson (Stinson 1990) one obtains families of cryptocodes which achieve perfect 2-fold secrecy and are 1-fold secure against spoofing (Corollary 1.3).