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).
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
References
Bierbrauer, J., and van Trung, T. Some highly symmetric authentication perpendicular arrays, (manuscript).
O'Nan, M.E., 1985. Sharply 2-transitive sets of permutations. Proc. Rutgers Group Theory Year 1983–1984 (ed. M. Aschbacher et. al.), Cambridge: Cambridge Univ. Press.
Stinson, D.R., 1990. The combinatorics of authentication and secrecy codes. Journal of Cryptology, 2:23–49.
Stinson, D.R., and Teirlinck, L., 1990. A construction for authentication/secrecy codes from 3-homogeneous permutation groups. Europ. J. Comb. 11:73–79.
Author information
Authors and Affiliations
Additional information
Communicated by D. Stinson
Rights and permissions
About this article
Cite this article
Bierbrauer, J., Van Trung, T. Halving PGL (2, 2f), f odd: A series of cryptocodes. Des Codes Crypt 1, 141–148 (1991). https://doi.org/10.1007/BF00157618
Received:
Revised:
Issue Date:
DOI: https://doi.org/10.1007/BF00157618