A Class of Cartesian Authentication Codes Constructed from PG(n,q)

Let P be a point of projective space PG(n,q) (n ≥ 3 ) over field GF(q), π be a S-space through P in PG(n,q) (S ≥ 1). Let S be the source set composed of r- spaces through π (s <; r ≤ n), E be the encode rules set composed of (n- s) -spaces which intersect π at P, M be the message set composed of (r - s) -spaces which intersect π at P. For any π₁∈ S, π₂ ∈ E, define f(π₁, π₂) = π₁ ∩ π₂ , we have a class of Cartesian authentication codes. We compute the parameters of these codes. Assume that the encoding rules are chosen according to a uniform probability distribution, the largest probabilities of a successful impersonation attack P_I and the largest probabilities of a successful substitution attack P_S of these codes are also computed.