Identifying flaws in the security of critical sets in latin squares via triangulations

In this paper we answer a question in theoretical cryptography by reducing it to a seemingly unrelated geometrical problem. Drápal (1991) showed that a given partition of an equilateral triangle of side n into smaller, integer-sided equilateral triangles gives rise to, under certain conditions, a latin trade within the latin square based on the addition table for the integers (mod n). We apply this result in the study of flaws within certain theoretical cryptographic schemes based on critical sets in latin squares. We classify exactly where the flaws occur for an infinite family of critical sets. Using Drápal’s result, this classification is achieved via a study of the existence of triangulations of convex regions that contain prescribed triangles. 244 DONOVAN, LEFEVRE, MCCOURT, CAVENAGH, KHODKAR