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

Algebraic completions of SBIBD

We study the minimal set of vectors required to make the (0,1) incidence matrix of an SBIBD(v,k,r). We devise algorithms to generate all possible vectors which could complete SBIBD and to find minimal sets of such vectors. The 2-(7,3,l) and 2-( 16,6,2) designs were studied to test our algorithms on the work of Greenhill and Street. We show the minimal defining set of the 5i5/BD(31,15, 7) constructed using the Paley difference set has 13 blocks. We give a minimal defining set with 19 blocks for the SBIBD (31, 15, 7) constructed using the Hall difference set. We show the smallest minimal defining set of teh Hall SBIBD(31,15,7) has between 16 and 20 elements. We investigate the influence of the removal of vectors from minimal sets and discuss the relationship with secret sharing schemes

Back circulant Latin squares and the influence of a set

We define the notions of nest and influence of a subset of a critical set of a back circulant latin square, and study their properties. We also show that a secret sharing scheme based on a critical set of a latin square is both compartmentalised, and hierachical