Let A be a rational n × n square matrix and b be a rational n-vector for some positive integer n. The linear complementarity problem (abbreviated by LCP) is to find a vector (x, y) in R^(2n) satisfying y = Ax + b (x, y) >= 0 and the complementarity condition: xi yi = 0 for all i = 1 ,... , n. The LCP is known to be NP-complete, but there are some known classes of matrices A for which the LCP is polynomially solvable, for example the class of positive semi-definite (PSD-) matrices. In this paper, we study the LCP from the view point of EP (existentially polynomial time) theorems due to Cameron and Edmonds. In particular, we investigate the LCP duality theorem of Fukuda and Terlaky in EP form, and show that this immediately yields a simple modification of the criss- cross method with a nice practical feature. Namely, this algorithm can be applied to any given A and b, and terminates in one of the three states: (1) a solution x is found; (2) a solution to the dual LCP is found (implying the nonexistence of a solution to the LCP); or (3) a succinct certificate is given to show that the input matrix A is not “sufficient”. Note that all PSD-matrices and P- matrices are sufficient matrices.