On the Multi-Chi-square Tests and Their Data Complexity

Chi-square tests are generally used for distinguishing purposes; however when they are combined to simultaneously test several independent variables, extra notation is required. In this study, the chi-square statistics in some previous works is revealed to be computed half of its real value. Therefore, the notion of Multi-Chi-square tests is formulated to avoid possible future confusions. In order to show the application of Multi-Chi-square tests, two new tests are introduced and applied to reduced round Trivium as a special case. These tests are modifications of the ANF monomial test, and when applied to Trivium with the same number of rounds, the data complexity of them is roughly 24 times smaller than that of former ANF monomial test. In a Multi-Chi-square test the critical degrees of freedom is defined to be the minimum value of the degrees of freedom for which the test is successful at distinguishing the samples set from random. This study investigates the relation between this critical value and the chi-square statistic of a Multi-Chi-square test. In the sequel, by exploiting this relation, a method to approximate the data complexity of a distinguishing Multi-Chi-square test is introduced and shown to perform properly in the special case of reduced round Trivium