ON IDEAL t-TUPLE DISTRIBUTION OF ORTHOGONAL FUNCTIONS IN FILTERING DE BRUIJN GENERATORS

Uniformity in binary tuples of various lengths in a pseudorandom sequence is an important randomness property. We consider ideal t-tuple distribution of a filtering de Bruijn generator consisting of a de Bruijn sequence of period 2n and a filtering function in m variables. We restrict ourselves to the family of orthogonal functions, that correspond to binary sequences with ideal 2-level autocorrelation, used as filtering functions. After the twenty years of discovery of Welch-Gong (WG) transformations, there are no much significant results on randomness of WG transformation sequences. In this article, we present new results on uniformity of the WG transform of orthogonal functions on de Bruijn sequences. First, we introduce a new property, called invariant under the WG transform, of Boolean functions. We have found that there are only two classes of orthogonal functions whose WG transformations preserve t-tuple uniformity in output sequences, up to t = (n - m + 1). The conjecture of Mandal et al. in [29] about the ideal tuple distribution on the WG trans-formation is proved. It is also shown that the Gold functions and quadratic functions can guarantee (n-m+1)-tuple distributions. A connection between the ideal tuple distribution and the invariance under WG transform property is established.