A lower bound on the average entropy of a function determined up to a diagonal linear map on F_q^n

Abstract. In this note, it is shown that if f : Fq → F n q is any function and A = (A1, . . . , An) is uniformly distributed over F n q , then the average over (k1, . . . , kn) ∈ Fq of the Rényi (and hence, of the Shannon) entropy of f(A)+(k1A1, . . . , knAn) is at least about log 2 (q) − n bits. In fact, it is shown that the average collision probability of f(A) + (k1A1, . . . , knAn) is at most about 2 /q.