Further enumerating Boolean functions of cryptographic significance (Q1895962)

A function from \(GF(2)^m\) to \(GF(2)^n\) (where \(m \geq n \geq 1)\) is said to be an \((m,n)\) Boolean function or abbreviated an \((m,n) - LUT\). In this paper the following properties of such functions are considered: (P1) For any \(y \in GF(2)^n\), \(|f^{-1} (y) |= 2^{m - n}\). (P2) For every \(i\) \((1 \leq i \leq n)\) there must not be a vector \(h \in GF (2)^m\) and a fixed scalar \(a\) such that \((f(x))_i = xh + a\) for every \(x \in GF(2)^m\). (P3) Each of the \(n\) outputs of \(f\) must depend on all the inputs. (P4) Given any vectors \(x \in GF(2)^m\) and \(y \in GF(2)^n\) such that \(f(x) = y\), then \(P(x_i = y_j) = 0.5\) for any \(1 \leq i \leq m\), \(1 \leq j \leq n\). (P5) If \(P\) is any permutation matrix, then \(f(x) = f(xP)\) for any \(x \in GF(2)^m\). Let \(A_{mn} (i_1, \dots, i_s)\) be the number of \((m,n) - LUTs\) having the properties \(P_{i_1}, \dots, P_{i_s}\). In this paper some exact values, lower or upper bounds and recurrences are deduced for \(A_{mn}(4)\), \(A_{m1} (1,3)\), \(A_{mn} (2,3)\), \(A_{mn} (2,4)\), \(A_{mn} (3,4)\), \(A_{mn} (4,5)\), \(A_{m1} (1,2,4)\), \(A_{m1} (1,2,5)\), \(A_{m1} (1,3,4)\), \(A_{mn} (1,4,5)\), \(A_{mn} (2,3,4)\), \(A_{mn} (2,4,5)\), \(A_{mn} (3,4,5)\), \(A_{m1} (1,2,3,4)\), \(A_{m1} (1,2,4,5)\), \(A_{m1} (1,3,4,5)\), \(A_{mn} (2,3,4,5)\) and \(A_{m1} (1,2,3,4,5)\).