B More about Lloyd's Formula

Formula (1) was used in 6] in order to count all functions which satisfy the so-called Strict Avalanche Criterion (SAC) of order (m ? 2). The SAC for a boolean function g(x) means that if the vector x is chosen with uniform probability and any one of its bits is ipped, then with probability exactly 1 2 the value of the function is ipped as well. The SAC of order (m?2) means, that any function obtained from f by keeping (m ?2) or less of its input bits constant satisses the SAC (for any choice of the positions and of the values of the constant bits). Note that a function cannot satisfy the SAC of order (m ? 1) (since it would require the existence of a function g : Z 2 ! Z 2 which satisses SAC, and no such function exists). The function f(x; K), with a xed K, satisses the SAC. This follows easily from the fact that, when x i changes its value from zero to one, the diierence in the control setting function is: (6) Here e i is a binary vector of m bits, whose bits are all zero, except for the i th bit. (This can be proved by considering the 4 cases: h(x) 0; 1; 2; 3 (mod 4).) Since k 0 k i is a constant for the xed key, and h(x) (mod 2) = 0 or 1 for exactly one half of the x-s, f(x; K) satisses SAC.