  Consider a memoryless degraded broadcast channel (DBC) in which the channel
output is a single-letter function of the channel input and the channel noise.
As examples, for the Gaussian broadcast channel (BC) this single-letter
function is regular Euclidian addition and for the binary-symmetric BC this
single-letter function is Galois-Field-two addition. This paper identifies
several classes of discrete memoryless DBCs for which a relatively simple
encoding scheme, which we call natural encoding, achieves capacity. Natural
Encoding (NE) combines symbols from independent codebooks (one for each
receiver) using the same single-letter function that adds distortion to the
channel. The alphabet size of each NE codebook is bounded by that of the
channel input.
  Inspired by Witsenhausen and Wyner, this paper defines the conditional
entropy bound function $F^*$, studies its properties, and applies them to show
that NE achieves the boundary of the capacity region for the multi-receiver
broadcast Z channel. Then, this paper defines the input-symmetric DBC,
introduces permutation encoding for the input-symmetric DBC, and proves its
optimality. Because it is a special case of permutation encoding, NE is
capacity achieving for the two-receiver group-operation DBC. Combining the
broadcast Z channel and group-operation DBC results yields a proof that NE is
also optimal for the discrete multiplication DBC. Along the way, the paper also
provides explicit parametric expressions for the two-receiver binary-symmetric
DBC and broadcast Z channel.
