§III, unnumbered auxiliary-cardinality bounds

Let $S,X,Y,Z$ be finite nonempty alphabets and let $W(x,y,z\mid s)$ be a normalized discrete memoryless broadcast channel. All entropies and information quantities are in bits. For an input law $p$, information quantities below use exactly the induced law $p(s)W(x,y,z\mid s)$.

Alice draws a uniform finite message $M$, uses a possibly stochastic encoder to excite $n$ channel uses, and observes $(M,X^n)$. A public forward message $\Phi=\phi(M,X^n)$ is visible to Bob and Eve. Their keys are $K_A=k_A(M,X^n)$ and $K_B=k_B(Y^n,\Phi)$. All design maps are fixed publicly before observations. Message and public alphabets follow the fixed-rate ceiling convention; the key alphabet at rate $R$ has exactly $2^{\lceil nR\rceil}$ labels. A weakly achievable rate $R$ has one sequence of such codes with disagreement tending to zero, $I(K_A;Z^n,\Phi)/n\to0$, and $\liminf H(K_A)/n\ge R$.

Let $\Lambda:S\to[0,L]$ be bounded and nonnegative, and let $\Gamma>0$ be admissible, meaning that a normalized input law of expected cost at most $\Gamma$ exists. Every operational code obeys $n^{-1}\sum_i\Lambda(S_i)\le\Gamma$ almost surely; each single-letter optimization obeys $\sum_s p(s)\Lambda(s)\le\Gamma$. Excitation and public rates are unrestricted.

The unrestricted auxiliary maximum in Proposition 1 is attained with

The maximizer obeys the fixed-channel factorization and the expected-cost constraint.

These bounds reduce the auxiliary optimization to bounded finite alphabets.