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IEEE International Symposium on Information Theory (ISIT)

Secret Sharing Over a Two Receiver Classical-Quantum Broadcast Channel

Truman Welling, Rémi A. Chou, Aylin Yener

Published Jun 22, 2025• 1 citation
Publisher

Abstract

This work considers a secret sharing problem where the dealer uses a classical-quantum channel to divide and distribute shares to two users by encoding the secret so that the channel observation of each user is their share. The channel observations of a single user contains very little information about the secret, but the two users together can recover the secret. We derive a one-shot and an asymptotic achievability result for secret sharing over a quantum broadcast channel. Additionally, our achievability results coincide with the secret sharing capacity for the classical broadcast channel.

DOI
10.1109/isit63088.2025.11195606

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This work considers a secret sharing problem where the dealer uses a classical-quantum channel to divide and distribute shares to two users by encoding the secret so that the channel observation of each user is their share. The channel observations of a single user contains very little information about the secret, but the two users together can recover the secret. We derive a one-shot and an asymptotic achievability result for secret sharing over a quantum broadcast channel. Additionally, our achievability results coincide with the secret sharing capacity for the classical broadcast channel.
We consider a secret sharing setting with a monotone access structure involving a control node and LLL users, connected via a classical-quantum broadcast channel whose input is controlled by the control node, referred to as the dealer. Unlike traditional secret sharing settings, where the dealer fully controls the shares given to each user, in our model, the dealer encodes the secret for transmission over the broadcast channel. This means that the shares received by users are perturbed by the channel and are not fully controlled by the dealer. Our main results are achievable one-shot secret sharing rates, as well as converse bounds for arbitrary monotone access structures. We further derive second-order and asymptotic achievable rates for arbitrary monotone access structures. In the special case where all shares are required to recover the secret, we show that our result coincides with the existing secret sharing capacity over classical channels.
We study the transmission of two independent confidential classical messages over a quantum broadcast channel, one for each receiver. Each message must remain secret from the other receiver, including when that receiver knows its own message. For classical inputs and quantum outputs, we establish the classical Marton-type inner bound under average reliability and conditional strong secrecy. The encoder selects pairs of codewords from independently generated codebooks using normalized likelihood weights. We prove reliability through a change-of-distribution argument. Our main technical result is a bipartite classical-quantum resolvability theorem that accounts for the dependence created by pair selection and establishes secrecy for the same encoder. We also obtain a multi-letter capacity characterization and extend it to arbitrary quantum inputs under secrecy against the other receiver, together with the Stinespring environment. We recover confidential capacity regions for deterministic classical and degraded classical-quantum channels, with an explicit evaluation for the classical Blackwell channel. For coherent isometric extensions of injective deterministic classical broadcast channels, we show that the confidential classical capacity region equals that of the corresponding classical channel. We then compare confidential classical communication with quantum transmission. For the coherent isometric extension of the Blackwell channel, we determine the unassisted quantum-capacity region and show that some achievable confidential classical rate pairs lie outside it. The Platypus channel provides another example of this separation.
Secret-sharing building blocks based on quantum broadcast communication are studied. The confidential capacity region of the pure-loss bosonic broadcast channel is determined with key assistance, under the assumption of the long-standing minimum output-entropy conjecture. If the main receiver has a transmissivity of η<12\eta\lt\frac{1}{2}η<21​, then confidentiality solely relies on the key-assisted encryption of the one-time pad. We also address conference key agreement for the distillation of two keys, a public key and a secret key. A regularized formula is derived for the key-agreement capacity region. In the pure-loss bosonic case, the key-agreement region is included within the capacity region of the corresponding broadcast channel with confidential messages. We then consider a network with layered secrecy, where three users with different security ranks communicate over the same broadcast network. We derive an achievable layered-secrecy region for a pure-loss bosonic channel that is formed by the concatenation of two beam splitters.
The two-receiver broadcast channel with primary and third party receivers is studied. The sender wishes to reliably communicate a common (or public) message to both receivers as well as individualized and confidential messages to the primary receiver only. The third party receiver must be kept completely ignorant of the confidential message but there are no secrecy requirements associated to the individualized message. A trade-off arises between the rates of the three messages: when one of the rates is high, the other rates may need to back off to guarantee the reliable transmission of all three messages. In addition, the confidentiality requirement implies availability of local randomness at the transmitter in order to implement a stochastic encoding. This article studies the trade-off between the rates of the common, individualized and confidential messages as well as that of the local randomness in the one-shot regime of a quantum broadcast channel. We provide an achievability region, by proving a conditional version of the convex-split lemma combined with the position-based decoding, as well as a (weak) converse region. We study the asymptotic behaviour of our bounds and recover several well-known asymptotic results in the literature, including simultaneous transmission of classical and quantum information.
In this paper, we investigate secret communications over a two-user semi-deterministic broadcast channel, in which user 1's channel is deterministic, while user 2's channel is a discrete memoryless channel. The transmitter has two inde­pendent confidential messages, each of which is intended for one of the receivers, but is kept completely secret from the other. We first establish the secrecy capacity region of the semi-deterministic broadcast channel. Next, as a special case, we consider both channels are deterministic and compare the secrecy capacity region with the well-known general capacity region of the deterministic broadcast channel without secrecy constraint. Our result illustrates the fundamental loss of the transmission rates due to keeping the messages secret as well as the tradeoff between the communication rate and the perfect secrecy requirement.