Message-Specific Common Randomness

• Message-specific common randomness is shared randomness whose realization depends on the message index or transmitted codeword, influencing coordination and rate constraints.
• It enables tailored schemes such as SVD-based MIMO channel synthesis, color mapping in identification, and message-aligned one-time pads in SPIR systems.
• Its varied implementation—from Gaussian channels to interactive protocols—offers improved communication efficiency and optimized tradeoffs in secrecy and reliability.

Message-specific common randomness denotes shared or agreed randomness whose realization may depend on a message index, a transmitted codeword, or a public transcript, rather than being modeled as a seed independent of the source sequence or message. Across the cited literature, this notion appears in several non-equivalent forms: as common randomness generated from correlated sources and then specialized to individual identification hypotheses over Gaussian channels (Ezzine et al., 2020); as outage common randomness over slow-fading MIMO channels where independence from the transmitted message is not required (Ezzine et al., 2022); as message-specific shared randomness in strong channel synthesis, where pairwise seeds may be chosen after observing the common message (Managoli et al., 2024); and as a deliberate contrast to models in which the shared seed must remain independent of the source, so that allowing message-specific randomness would change the rate region itself (0805.0065).

1. Conceptual distinctions and modeling choices

In the common-randomness literature, the base object is usually a pair of random variables $(K,L)$ generated at two terminals from correlated observations and communication, with reliability measured by agreement probability and rate measured by entropy. In the Gaussian-source model of common-randomness generation, an achievable CR rate $H$ is defined by the existence, for sufficiently large blocklength, of permissible $(K,L)$ with $P[K \neq L] \le \alpha$ and $(1/n)H(K) \ge H-\delta$; secrecy is not imposed, in contrast with secret-key generation (Ezzine et al., 2020). This already separates message-specific CR from secret keys: message dependence is compatible with CR, whereas secrecy constraints impose additional structure.

A second distinction concerns whether the shared randomness is exogenous or message-derived. In the rate-distortion-perception model with common randomness, the shared seed $J$ is uniform, known to both terminals, and independent of $X^n$; if no such seed is available, then any decoder-side randomness correlated with the encoder must be conveyed through the message, and the no-common-randomness region requires $R \ge \max\{I(X;U), I(U;Y)\}$ (Wagner, 2022). The channel-simulation formulation makes the same independence point more sharply: the encoder and decoder share $J \in \{1,\ldots,2^{nC}\}$, uniformly distributed and independent of $X^n$, and the rate region

$H$0

holds only under input-independent common randomness (0805.0065).

A third distinction is whether message specificity is merely permitted, explicitly exploited, or ruled out by definition. In slow-fading MIMO CR generation, independence from the transmitted message is not required; the constructive scheme sends a short index over the channel, so the agreed CR can be a function of that transmitted index, and message-specific CR is therefore covered by the model (Ezzine et al., 2022). By contrast, in finite arbitrarily varying quantum channels, the shared randomness variable is explicitly independent of the message index and jammer state, and the paper does not define or use message-specific randomness (Boche et al., 2013). In the classical-quantum informed-jammer setting, the construction likewise generates a universal seed in a preamble independent of the message, and the paper states that allowing message-specific CR causes no capacity gain there because it can be simulated from a universal seed (Boche et al., 2019).

2. Common-randomness generation over Gaussian and MIMO channels

A central formulation of message-specific common randomness arises from correlated-source CR generation with one-way communication over Gaussian channels. Two terminals observe i.i.d. samples $H$1 from a discrete memoryless multiple source $H$2, terminal A sends $H$3 over a Gaussian link, terminal A forms $H$4, and terminal B forms $H$5 (Ezzine et al., 2020). For the normalized SISO channel $H$6 with $H$7 and $H$8, the Shannon capacity is $H$9 for Gaussian input. Proposition 1 gives the CR capacity

$(K,L)$0

For MIMO,

$(K,L)$1

with

$(K,L)$2

where the singular values $(K,L)$3 arise from the SVD $(K,L)$4 and the powers $(K,L)$5 follow water-filling (Ezzine et al., 2020).

The optimal signal processing is therefore explicit: SVD-based linear precoding and combining diagonalize the MIMO channel into orthogonal eigenmodes, water-filling allocates power across eigenmodes, and a typicality-based source code selects a $(K,L)$6 sequence jointly typical with $(K,L)$7, transmits its index over the Gaussian channel, and reconstructs a matching sequence from $(K,L)$8 at the receiver (Ezzine et al., 2020). The source-side auxiliary $(K,L)$9 carries the source dependence, while the channel contributes only the reconciliation budget $P[K \neq L] \le \alpha$0.

This framework already exhibits a limited form of message specificity. In the slow-fading MIMO extension with CSIR-only and arbitrary channel-state distribution, the outage CR protocol again uses a codebook indexed by $P[K \neq L] \le \alpha$1, sends a short index $P[K \neq L] \le \alpha$2 over the channel, and reconstructs the agreed $P[K \neq L] \le \alpha$3 at the receiver. The paper states that independence from the transmitted message is not required in the definition, so the agreed CR can be a function of the transmitted codeword or index; however, allowing this dependence does not change the rate constraint, because the bottleneck remains the outage-limited reconciliation budget

$P[K \neq L] \le \alpha$4

for the lower bound and

$P[K \neq L] \le \alpha$5

for the upper bound (Ezzine et al., 2022). Under a positive channel-state density off a null set, the bounds coincide for all $P[K \neq L] \le \alpha$6, giving

$P[K \neq L] \le \alpha$7

Several structural consequences follow directly. In the fixed-channel Gaussian model, there exists $P[K \neq L] \le \alpha$8 with $P[K \neq L] \le \alpha$9 such that for all $(1/n)H(K) \ge H-\delta$0, $(1/n)H(K) \ge H-\delta$1, so the CR rate saturates once the forward link can carry the conditional entropy (Ezzine et al., 2020). In the outage model, exactness may fail only at countably many discontinuity points of the outage transmission capacity, and practical operation is recommended slightly below such points (Ezzine et al., 2022).

3. Identification, message specificity, and secure identification

Message-specific common randomness becomes operationally central in identification over channels. In the identification framework, the decoder is interested in deciding whether a specific message of interest was sent, rather than recovering the entire transmitted message. A CR-assisted $(1/n)H(K) \ge H-\delta$2 identification code over the Gaussian channel is defined by codewords $(1/n)H(K) \ge H-\delta$3 and decision regions $(1/n)H(K) \ge H-\delta$4 satisfying

$(1/n)H(K) \ge H-\delta$5

with $(1/n)H(K) \ge H-\delta$6 (Ezzine et al., 2020). The CR-assisted identification capacity is measured on a log-log scale through the requirement

$(1/n)H(K) \ge H-\delta$7

The role of CR is not merely generic coordination. The paper’s construction uses coloring maps $(1/n)H(K) \ge H-\delta$8 tied to each hypothesis $(1/n)H(K) \ge H-\delta$9. The encoder maps its common randomness $J$0 to the color $J$1, the decoder reconstructs $J$2, and then checks whether the received color matches $J$3 for the specific hypothesis under test (Ezzine et al., 2020). This is message-specific common randomness in a literal sense: the same shared random variable is post-processed differently for each message index, and the resulting randomized tests are indexed by the queried message.

The secure version adds a Gaussian wiretap channel and requires that the wiretapper cannot identify the message hypothesis. Proposition 3 states that if the secrecy capacity $J$4, then

$J$5

The proof uses a concatenated scheme: first, a CR index derived from $J$6 is sent over the main channel using an error-correcting code at rate approximately $J$7; second, the color $J$8 is sent using a wiretap code of length $J$9 (Ezzine et al., 2020). By the Transformator-Lemma, positive secrecy capacity suffices to convert the CR rate into a secure-identification lower bound.

The paper records several concrete consequences. If $X^n$0 and $X^n$1, then $X^n$2. In the numerical binary-source example with $X^n$3 and $X^n$4, at $X^n$5 the lower bound exceeds the main-channel identification capacity with randomized encoding by at least $X^n$6 bits on the log-log scale. It also states that if CR is unavailable and encoding is deterministic for Gaussian channels, identification capacity on the log-log scale is zero (Ezzine et al., 2020). These statements locate message-specific CR as a rate-critical resource for post-Shannon identification, especially in machine-to-machine systems, human-to-machine systems, and the tactile Internet.

4. Simulation, synthesis, and the distinction between message-derived and independent randomness

In channel simulation and synthesis, the status of message-specific randomness determines the communication rate itself. For generating correlated random variables through strong simulation of a discrete memoryless channel, the minimum description rate without common randomness is Wyner’s common information

$X^n$7

whereas with sufficiently large common randomness independent of the input the minimum description rate drops to $X^n$8 (0805.0065). The exact tradeoff is the region

$X^n$9

over auxiliaries satisfying $R \ge \max\{I(X;U), I(U;Y)\}$0. The paper explicitly notes that if the common randomness were allowed to depend on $R \ge \max\{I(X;U), I(U;Y)\}$1 or the message, it would effectively become an additional communication pipe and could drive $R \ge \max\{I(X;U), I(U;Y)\}$2 toward zero, so the independence of $R \ge \max\{I(X;U), I(U;Y)\}$3 from $R \ge \max\{I(X;U), I(U;Y)\}$4 is fundamental (0805.0065).

The rate-distortion-perception tradeoff gives an allied but distinct picture. The shared seed $R \ge \max\{I(X;U), I(U;Y)\}$5 is again required to be independent of $R \ge \max\{I(X;U), I(U;Y)\}$6, and the achievable region with perfect realism is

$R \ge \max\{I(X;U), I(U;Y)\}$7

When $R \ge \max\{I(X;U), I(U;Y)\}$8, the specialization becomes

$R \ge \max\{I(X;U), I(U;Y)\}$9

which formalizes the penalty for replacing an independent seed by message-derived randomness (Wagner, 2022). In the quadratic Gaussian case with $J \in \{1,\ldots,2^{nC}\}$0 and $J \in \{1,\ldots,2^{nC}\}$1, the exact tradeoff satisfies

$J \in \{1,\ldots,2^{nC}\}$2

where $J \in \{1,\ldots,2^{nC}\}$3 solves

$J \in \{1,\ldots,2^{nC}\}$4

The special cases are

$J \in \{1,\ldots,2^{nC}\}$5

and

$J \in \{1,\ldots,2^{nC}\}$6

which the paper identifies as a $J \in \{1,\ldots,2^{nC}\}$7 dB penalty relative to the classical Gaussian RDF (Wagner, 2022).

The strongest explicit use of message-specific common randomness appears in broadcast channel synthesis. There, message-oblivious pairwise shared randomness means seeds $J \in \{1,\ldots,2^{nC}\}$8 are chosen before seeing $J \in \{1,\ldots,2^{nC}\}$9 and are independent of the common message $X^n$0, whereas message-specific common randomness permits choosing the shared randomness after observing the message and allowing it to depend on the message (Managoli et al., 2024). The paper states that with MSCR the minimum common message rate collapses to the reverse-Shannon benchmark

$X^n$1

while in the message-oblivious model a broadcast Wyner-type lower bound appears,

$X^n$2

with

$X^n$3

This is one of the clearest instances in which message specificity does not merely alter implementation, but changes the optimal communication law itself (Managoli et al., 2024).

5. Interaction, adversaries, and transcript-tied common randomness

A different line of work studies whether common randomness must be message-independent when channels are adversarial or interactive. For finite arbitrarily varying quantum channels, the main result is that asymptotically perfect common randomness is not needed: access to a bipartite source $X^n$4 with $X^n$5, even at an arbitrarily small nonzero fraction per channel use, suffices to achieve the randomness-assisted capacities for message and entanglement transmission (Boche et al., 2013). At the same time, the paper emphasizes that CR is a costly resource: for full-support bipartite distributions, asymptotically perfect CR cannot be obtained by local processing alone, and its model of shared randomness remains message-independent throughout (Boche et al., 2013).

In arbitrarily varying classical-quantum channels with an informed jammer, correlation again substitutes for explicit common randomness. If $X^n$6, the correlation-assisted message-transmission capacity is

$X^n$7

The construction first generates a short universal seed $X^n$8 in a preamble and then uses it to coordinate a randomness-assisted payload code. The paper states that if one insists on message-specific CR depending on the message index $X^n$9, this can be simulated at no rate loss by defining $H$00, so allowing message-specific CR does not improve the capacity in that model (Boche et al., 2019).

Interactive round complexity provides another notion of specificity, now at the transcript level. For the pointer-chasing source $H$01, there is an $H$02-round protocol achieving $H$03 for CRG and SKG, while for $H$04 any $H$05-round protocol achieving $H$06 must have communication at least $H$07 up to polylogarithmic factors (Golowich et al., 2019). The paper frames the gain as coming from transcript-adaptive interaction: each message is tailored to realized inputs and prior messages, and insufficient rounds force substantially higher communication.

Transcript-tied common randomness also appears in query-resolution complexity. In the multiterminal source model, an external observer sees the public transcript $H$08 and tries to resolve the realized CR value $H$09 by yes/no queries. The optimum query exponent is

$H$10

and the paper explicitly interprets message-specific CR under this framework as CR tied to the particular public transcript $H$11 generated in a protocol instance (Tyagi et al., 2013). Thus, even when the randomness is not message-indexed in advance, the realized transcript can make it instance-specific in a strong operational sense.

The resource-efficient CR literature makes a related implementation observation. Message-indexed colorings or permutations can be introduced into one-way CR schemes by enlarging the communicated color index, and the transmitted prefix can be separated from a nearly uniform suffix key; this supports message-specific or message-independent common randomness with explicit tradeoffs among communication, agreement probability, and secrecy (Ghazi et al., 2017). A plausible implication is that, in several interactive and constructive settings, message specificity is best understood as a coding-layer transformation applied to a more primitive CR resource.

6. Message-specific randomness in private retrieval and replicated systems

In private information retrieval, message-specific common randomness appears as per-message one-time pads or as user-side knowledge of a message-aligned subset of server randomness. In SPIR on graph-replicated databases, each message corresponds to an edge of a graph and is stored at exactly two servers. Under graph-replicated common randomness, the two servers storing message $H$12 share an independent random variable $H$13, and the message-specific randomness ratio is $H$14 (Meel et al., 18 Feb 2026). The paper proves that for any graph there exists an SPIR scheme with

$H$15

achieved with $H$16, and that $H$17 is necessary for feasibility. For path graphs and regular graphs the converse matches, giving

$H$18

This is message-specific common randomness in a literal storage-theoretic form: each message has its own pad, shared only by the two servers that replicate it (Meel et al., 18 Feb 2026).

The same paper contrasts this with fully replicated common randomness, where all servers share the same private random seed. In that setting the capacity can be strictly larger than in the message-specific graph-replicated model. For paths and cycles, the achievable rates are

$H$19

with corresponding lower bounds on the required total randomness, while for the $H$20-node path the exact result is $H$21 with $H$22 (Meel et al., 18 Feb 2026). The comparison isolates a tradeoff between locality of randomness replication and achievable download efficiency.

A different SPIR model gives the user access to a random subset $H$23 of the shared database randomness $H$24, unknown to the databases except through its cardinality. The exact capacity region for the normalized triple $H$25 with $H$26 is given by

$H$27

$H$28

$H$29

The paper shows that the smallest user-side common randomness achieving the PIR download cost is $H$30, and at that point the exact server-side requirement is

$H$31

It also states that single-database SPIR becomes feasible in this model (Wang et al., 2021).

The paper interprets the user-side seed as a message-specific one-time pad in effect: the desired-message symbols are aligned with the seed pad known to the user, while pads attached to undesired content remain unknown and protect database privacy (Wang et al., 2021). This does not make the server-side randomness message-dependent in the channel-simulation sense; instead, it makes the user’s cancellation capability message-specific at retrieval time. That distinction is characteristic of the broader literature: “message-specific common randomness” can mean message-indexed shared seeds, message-derived coordination, transcript-tied CR, or message-aligned privacy masks, depending on the operational problem.

Taken together, these works show that message-specific common randomness is not a single theorem but a family of modeling choices. In some settings it is essential and rate-improving, as in broadcast synthesis (Managoli et al., 2024). In some it is permitted but rate-neutral because the same reconciliation constraint remains binding, as in outage CR generation over slow-fading MIMO channels (Ezzine et al., 2022). In others it is excluded by definition because independence from the source or message is fundamental to the converse, as in strong channel simulation and RDP with an independent seed (0805.0065). And in identification, SPIR, and graph-replicated systems, it functions as a sharply targeted coordination resource that attaches shared randomness to a specific hypothesis, transcript, or message replica (Ezzine et al., 2020).