# A Novel Scalable Quantum Protocol for the Dining Cryptographers Problem

^{*}

^{†}

## Abstract

**:**

## 1. Introduction

## 2. Background Notions

**Definition**

**1.**

- There are n quantum registers ${r}_{0},{r}_{1},\dots ,{r}_{n-1}$;
- Each register contains m qubits;
- The n qubits in the ${j}^{th}$ position of every register, $0\le j\le m-1$, are entangled together in the $|GH{Z}_{n}\rangle $ state.

- $\mathbb{B}$ is the binary set $\{0,1\}$.
- To distinguish bit vectors from bits, we write bit vectors $\mathbf{x}\in {\mathbb{B}}^{m}$ in boldface. A bit vector $\mathbf{x}$ of length m corresponds to a sequence of m bits: $\mathbf{x}={x}_{m-1}\dots {x}_{0}$. In the special case where all bits are zero, i.e., $0\dots 0$, we have the zero bit vector, denoted by $\mathbf{0}$.
- In this setting, a bit vector $\mathbf{x}\in {\mathbb{B}}^{m}$ stands for the binary representation of one of the ${2}^{m}$ basis kets that form the computational basis of the Hilbert space at hand.
- To eliminate any source of ambiguity, we rely on the indices $i,\phantom{\rule{4pt}{0ex}}0\le i\le n-1$, to emphasize that ${|\mathbf{x}\rangle}_{i}$ is the state of the ith quantum register.

## 3. The Scalable Quantum Protocol for the Dining Cryptographers Problem

- (
**N**_{1}) - Scalability. In the SQDCP protocol, the notion of scalability encompasses both parameters n and m. The number of cryptographers n can be any large positive integer. In addition to the scalability of players, our protocol can seamlessly scale in terms of the amount of anonymous information it conveys. Initially, the Dining Cryptographers problem was about just one bit of information, namely, whether or not one of the cryptographers paid for the dinner. In the SQDCP protocol, the number m of qubits in each register can also be any large positive integer. This number reflects the amount of information that can be transmitted. For instance, the cryptographer that actually paid the bill may also disclose how much it cost or when the arrangement was made, etc.
- (
**N**_{2}) - Local and Distributed Mode. In its initial formulation in [1] and in the subsequent treatments, the cryptographers’ dinner was a localized event, in the sense that all cryptographers were together at the same spatial location at a specific point in time. The protocol introduced in this work can address not only this localized situation but also a distributed version of the Dining Cryptographers problem, in which the cryptographers are in different spatial locations.
- (
**N**_{3}) - Uniformity and Ease of Implementation. All cryptographers employ identical quantum circuits, that is, the quantum implementation is completely modular, with all modules being the same. Furthermore, each quantum circuit can be easily implemented by a contemporary quantum computer because it only uses the ubiquitous Hadamard and CNOT quantum gates.

**Definition**

**2**

**.**The localized setup is described below.

- Alice gathers together her $n-1$ cryptographer colleagues ${C}_{0}$, …, ${C}_{n-2}$ for a friendly dinner in a nearby restaurant.
- For all n players, the dinner event takes place simultaneously and at the same location.
- Each player employs a quantum circuit where she secretly embeds the desired information, namely, whether or not she paid for the dinner.
- Upon measuring their quantum registers and publicly combining the obtained results, all the players know whether the dinner was paid for by one of them or by their employer, and, possibly, some additional information, e.g., the cost of the dinner or the date of the payment, etc.
- The identity of the one who paid the bill remains unknown to all other cryptographers.

**Definition**

**3**

**.**Let us now envision a more general situation.

- Alice and her $n-1$ cryptographer colleagues ${C}_{0}$, …, ${C}_{n-2}$ have made arrangement for dinner.
- There is a complication now compared to the previous case because all n agents reside at different geographical locations.
- Nevertheless, they are determined to dine at the same time, albeit in different restaurants, and be in constant audio and visual contact via teleconference.
- Each player employs a local quantum circuit where she secretly embeds the desired information, namely, whether or not she paid for the dinner.
- Upon measuring their quantum registers, they publicly exchange their measurements via classical channels. Subsequently, each player uses the received results to find out whether the dinner was paid for by one of them or by their employer, and, possibly, some additional information.
- The identity of the one who paid the bill remains unknown to all other cryptographers.

- Although there is no theoretical limitation on the number n of cryptographers that can be an arbitrarily large integer, contemporary quantum apparatus may impose constraints to the generation of $|GH{Z}_{n}\rangle $ tuples, whenever n exceeds some threshold.
- We assume that, prior to the execution of the SQDCP protocol, certain arrangements have taken place among the cryptographers regarding the amount and nature of the desired information. This is necessary in order to fix the number of m, corresponding to the amount of information, and the proper interpretation of the outcome.
- In the distributed version, we also assume the existence of pairwise authenticated channels that enable the transmission of classical information.

**Example**

**1.**

## 4. Execution of the SQDCP Protocol in Three Phases

#### 4.1. Entanglement Distribution Phase

- (
**ED**_{1}) - Alice or perhaps a third party, trusted by all cryptographers, generates a sequence of m$|GH{Z}_{n}\rangle $ tuples, $mn$ qubits in total, which are necessary for the execution of the protocol and the private transmission of the required information. For the SQDCP protocol, the exact source responsible for the production of the $|GH{Z}_{n}\rangle $ tuples is not important; the only thing that matters is that they are faithfully created and uniformly distributed among the cryptographers.
- (
**ED**_{2}) - Say for convenience that in every $|GH{Z}_{n}\rangle $ tuple the qubits are numbered from 0 to $n-1$. Their distribution adheres to the following pattern, which guarantees the even and uniform distribution of entanglement among the cryptographers.
- (
**ED**_{3}) - In addition to her input register, Alice utilizes a single-bit output register designated by $AOR$ in Figure 4, which is initialized at state $H|1\rangle =\frac{|0\rangle -|1\rangle}{\sqrt{2}}=|-\rangle $. Likewise, all her cryptographer colleagues ${C}_{i}$, $0\le i\le n-2$, possess a similar single-bit output register denoted by $O{R}_{i}$ in Figure 4. The output registers are crucial for the embedding of private information into the entangled state of the composite circuit.

#### 4.2. Private Information Embedding Phase

- (
**PIE**_{1}) - If it was Alice who secretly paid for the dinner and the binary representation of the amount she paid is ${\mathbf{p}}_{A}$ in euros (EUR), then she will insert ${\mathbf{p}}_{A}$ into the global entangled state of the circuit via her private unitary transform ${U}_{{f}_{A}}$. Since ${U}_{{f}_{A}}$ is only known to her, the required information will be embedded secretly, privately, and none will be able to be trace it back to Alice.
- (
**PIE**_{2}) - If Alice did not pay for the dinner, then she uses the zero bit vector $\mathbf{0}$ in her private unitary transform ${U}_{{f}_{A}}$, which in effect leaves the global state of the system unchanged.
- (
**PIE**_{3}) - Entirely analogously, if it was cryptographer ${C}_{i}$, $0\le i\le n-2$, who secretly paid for the dinner and the binary representation of the amount paid is ${\mathbf{p}}_{i}$, then she will insert ${\mathbf{p}}_{i}$ into the global entangled state of the circuit via her private unitary transform ${U}_{{f}_{i}}$. Since ${U}_{{f}_{i}}$ is only known to cryptographer ${C}_{i}$, this information will be embedded secretly, and privately, and none will be able to trace it back to ${C}_{i}$. Obviously, if ${C}_{i}$ did not pay for the dinner, then she uses the zero bit vector $\mathbf{0}$ in her private unitary transform ${U}_{{f}_{i}}$.
- (
**PIE**_{4}) - The quantum part of the protocol is completed when the cryptographers measure their input registers. The obtained measurements are added together using addition modulo 2, i.e., they are XOR-ed together. The final outcome $\mathbf{p}$ gives the desired information in the following sense.
- ▸
- If $\mathbf{p}$ is nonzero, this means that the dinner was paid by one of the cryptographers. We also find out how much the dinner cost, because $\mathbf{p}$ is the binary representation of the cost in euros. The identity of the cryptographer who paid cannot be inferred from $\mathbf{p}$; it remains unknown and untraceable.
- ▸
- If $\mathbf{p}$ is the zero bit vector $\mathbf{0}$, this means that the dinner was paid for by their employer and not by one of the cryptographers.

- (
**PIE**_{5}) - The SQDCP protocol will work even if all the players are in different geographical locations. This is because, even if the quantum input registers are spatially separated, they still constitute one composite distributed quantum system due to the strong correlations among their qubits originating from the $|GH{Z}_{n}\rangle $ entanglement. The only difference in the distributed case is that each cryptographer must communicate the obtained measurements to each other cryptographer using pairwise authenticated classical channels.

- $AIR$ is Alice’s input register.
- $I{R}_{i}$ is the input register of cryptographer ${C}_{i}$, $0\le i\le n-2$.
- In total, there are n input registers, each containing m qubits. The corresponding qubits in each of the n registers are entangled in the $|GH{Z}_{n}\rangle $ state.
- $AOR$ is Alice’s output register.
- $O{R}_{i}$ is the output register of cryptographer ${C}_{i}$, $0\le i\le n-2$.
- All output registers contain just a single qubit in the $|-\rangle $ state.
- ${U}_{{f}_{A}}$ is Alice’s unitary transform.
- ${U}_{{f}_{i}}$ is the unitary transform of cryptographer ${C}_{i}$, $0\le i\le n-2$.
- ${H}^{\otimes m}$ is the m-fold Hadamard transform.

#### 4.3. Deciphering Phase

- If ${\mathbf{p}}_{A}\oplus {\mathbf{p}}_{n-2}\oplus \dots \oplus {\mathbf{p}}_{0}\oplus \mathbf{a}\oplus {\mathbf{c}}_{n-2}\oplus \dots \oplus {\mathbf{c}}_{0}\ne \mathbf{0}$ or, equivalently, $\mathbf{a}\oplus {\mathbf{c}}_{n-2}\oplus \dots \oplus {\mathbf{c}}_{0}$≠${\mathbf{p}}_{A}\oplus {\mathbf{p}}_{n-2}\oplus \dots \oplus {\mathbf{p}}_{0}$, the sum ${\sum}_{\mathbf{x}\in {\mathbb{B}}^{m}}$${(-1)}^{({\mathbf{p}}_{A}\oplus {\mathbf{p}}_{n-2}\oplus \dots \oplus {\mathbf{p}}_{0}\oplus \mathbf{a}\oplus {\mathbf{c}}_{n-2}\oplus \dots \oplus {\mathbf{c}}_{0})\xb7\mathbf{x}}$${|-\rangle}_{A}$${|\mathbf{a}\rangle}_{A}$${|-\rangle}_{n-2}{|{\mathbf{c}}_{n-2}\rangle}_{n-2}$⋯${|-\rangle}_{0}{|{\mathbf{c}}_{0}\rangle}_{0}$ appearing in (14) becomes just 0.
- If, on the other hand, ${\mathbf{p}}_{A}\oplus {\mathbf{p}}_{n-2}\oplus \dots \oplus {\mathbf{p}}_{0}\oplus \mathbf{a}\oplus {\mathbf{c}}_{n-2}\oplus \dots \oplus {\mathbf{c}}_{0}=\mathbf{0}$ or, equivalently, $\mathbf{a}\oplus {\mathbf{c}}_{n-2}\oplus \dots \oplus {\mathbf{c}}_{0}$=${\mathbf{p}}_{A}\oplus {\mathbf{p}}_{n-2}\oplus \dots \oplus {\mathbf{p}}_{0}$, the sum ${\sum}_{\mathbf{x}\in {\mathbb{B}}^{m}}$${(-1)}^{({\mathbf{p}}_{A}\oplus {\mathbf{p}}_{n-2}\oplus \dots \oplus {\mathbf{p}}_{0}\oplus \mathbf{a}\oplus {\mathbf{c}}_{n-2}\oplus \dots \oplus {\mathbf{c}}_{0})\xb7\mathbf{x}}$${|-\rangle}_{A}$${|\mathbf{a}\rangle}_{A}$${|-\rangle}_{n-2}{|{\mathbf{c}}_{n-2}\rangle}_{n-2}$⋯${|-\rangle}_{0}{|{\mathbf{c}}_{0}\rangle}_{0}$ becomes ${2}^{m}$${|-\rangle}_{A}$${|\mathbf{a}\rangle}_{A}$${|-\rangle}_{n-2}{|{\mathbf{c}}_{n-2}\rangle}_{n-2}$⋯${|-\rangle}_{0}{|{\mathbf{c}}_{0}\rangle}_{0}$.

- (
**D**_{1}) - Every cryptographer communicates to every other cryptographer the measured contents of her input register. That is, Alice sends $\mathbf{a}$ to her $n-1$ cryptographer colleagues, and each ${C}_{i}$, $0\le i\le n-2$, sends ${\mathbf{c}}_{i}$ to Alice and every other cryptographer.
- (
**D**_{2}) - In a localized setting, this step is quite trivial. In a distributed setting, it is also easily achievable, as it only requires the use of pairwise authenticated classical communication channels.
- (
**D**_{3}) - At this point, every player knows all bit vectors $\mathbf{a},{\mathbf{c}}_{n-2},\dots ,{\mathbf{c}}_{0}$. This allows each cryptographer to compute the modulo 2 sum $\mathbf{a}\oplus {\mathbf{c}}_{n-2}\oplus \dots \oplus {\mathbf{c}}_{0}$, which, according to (18), produces the modulo 2 sum ${\mathbf{p}}_{A}\oplus {\mathbf{p}}_{n-2}\oplus \dots \oplus {\mathbf{p}}_{0}$.
- (
**D**_{4}) - The modulo 2 sum ${\mathbf{p}}_{A}\oplus {\mathbf{p}}_{n-2}\oplus \dots \oplus {\mathbf{p}}_{0}$ conveys the information the cryptographers wanted to uncover in the first place. Here is why.
- In case none of the cryptographers paid for the dinner, then, according to (
**PIE**_{2}) and (**PIE**_{3}), ${\mathbf{p}}_{A}={\mathbf{p}}_{n-2}=\dots ={\mathbf{p}}_{0}=\mathbf{0}$. Consequently, their modulo 2 sum is $\mathbf{0}$, which means that the computed modulo 2 sum $\mathbf{a},{\mathbf{c}}_{n-2},\dots ,{\mathbf{c}}_{0}$ is also $\mathbf{0}$. Hence, the cryptographers infer that the dinner was paid for by their employer. - In case ${C}_{i}$, $0\le i\le n-2$, paid for the dinner a certain amount of money, then, considering (
**PIE**_{1}) – (**PIE**_{3}), ${\mathbf{p}}_{i}$, the binary representation of this amount, is nonzero, whereas ${\mathbf{p}}_{A}$ and all other ${\mathbf{p}}_{j}$, $j\ne i$, are zero. This implies that the computed modulo 2 sum $\mathbf{a},{\mathbf{c}}_{n-2},\dots ,{\mathbf{c}}_{0}$ is ${\mathbf{p}}_{i}$. Therefore, the cryptographers infer that it was one of them who paid for the dinner and, as an added bonus, they also get to know how much the dinner cost. Obviously, the same argument goes verbatim in case it was Alice who paid for the dinner.

- (
**D**_{5}) - The above explanation also shows that the original source of the information remains unknown and untraceable. The private information, be it ${\mathbf{p}}_{A}$ or some ${\mathbf{p}}_{i}$, $0\le i\le n-2$, has been absorbed into the sum ${\mathbf{p}}_{A}\oplus {\mathbf{p}}_{n-2}\oplus \dots \oplus {\mathbf{p}}_{0}$ and there is no way that it can be retrieved.

**Example**

**2.**

^{16}= 65,536 equiprobable outcomes. For obvious technical limitations, we cannot show all these outcomes, since this would result in an unintelligible figure. Hence, we depict only 16 of them in Figure 6. It is straightforward to check that every possible outcome satisfies the Fundamental Correlation Property and verifies Equations (16) and (18). For example, we may examine the label of the first bar of the histogram contained in Figure 6, which is 0001 1000 0111 0010. This means that upon measurement the contents of Alice, Bob, Charlie, and Dave’s input registers are $\mathbf{a}$ = 0001, $\mathbf{b}$ = 1000, $\mathbf{c}$ = 0111, and $\mathbf{d}$ = 0010, respectively. These contents are shared among the four cryptographers, according to (

**D**

_{1}) and (

**D**

_{2}), and become common knowledge to all of them. Finally, they XOR them together to uncover the secret information, i.e., $\mathbf{p}=\mathbf{a}\oplus \mathbf{b}\oplus \mathbf{c}\oplus \mathbf{d}$ = 1100, which leads them to infer that one of them paid EUR 12 for the dinner. The crucial thing is that neither the measured contents $\mathbf{a}$, $\mathbf{b}$, $\mathbf{c}$, and $\mathbf{d}$ of the input registers, nor the final private information $\mathbf{p}$ can reveal the identity of the cryptographer who paid the bill.

**D**

_{1}) and (

**D**

_{2}), and become common knowledge to all of them. Finally, they XOR them together to uncover the secret information, i.e., $\mathbf{p}=\mathbf{a}\oplus \mathbf{b}\oplus \mathbf{c}\oplus \mathbf{d}$ = 0000, from which they deduce that none of them paid for the dinner, so it must have been their employer.

## 5. Discussion and Conclusions

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Conflicts of Interest

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**Figure 1.**This figure draws the n qubits that populate the same position in the ${r}_{0},\dots ,{r}_{n-1}$ registers with the same color so as to emphasize that they belong to the same $|GH{Z}_{n}\rangle $n-tuple.

**Figure 2.**This figure visualizes an example of the localized scenario. Four cryptographers, Alice, Bob, Charlie, and Dave, have gathered together at a restaurant for dinner. They want to find out if one of them has paid for this dinner, but without disclosing her or his identity.

**Figure 3.**The above figure illustrates an example of the distributed scenario. Four cryptographers, Alice, Bob, Charlie, and Dave, have arranged a virtual dinner using state-of-the-art technology because they are at different geographical locations. Of course, they still want to find out if one of them has paid for this dinner, but without disclosing her or his identity.

**Figure 4.**The above figure shows the composite quantum circuit used by the dining cryptographers, composed of the individual local circuits Alice and her colleagues possess. Even if these local circuits are spatially separated, they still constitute one composite system because they are linked due to entanglement. The state vectors $|{\psi}_{0}\rangle $, $|{\psi}_{1}\rangle $, $|{\psi}_{2}\rangle $, and $|{\psi}_{f}\rangle $ describe the evolution of this composite system.

**Figure 5.**The above quantum circuit simulates the SQDCP protocol corresponding to the case where Alice paid for the dinner, as outlined in Example 2.

**Figure 6.**Some of the possible measurements and their corresponding probabilities for the circuit in Figure 5.

**Figure 7.**The above quantum circuit simulates the SQDCP protocol corresponding to the case where none of the four cryptographers paid for the dinner, as outlined in Example 2.

**Figure 8.**Some of the possible outcomes and their corresponding probabilities for the circuit in Figure 7.

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Karananou, P.; Andronikos, T.
A Novel Scalable Quantum Protocol for the Dining Cryptographers Problem. *Dynamics* **2024**, *4*, 170-191.
https://doi.org/10.3390/dynamics4010010

**AMA Style**

Karananou P, Andronikos T.
A Novel Scalable Quantum Protocol for the Dining Cryptographers Problem. *Dynamics*. 2024; 4(1):170-191.
https://doi.org/10.3390/dynamics4010010

**Chicago/Turabian Style**

Karananou, Peristera, and Theodore Andronikos.
2024. "A Novel Scalable Quantum Protocol for the Dining Cryptographers Problem" *Dynamics* 4, no. 1: 170-191.
https://doi.org/10.3390/dynamics4010010