#
Multi-User PIR with Cyclic Wraparound Multi-Access Caches^{ †}

^{*}

^{†}

## Abstract

**:**

## 1. Introduction

**Notation**

**1.**

#### 1.1. Private Information Retrieval

**Note**: In the literature, the term rate is used for the transmission cost (e.g., [6]) as we use it here, whereas sometimes (e.g., [2]) the term rate is used for teh mean the inverse of the transmission cost as used by us. We use the term “transmission cost” instead of rate in this paper as in most of the coded caching literature [6,14,15].

#### 1.2. Coded Caching

#### 1.3. Multi-Access Coded Caching with Cyclic Wraparound Cache Access

#### 1.4. Dedicated Cache Aided MuPIR

#### 1.5. Our Contributions

- The paper comprehensively describes the system model for the MACAMuPIR setup with cyclic wraparound cache access. It outlines the key components and mechanisms involved in the scheme.
- The paper presents an achievable scheme for the multi-access problem described above and characterizes its transmission cost.
- A comparison is made between the transmission costs of the multi-access setup and a dedicated cache setup proposed in previous work. The results show that the multi-access setup outperforms the dedicated cache setup.
- The paper includes proofs that validate the privacy guarantees and transmission costs mentioned in the scheme description. These proofs demonstrate the scheme’s ability to preserve user privacy and ensure accurate retrieval of requested data.

#### 1.6. Paper Organization

- In Section 2, the problem statement is described, along with formal descriptions of transmission cost, privacy and correctness conditions.
- Then, in Section 3, the main results of the paper are summarized. The achieved rate is mentioned in this section.
- Section 4 has the scheme to achieve the transmission load mentioned in Section 3. We first explain the scheme using a concrete example in Section 4.1. Then, we extend the description to encompass general parameters in Section 4.2. We then specialize the scheme to the context of cyclic wraparound cache access in Section 4.3. Then, proof of privacy and calculation of subpacketization level follows.
- After the specialized description of Section 4.3, we arrive at the critical observation that to calculate the rate, it is essential to characterize the number of $t+L$-sized subsets of $\left[K\right]$ that contain at least L consecutive integers, with wrapping around K allowed. Here, $t,K,L\in \mathbb{Z}$. This is calculated in Section 4.4 onward.

## 2. System Model: MACAMuPIR with Cyclic Wraparound Caches

## 3. Main Results: Achievable Rate and Comparison

#### 3.1. Achievable Rate

**Theorem**

**1.**

**Proof.**

#### 3.2. Comparison with the Dedicated Cache Setup of [7]

## 4. Achievable Scheme: Proof of Theorem 1

#### 4.1. Example

**Placement Phase**: We let $t=\frac{CM}{N}=1$. We divide each file into $\left(\genfrac{}{}{0pt}{}{8}{1}\right)=8$ subfiles, each indexed by integers in $\left[8\right]$.

**Delivery Phase**: In this phase, every user chooses one of the file indexes. We enumerate the demands of the users:

#### Decoding

#### 4.2. General Scheme: K Users, Each Connected to a Unique Arbitrary Set of L Caches

**Placement Phase**: We let $t=\frac{CM}{N}$ be an integer. Then, we divide each file into $\left(\genfrac{}{}{0pt}{}{C}{t}\right)$ subfiles, each indexed by a t-sized subset of $\left[C\right]$.

**Delivery Phase**: In this phase, every user chooses one of the file indexes. We let user $\mathcal{K},\forall \mathcal{K}\in \mathcal{U}$ choose index ${d}_{\mathcal{K}}\in \left[N\right]$. User $\mathcal{K}$ then wishes to retrieve file ${W}_{{d}_{\mathcal{K}}}$ from the servers without reveling the index of the demanded file to the servers. We let $\mathbf{d}={\left({d}_{\mathcal{K}}\right)}_{\mathcal{K}\in \mathcal{U}}$ be the demand vector. Users do not want the servers to obtain any information about the demand vector. For privately retrieving the files, the users cooperatively generate S queries ${\mathbf{Q}}_{s}^{\mathbf{d}}$ as follows. For every $\mathcal{S}\in \left(\genfrac{}{}{0pt}{}{\left[C\right]}{t+L}\right)$, such that $\mathcal{S}\supset \mathcal{K}$ for at least one $\mathcal{K}\in \mathcal{U}$, the users generate sub-queries

#### 4.2.1. Decoding

#### 4.2.2. Proof of Privacy

#### 4.2.3. Subpacketization

#### 4.3. General Scheme: Cyclic Wraparound Cache Access

#### 4.4. Proving the Expression for $cyc(n,k,m)$

- $1\in \mathcal{K}$ and $n\notin \mathcal{K}$. This corresponds to sequences of the form ${i}_{1},{o}_{1},\dots ,{i}_{r},{o}_{r}$ where ${i}_{l},{o}_{l}\ge 1$ for all $l\in \left[r\right]$, ${\sum}_{l\in \left[r\right]}{i}_{l}=k$, ${\sum}_{l\in \left[r\right]}{o}_{l}=n-k$, $\exists l\in \left[r\right]$ such that ${i}_{l}\ge m$, $\forall r\in [k-m+1]$. We let the set of all such k-sized subsets be ${\mathcal{K}}_{1}$.
- $1\notin \mathcal{K}$ and $n\in \mathcal{K}$. This corresponds to sequences of the form ${o}_{1},{i}_{1},\dots ,{o}_{r},{i}_{r}$ where ${i}_{l},{o}_{l}\ge 1$ for all $l\in \left[r\right]$, ${\sum}_{l\in \left[r\right]}{i}_{l}=k$, ${\sum}_{l\in \left[r\right]}{o}_{l}=n-k$, $\exists l\in \left[r\right]$ such that ${i}_{l}\ge m$, $\forall r\in [k-m+1]$. We let the set of all such k-sized subsets be ${\mathcal{K}}_{2}$.
- $1\notin \mathcal{K}$ and $n\notin \mathcal{K}$. This corresponds to sequences of the form ${o}_{1},{i}_{1},\dots ,{o}_{r},{i}_{r},{o}_{r+1}$ where ${i}_{l},{o}_{l}\ge 1$ for all $l\in [r+1]$, ${\sum}_{l\in \left[r\right]}{i}_{l}=k$, ${\sum}_{l\in [r+1]}{o}_{l}=n-k$ and $\exists l\in \left[r\right]$ such that ${i}_{l}\ge m$, $\forall r\in [k-m+1]$. The set of all such k-sized subsets is denoted by ${\mathcal{K}}_{3}$.
- $1\in \mathcal{K}$ and $n\in \mathcal{K}$. This corresponds to sequences of the form ${i}_{1},{o}_{1},\dots ,{o}_{r-1},{i}_{r}$ where ${i}_{l},{o}_{l}\ge 1$ for all $l\in \left[r\right]$, ${\sum}_{l\in \left[r\right]}{i}_{l}=k$, ${\sum}_{l\in \left[r\right]}{o}_{l}=n-k$ and $\exists l\in [2:r-1]$ such that ${i}_{l}\ge m$ or ${x}_{1}+{x}_{r}\ge m$, $\forall r\in [k-m+1]$. We let the set of all such k-sized subsets be denoted by ${\mathcal{K}}_{4}$.

#### 4.4.1. Calculation of $\left|{\mathcal{K}}_{1}\right|$

#### 4.4.2. Calculation of $\left|{\mathcal{K}}_{2}\right|$

#### 4.4.3. Calculation of $\left|{\mathcal{K}}_{3}\right|$

#### 4.4.4. Calculation of $\left|{\mathcal{K}}_{4}\right|$

#### Calculation of $\left|{\mathcal{K}}_{41}\right|$

#### Calculation of $\left|{\mathcal{K}}_{42}\right|$

## 5. Discussion

## 6. Conclusions

## Author Contributions

## Funding

## Institutional Review Board Statement

## Data Availability Statement

## Conflicts of Interest

## References

- Chor, B.; Goldreich, O.; Kushilevitz, E.; Sudan, M. Private information retrieval. In Proceedings of the IEEE 36th Annual Foundations of Computer Science, Milwaukee, WI, USA, 23–25 October 1995; pp. 41–50. [Google Scholar] [CrossRef]
- Sun, H.; Jafar, S.A. The Capacity of Private Information Retrieval. IEEE Trans. Inf. Theory
**2017**, 63, 4075–4088. [Google Scholar] [CrossRef] - Sun, H.; Jafar, S.A. The Capacity of Robust Private Information Retrieval With Colluding Databases. IEEE Trans. Inf. Theory
**2018**, 64, 2361–2370. [Google Scholar] [CrossRef] - Lin, H.Y.; Kumar, S.; Rosnes, E.; Amat, A.G.i.; Yaakobi, E. Weakly-Private Information Retrieval. In Proceedings of the 2019 IEEE International Symposium on Information Theory (ISIT), Mutualité, France, 7–12 July 2019; pp. 1257–1261. [Google Scholar] [CrossRef]
- Chen, Z.; Wang, Z.; Jafar, S.A. The Capacity of T-Private Information Retrieval with Private Side Information. IEEE Trans. Inf. Theory
**2020**, 66, 4761–4773. [Google Scholar] [CrossRef] - Maddah-Ali, M.A.; Niesen, U. Fundamental Limits of Caching. IEEE Trans. Inf. Theory
**2014**, 60, 2856–2867. [Google Scholar] [CrossRef] - Zhang, X.; Wan, K.; Sun, H.; Ji, M.; Caire, G. On the Fundamental Limits of Cache-Aided Multiuser Private Information Retrieval. IEEE Trans. Commun.
**2021**, 69, 5828–5842. [Google Scholar] [CrossRef] - Hachem, J.; Karamchandani, N.; Diggavi, S.N. Coded Caching for Multi-level Popularity and Access. IEEE Trans. Inf. Theory
**2017**, 63, 3108–3141. [Google Scholar] [CrossRef] - Reddy, K.S.; Karamchandani, N. Rate-Memory Trade-off for Multi-Access Coded Caching With Uncoded Placement. IEEE Trans. Commun.
**2020**, 68, 3261–3274. [Google Scholar] [CrossRef] - Trinadh, P.; Dutta, M.; Thomas, A.; Rajan, B.S. Decentralized Multi-access Coded Caching with Uncoded Prefetching. In Proceedings of the 2021 IEEE Information Theory Workshop (ITW), Virtual, 17–21 October 2021; pp. 1–6. [Google Scholar] [CrossRef]
- Cheng, M.; Wan, K.; Liang, D.; Zhang, M.; Caire, G. A Novel Transformation Approach of Shared-Link Coded Caching Schemes for Multiaccess Networks. IEEE Trans. Commun.
**2021**, 69, 7376–7389. [Google Scholar] [CrossRef] - Sasi; Shanuja; Rajan, B.S. An improved multi-access coded caching with uncoded placement. arXiv
**2020**, arXiv:2009.05377. [Google Scholar] - Serbetci, B.; Parrinello, E.; Elia, P. Multi-access coded caching: Gains beyond cache-redundancy. In Proceedings of the 2019 IEEE Information Theory Workshop (ITW), Visby, Sweden, 25–28 August 2019; pp. 1–5. [Google Scholar] [CrossRef]
- Muralidhar, P.N.; Katyal, D.; Rajan, B.S. Maddah-Ali-Niesen Scheme for Multi-access Coded Caching. In Proceedings of the IEEE Information Theory Workshop, (ITW2021), Kanazawa, Japan, 17–21 October 2021. [Google Scholar]
- Katyal, D.; Muralidhar, P.N.; Rajan, B.S. Multi-access Coded Caching Schemes From Cross Resolvable Designs. IEEE Trans. Inf. Theory
**2021**, 69, 2997–3010. [Google Scholar] [CrossRef] - Brunero, F.; Elia, P. Fundamental Limits of Combinatorial Multi-access Caching. IEEE Trans. Inf. Theory
**2022**, 69, 1037–1056. [Google Scholar] [CrossRef] - Somekh, O.; Zaidel, B.M.; Shamai, S. Spectral Efficiency of Joint Multiple Cell-Site Processors for Randomly Spread DS-CDMA Systems. IEEE Trans. Inf. Theory
**2007**, 53, 2625–2637. [Google Scholar] [CrossRef] - Wyner, A.D. Shannon-theoretic approach to a Gaussian cellular multiple-access channel. IEEE Trans. Inf. Theory
**1994**, 40, 1713–1727. [Google Scholar] [CrossRef] - Wigger, M.; Timo, R.; Shamai, S. Complete interference mitigation through receiver-caching in Wyner’s networks. In Proceedings of the 2016 IEEE Information Theory Workshop (ITW), Cambridge, UK, 11–14 September 2016; pp. 335–339. [Google Scholar] [CrossRef]
- Sanderovich, A.; Somekh, O.; Poor, H.V.; Shamai, S. Uplink Macro Diversity of Limited Backhaul Cellular Network. IEEE Trans. Inf. Theory
**2009**, 55, 3457–3478. [Google Scholar] [CrossRef] - Vaidya, K.; Rajan, B.S. Multi-Access Cache-Aided Multi-User Private Information Retrieval. arXiv
**2022**, arXiv:2201.11481. [Google Scholar] - Vaidya, K.; Rajan, B.S. Cache-Aided Multi-Access Multi-User Private Information Retrieval. In Proceedings of the 2022 20th International Symposium on Modeling and Optimization in Mobile, Ad hoc, and Wireless Networks (WiOpt), Torino, Italy, 19–23 September 2022; pp. 246–253. [Google Scholar] [CrossRef]

**Figure 1.**Multi-access coded caching setup with cyclic wraparound cache access with four users, four helper cache and two servers. Each user is accessing two adjacent helper caches.

**Figure 2.**Comparison of transmission costs for dedicated cache (dotted lines) and multi-access (solid lines) with cyclic wraparound cache access. Here, we take $K=8$ users and cache nodes.

**Figure 3.**Transmission cost for $K=8,L=2,S=2,N=3$. Multi-access setup with cyclic wraparound cache access incur transmission cost only as high as dedicated cache setup with equal total memory in both systems.

Disclaimer/Publisher’s Note: The statements, opinions and data contained in all publications are solely those of the individual author(s) and contributor(s) and not of MDPI and/or the editor(s). MDPI and/or the editor(s) disclaim responsibility for any injury to people or property resulting from any ideas, methods, instructions or products referred to in the content. |

© 2023 by the authors. Licensee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC BY) license (https://creativecommons.org/licenses/by/4.0/).

## Share and Cite

**MDPI and ACS Style**

Vaidya, K.; Rajan, B.S.
Multi-User PIR with Cyclic Wraparound Multi-Access Caches. *Entropy* **2023**, *25*, 1228.
https://doi.org/10.3390/e25081228

**AMA Style**

Vaidya K, Rajan BS.
Multi-User PIR with Cyclic Wraparound Multi-Access Caches. *Entropy*. 2023; 25(8):1228.
https://doi.org/10.3390/e25081228

**Chicago/Turabian Style**

Vaidya, Kanishak, and Balaji Sundar Rajan.
2023. "Multi-User PIR with Cyclic Wraparound Multi-Access Caches" *Entropy* 25, no. 8: 1228.
https://doi.org/10.3390/e25081228