Hua Sun
Abstract
A private information retrieval (PIR) scheme is a mechanism that allows a user to retrieve any one out of <inline-formula> <tex-math notation="LaTeX">$K$ </tex-math></inline-formula> messages from <inline-formula> <tex-math notation="LaTeX">$N$ </tex-math></inline-formula> non-communicating replicated databases, each of which stores all <inline-formula> <tex-math notation="LaTeX">$K$ </tex-math></inline-formula> messages, without revealing anything (in the information theoretic sense) about the identity of the desired message index to any individual database. If the size of each message is <inline-formula> <tex-math notation="LaTeX">$L$ </tex-math></inline-formula> bits and the total download required by a PIR scheme from all <inline-formula> <tex-math notation="LaTeX">$N$ </tex-math></inline-formula> databases is <inline-formula> <tex-math notation="LaTeX">$D$ </tex-math></inline-formula> bits, then <inline-formula> <tex-math notation="LaTeX">$D$ </tex-math></inline-formula> is called the download cost and the ratio <inline-formula> <tex-math notation="LaTeX">$L/D$ </tex-math></inline-formula> is called an achievable rate. For fixed <inline-formula> <tex-math notation="LaTeX">$K,N\in \mathbb {N}$ </tex-math></inline-formula>, the capacity of PIR, denoted by <inline-formula> <tex-math notation="LaTeX">$C$ </tex-math></inline-formula>, is the supremum of achievable rates over all PIR schemes and over all message sizes, and was recently shown to be <inline-formula> <tex-math notation="LaTeX">$C=(1+1/N+1/N^{2}+\cdots +1/N^{K-1})^{-1}$ </tex-math></inline-formula>. In this paper, for arbitrary <inline-formula> <tex-math notation="LaTeX">$K$ </tex-math></inline-formula> and <inline-formula> <tex-math notation="LaTeX">$N$ </tex-math></inline-formula>, we explore the minimum download cost <inline-formula> <tex-math notation="LaTeX">$D_{L}$ </tex-math></inline-formula> across all PIR schemes (not restricted to linear schemes) for arbitrary message lengths <inline-formula> <tex-math notation="LaTeX">$L$ </tex-math></inline-formula> under arbitrary choices of alphabet (not restricted to finite fields) for the message and download symbols. If the same <inline-formula> <tex-math notation="LaTeX">$M$ </tex-math></inline-formula>-ary alphabet is used for the message and download symbols, then we show that the optimal download cost in <inline-formula> <tex-math notation="LaTeX">$M$ </tex-math></inline-formula>-ary symbols is <inline-formula> <tex-math notation="LaTeX">$D_{L}=\lceil \frac {L}{C}\rceil$ </tex-math></inline-formula>. If the message symbols are in <inline-formula> <tex-math notation="LaTeX">$M$ </tex-math></inline-formula>-ary alphabet and the downloaded symbols are in <inline-formula> <tex-math notation="LaTeX">$M'$ </tex-math></inline-formula>-ary alphabet, then we show that the optimal download cost in <inline-formula> <tex-math notation="LaTeX">$M'$ </tex-math></inline-formula>-ary symbols, <inline-formula> <tex-math notation="LaTeX">$D_{L}\in \{\lceil ~({L'}/{C})\rceil, \lceil ~({L'}/{C}\rceil -1,\lceil ~({L'}/{C})\rceil -2\}$ </tex-math></inline-formula>, where <inline-formula> <tex-math notation="LaTeX">$L'= \lceil L \log _{M'} M\rceil$ </tex-math></inline-formula>, i.e., the optimal download cost is characterized to within two symbols.
- [1] , “Private information retrieval,” in Proc. 36th Annu. Symp. Found. Comput. Sci.,
1995 , pp. 41–50.Google Scholar
- [2] , “Private information retrieval,” J. ACM, vol. 45, no. 6, pp. 965–981, 1998.Google ScholarDigital Library
- [3] , “
Upper bound on the communication complexity of private information retrieval ,” in Automata, Languages and Programming. London, U.K.: Springer-Verlag, 1997, pp. 401–407.Google Scholar - [4] , “General constructions for information-theoretic private information retrieval,” J. Comput. Syst. Sci., vol. 71, no. 2, pp. 213–247, 2005.Google ScholarDigital Library
- [5] , “Breaking the
$O(n^{1/(2k-1)})$
barrier for information-theoretic private information retrieval,” in Proc. 43rd Annu. IEEE Symp. Found. Comput. Sci.,
Nov. 2002 , pp. 261–270.Google Scholar - [6] , “Locally decodable codes and private information retrieval schemes,” Ph.D. dissertation,
Dept. Elect. Eng. Comput. Sci.,
Massachusetts Inst. Technol., Cambridge, MA, USA, 2007.Google Scholar
- [7] , “2-Server PIR with sub-polynomial communication,” in Proc. 47th Annu. ACM Symp. Theory Comput. (STOC),
2015 , pp. 577–584.Google Scholar - [8] , “
Reducing the servers computation in private information retrieval: PIR with preprocessing ,” in Advances in Cryptology—CRYPTO. Berlin, Germany: Springer-Verlag, 2000, pp. 55–73.Google Scholar - [9] , “
A random server model for private information retrieval ,” in Randomization and Approximation Techniques in Computer Science. Berlin, Germany: Springer-Verlag, 1998, pp. 200–217.Google Scholar - [10] , “Universal service-providers for database private information retrieval,” in Proc. 17th Annu. ACM Symp. Principles Distrib. Comput.,
1998 , pp. 91–100.Google Scholar - [11] , “One extra bit of download ensures perfectly private information retrieval,” in Proc. IEEE Int. Symp. Inf. Theory (ISIT),
Mar. 2014 , pp. 856–860.Google Scholar - [12] , “Private information retrieval for coded storage,” in Proc. IEEE Int. Symp. Inf. Theory (ISIT),
Nov. 2015 , pp. 2842–2846.Google Scholar - [13] , “Codes for distributed PIR with low storage overhead,” in Proc. IEEE Int. Symp. Inf. Theory (ISIT),
Nov. 2015 , pp. 2852–2856.Google Scholar - [14] . (Feb. 2016). “Private information retrieval from MDS coded data in distributed storage systems.” [Online]. Available: https://arxiv.org/abs/1602.01458Google Scholar
- [15] . (May 2016). “Lower bound on the redundancy of PIR codes.” [Online]. Available: https://arxiv.org/abs/1605.01869Google Scholar
- [16] . (Jul. 2016). “PIR array codes with optimal PIR rate.” [Online]. Available: https://arxiv.org/abs/1607.00235Google Scholar
- [17] . (Sep. 2016). “PIR schemes with small download complexity and low storage requirements.” [Online]. Available: https://arxiv.org/abs/1609.07027Google Scholar
- [18] . (Sep. 2016). “The capacity of private information retrieval from coded databases.” [Online]. Available: https://arxiv.org/abs/1609.08138Google Scholar
- [19] . (Sep. 2016). “On private information retrieval array codes.” [Online]. Available: https://arxiv.org/abs/1609.09167Google Scholar
- [20] . (Feb. 2016). “The capacity of private information retrieval.” [Online]. Available: https://arxiv.org/abs/1602.09134Google Scholar
- [21] . (Jan. 2016). “Blind interference alignment for private information retrieval.” [Online]. Available: https://arxiv.org/abs/1601.07885Google Scholar
- [22] . (May 2016). “The capacity of robust private information retrieval with colluding databases.” [Online]. Available: https://arxiv.org/abs/1605.00635Google Scholar
Index Terms
- Optimal Download Cost of Private Information Retrieval for Arbitrary Message Length
Recommendations
Private Information Delivery
We introduce the problem of private information delivery (PID), comprised of <inline-formula> <tex-math notation="LaTeX">$K$ </tex-math></inline-formula> messages, a user, and <inline-formula> <tex-math notation="LaTeX">$N$ </tex-math></inline-formula> ...
Private Set Intersection Using Multi-Message Symmetric Private Information Retrieval
2020 IEEE International Symposium on Information Theory (ISIT)We study the problem of private set intersection (PSI). In PSI, there are two entities, each storing a set ${\mathcal{P}_i}$, whose elements are picked from a finite set ${\mathbb{S}_K}$, on N<inf>i</inf> replicated and non-colluding databases. It is ...
On Single Server Private Information Retrieval With Private Coded Side Information
Motivated by an open problem and a conjecture, this work studies the problem of single server private information retrieval with private coded side information (PIR-PCSI) that was recently introduced by Heidarzadeh et al. The goal of PIR-PCSI is to allow ...
Comments