Given a set P of n points in the plane, the unitdisk graph Gr(P) with respect to a parameter r is an undirected graph whose vertex set is P such that an edge connects two points p, q in P if the Euclidean distance between p and q is at most r (the weight of the edge is 1 in the unweighted case and is the distance between p and q in the weighted case). Given a value \lambda>0 and two points s and t of P, we consider the following reverse shortest path problem: computing the smallest r such that the shortest path length between s and t in Gr(P) is at most \lambda. In this paper, we present an algorithm of O(\lfloor \lambda \rfloor \cdot n log n) time and another algorithm of O(n^{5/4} log^{7/4} n) time for the unweighted case, as well as an O(n^{5/4} log^{5/2} n) time algorithm for the weighted case. We also consider the L1 version of the problem where the distance of two points is measured by the L1 metric; we solve the problem in O(n log^3 n) time for both the unweighted and weighted cases.
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Amortizing Rate1 OT and Applications to PIR and PSI
Recent new constructions of rate1 OT [Döttling, Garg, Ishai, Malavolta, Mour, and Ostrovsky, CRYPTO 2019] have brought this primitive under the spotlight and the techniques have led to new feasibility results for privateinformation retrieval, and homomorphic encryption for branching programs. The receiver communication of this construction consists of a quadratic (in the sender's input size) number of group elements for a single instance of rate1 OT. Recently [Garg, Hajiabadi, Ostrovsky, TCC 2020] improved the receiver communication to a linear number of group elements for a single stringOT. However, most applications of rate1 OT require executing it multiple times, resulting in large communication costs for the receiver.
In this work, we introduce a new technique for amortizing the cost of multiple rate1 OTs. Specifically, based on standard pairing assumptions, we obtain a twomessage rate1 OT protocol for which the amortized cost per stringOT is asymptotically reduced to only four group elements. Our results lead to significant communication improvements in PSI and PIR, special cases of SFE for branching programs.
 PIR: We obtain a rate1 PIR scheme with client communication cost of $O(\lambda\cdot\log N)$ group elements for security parameter $\lambda$ and database size $N$. Notably, after a onetime setup (or one PIR instance), any following PIR instance only requires communication cost $O(\log N)$ number of group elements.
 PSI with unbalanced inputs: We apply our techniques to private set intersection with unbalanced set sizes (where the receiver has a smaller set) and achieve receiver communication of $O((m+\lambda) \log N)$ group elements where $m, N$ are the sizes of the receiver and sender sets, respectively. Similarly, after a onetime setup (or one PSI instance), any following PSI instance only requires communication cost $O(m \cdot \log N)$ number of group elements. All previous sublinearcommunication nonFHE based PSI protocols for the above unbalanced setting were also based on rate1 OT, but incurred at least $O(\lambda^2 m \log N)$ group elements.
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 Award ID(s):
 2055358
 NSFPAR ID:
 10323377
 Editor(s):
 Nissim, K.; Waters, B.
 Date Published:
 Journal Name:
 19th Theory of Cryptography Conference (TCC)
 Volume:
 13044
 Format(s):
 Medium: X
 Sponsoring Org:
 National Science Foundation
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