**Efficient Range-Trapdoor Functions and Applications: Rate-1 OT and More**

*Sanjam Garg and Mohammad Hajiabadi and Rafail Ostrovsky*

**Abstract: **Substantial work on trapdoor functions (TDFs) has led to many powerful notions and applications. However, despite tremendous work and progress, all known constructions have prohibitively large public keys.

In this work, we introduce new techniques for realizing so-called range-trapdoor hash functions with short public keys. This notion, introduced by Döttling et al. [Crypto 2019], allows for encoding a range of indices into a public key in a way that the public key leaks no information about the range, yet an associated trapdoor enables recovery of the corresponding input part. We give constructions of range-trapdoor hash functions, where for a range $I$ the public key consists of $O(n)$ group elements, improving upon $O(n |I|)$ achieved by Döttling et al. Moreover, by designing our evaluation algorithm in a special way involving Toeplitz matrix multiplication and by showing how to perform fast-Fourier transforms in the exponent, we arrive at $O(n \log n)$ group operations for evaluation, improving upon $O(n^2)$, required of previous constructions. Our constructions rely on power-DDH assumptions in pairing-free groups. As applications of our results we obtain

(1) The first construction of (rate-1) lossy TDFs with public keys consisting of a linear number of group elements (without pairings).

(2) Rate-1 string OT with receiver communication complexity of $O(n)$ group elements, where $n$ is the sender's message size, improving upon $O(n^2)$ [Crypto 2019]. This leads to a similar result in the context of private-information retrieval (PIR).

(3) Two-round private-information retrieval protocols for one-bit records, where for a server of $N$ bits, the client's message consists of $O(\lambda) polylog(N)$ group elements, improving upon $O(\lambda^2) polylog(N)$.

(3) Semi-compact homomorphic encryption for branching programs: A construction of homomorphic encryption for branching programs, with ciphertexts consisting of $O(\lambda n d^2)$ group elements, improving upon $O(\lambda^2 n d^3)$. Here $\lambda$ denotes the security parameter, $n$ the input size and $d$ the depth of the program.

**Category / Keywords: **public-key cryptography / Trapdoor functions, range trapdoor hash, power DDH, rate-1 OT, Private information retrieval,

**Date: **received 31 Aug 2019, last revised 26 Sep 2019

**Contact author: **mdhajiabadi at berkeley edu

**Available format(s): **PDF | BibTeX Citation

**Version: **20190927:055311 (All versions of this report)

**Short URL: **ia.cr/2019/990

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