Cryptography with Constant Input Locality

Fast radio broadcasting with advice

We consider a fast fading 2-receiver Broadcast channel (BC), where one of the recievers is legitimate and the other is malicious/unintended. The instantaneous Channel State Information (CSI) is assumed to be known only at the receivers (CSIR). The secrecy capacity of this channel is characterized within 11 bits irrespective of fading statistics and Signal-to- Noise Ratios (SNRs). This is achieved by deriving a new upper bound to the secrecy capacity of the channel and two new lower bounds. The upper bound is derived by approximating Complementary Cumulative Distribution Functions (CCDFs) of the two links by corresponding staircase functions and has the desirable property of easy computability for arbitrary fading statistics. The smaller lower bound, although looser, has a form which can be analytically compared with the upper bound and facilitates the approximate secrecy capacity characterization. It is proved that, the so called Binary Expansion Signaling with Reverse Stripping (BES-RS) scheme can achieve a secrecy rate larger than both these lower bounds. This result establishes BES-RS scheme as a practically implementable coding scheme that can achieve the secrecy capacity of the FWC within a constant number of bits, where the constant is universal across all fading statistics and SNR. Explicit numerical computation of these bounds shows that the difference between the secrecy rate achievable by the BES-RS scheme and the upper bound is much smaller than the aforementioned universal constant. This in turn suggests that, much of the gap of 11 bits originates from the loosness introduced to bring the two bounds in a compatiable form.

We exhibit a two-prover perfect zero-knowledge proof system for 3-SAT. In this protocol, the verifier asks a single message to each prover, whose size grows logarithmically in the size of the 3-SAT formula. Each prover's answer consists of only a constant number of bits. The verifier will always accept correct proofs. Given an unsatisfiable formula S the verifier will reject with probability at least ?((|S| - max-sat(S))/|S|, where max-sat(S) denotes the maximum number of clauses of S that may be simultaneously satisfied, and |S| denotes the total number of clauses of S. Using a recent result by Arora et al [2], we can construct for any language in NP a protocol with the property that any nonmember of the language be rejected with constant probability.

Fast periodic graph exploration with constant memory

In a K-pair-user two-way Gaussian interference channel (IC), 2K messages and 2K transmitters/receivers form a Gaussian K-user IC in the forward direction (K messages) and another Gaussian K-user IC in the backward direction (K messages) which operate simultaneously in full-duplex mode. All nodes are permitted to interact, i.e. adapt current channel inputs to past received signals. We derive a new sum-rate outer bound for linear deterministic and Gaussian noise channels allowing for interaction, but show that for symmetric scenarios and certain interference regimes (moderately weak and strong), non-interactive schemes achieve to within a constant gap of this outer bound. That is, interaction for symmetric channels in certain interference regimes may only improve the sum-rate by a constant number of bits per channel use.

This paper deals with the problem of multicasting a set of discrete\nmemoryless correlated sources (DMCS) over a cooperative relay network.\nNecessary conditions with cut-set interpretation are presented. A \\emph{Joint\nsource-Wyner-Ziv encoding/sliding window decoding} scheme is proposed, in which\ndecoding at each receiver is done with respect to an ordered partition of other\nnodes. For each ordered partition a set of feasibility constraints is derived.\nThen, utilizing the sub-modular property of the entropy function and a novel\ngeometrical approach, the results of different ordered partitions are\nconsolidated, which lead to sufficient conditions for our problem. The proposed\nscheme achieves operational separation between source coding and channel\ncoding. It is shown that sufficient conditions are indeed necessary conditions\nin two special cooperative networks, namely, Aref network and finite-field\ndeterministic network. Also, in Gaussian cooperative networks, it is shown that\nreliable transmission of all DMCS whose Slepian-Wolf region intersects the\ncut-set bound region within a constant number of bits, is feasible. In\nparticular, all results of the paper are specialized to obtain an achievable\nrate region for cooperative relay networks which includes relay networks and\ntwo-way relay networks.\n

This paper studies a two-user Gaussian interference channel with an in-band-reception and out-of-band-transmission relay, where the link between the relay and the two receivers is modeled as a degraded broadcast channel. It is shown that a generalized hash-and-forward (GHF) relaying strategy can achieve the capacity region of this channel to within a constant number of bits in a certain weak relay regime, where the transmitter-to-relay link gains are not unboundedly stronger than the interference links between the transmitters and the receivers. In such a regime, the GHF relaying strategy can be performed in an incremental fashion, where the relay message to one receiver is a degraded version of the message to the other receiver. A generalized-degree-of-freedom analysis reveals that in the symmetric channel setting, each common relay bit can improve the sum rate roughly by about one bit or two bits asymptotically, and the rate gain can be thought as coming solely from the improvement of the common message rates.

Previously, the authors have shown that linear precoding with only private streams can have unbounded gap to the sum capacity of the two-user multi-input single-output (MISO) broadcast channel (BC), and that rate-splitting (RS) combined with zero-forcing (ZF) can reduce the gap to a constant number of bits per channel use. In this paper, we extend the investigation to the general K-user multiple-input multiple-output (MIMO) channel. We propose a RS scheme with minimum mean square error (MMSE) precoding. First, we show that the proposed scheme achieves the whole capacity region to within a constant gap in the two-user case. Second, we prove that the proposed scheme does not enjoy the same optimality in the three-user case in general. Third, we provide a simple pathological example showing why RS with linear precoding does not work beyond the two-user case. The current study reveals a fundamental gap between the transmitter-side and the receiver-side interference mitigation.

In this paper, we investigate the effect of feedback on two-user MIMO interference channels. At first, the capacity region of MIMO interference channels with feedback is characterized within a constant number of bits, where this constant is independent of the channel matrices. Further, the generalized degrees of freedom region for the MIMO interference channel with feedback is characterized.

We study the problem of constructing locally computable universal one-way hash functions (UOWHFs) $$\mathcal {H}:\{0,1\}^n \rightarrow \{0,1\}^m$$ . A construction with constant output locality, where every bit of the output depends only on a constant number of bits of the input, was established by Applebaum et al. (SIAM J Comput 36(4):845–888, 2006). However, this construction suffers from two limitations: (1) it can only achieve a sublinear shrinkage of $$n-m=n^{1-\epsilon }$$ and (2) it has a super-constant input locality, i.e., some inputs influence a large super-constant number of outputs. This leaves open the question of realizing UOWHFs with constant output locality and linear shrinkage of $$n-m= \epsilon n$$ , or UOWHFs with constant input locality and minimal shrinkage of $$n-m=1$$ . We settle both questions simultaneously by providing the first construction of UOWHFs with linear shrinkage, constant input locality and constant output locality. Our construction is based on the one-wayness of “random” local functions—a variant of an assumption made by Goldreich (Studies in Complexity and Cryptography, 76–87, 2011; ECCC 2010). Using a transformation of Ishai et al. (STOC, 2008), our UOWHFs give rise to a digital signature scheme with a minimal additive complexity overhead: signing n-bit messages with security parameter $$\kappa $$ takes only $$O(n+\kappa )$$ time instead of $$O(n\kappa )$$ as in typical constructions. Previously, such signatures were only known to exist under an exponential hardness assumption. As an additional contribution, we obtain new locally computable hardness amplification procedures for UOWHFs that preserve linear shrinkage.

We study the problem of constructing locally computable Universal One-Way Hash Functions (UOWHFs) \(\mathcal{H}:\{0,1\}^n \rightarrow \{0,1\}^m\). A construction with constant output locality, where every bit of the output depends only on a constant number of bits of the input, was established by [Applebaum, Ishai, and Kushilevitz, SICOMP 2006]. However, this construction suffers from two limitations: (1) It can only achieve a sub-linear shrinkage of n − m = n 1 − ε; and (2) It has a super-constant input locality, i.e., some inputs influence a large super-constant number of outputs. This leaves open the question of realizing UOWHFs with constant output locality and linear shrinkage of n − m = εn, or UOWHFs with constant input locality and minimal shrinkage of n − m = 1.We settle both questions simultaneously by providing the first construction of UOWHFs with linear shrinkage, constant input locality, and constant output locality. Our construction is based on the one-wayness of “random” local functions – a variant of an assumption made by Goldreich (ECCC 2000). Using a transformation of [Ishai, Kushilevitz, Ostrovsky and Sahai, STOC 2008], our UOWHFs give rise to a digital signature scheme with a minimal additive complexity overhead: signing n-bit messages with security parameter κ takes only O(n + κ) time instead of O(nκ) as in typical constructions. Previously, such signatures were only known to exist under an exponential hardness assumption. As an additional contribution, we obtain new locally-computable hardness amplification procedures for UOWHFs that preserve linear shrinkage.KeywordsHash FunctionLinear ShrinkageRandom StringPseudorandom GeneratorInput LocalityThese keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

The L-user additive white Gaussian noise multiway relay channel is investigated, where L users exchange information at the same rate through a single relay. A new achievable rate region, based on the functional-decode-forward coding strategy, is derived. For the case where there are three or more users, and all nodes transmit at the same power, the capacity is obtained. For the case where the relay power scales with the number of users, it is shown that both compress-forward and functional-decode-forward achieve rates within a constant number of bits of the capacity at all SNR levels; in addition, functional-decode-forward outperforms compress-forward and complete-decode-forward at high SNR levels.

We present a new construction of divisible e-cash that makes use of 1) a new generation method of the binary tree of keys; 2) a new way of using bounded accumulators. The transaction data sent to the merchant has a constant number of bits while spending a monetary value 2ℓ. Moreover, the spending protocol does not require complex zero-knowledge proofs of knowledge such as proofs about double discrete logarithms. We then propose the first strongly anonymous scheme with standard unforgeability requirement and realistic generation parameters while improving the efficiency of the spending phase.

The <i xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">K</i> -user cyclic Z-interference channel models a situation in which the <i xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">k</i> th transmitter causes interference only to the ( <i xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">k</i> -1)th receiver in a cyclic manner, e.g., the first transmitter causes interference only to the <i xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">K</i> th receiver. The impact of noiseless feedback on the capacity of this channel is studied by focusing on the Gaussian cyclic Z-interference channel. To this end, the symmetric feedback capacity of the linear shift deterministic cyclic Z-interference channel is completely characterized for all interference regimes. Using insights from the linear deterministic channel model, the symmetric feedback capacity of the Gaussian cyclic Z-interference channel is characterized up to within a constant number of bits. As a byproduct of the constant gap result, the symmetric generalized degrees of freedom with feedback for the Gaussian cyclic Z-interference channel are also characterized. These results highlight that the symmetric feedback capacities for both linear and Gaussian channel models are in general functions of <i xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">K</i> , the number of users. Furthermore, the capacity gain obtained due to feedback decreases as <i xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">K</i> increases.

A family of error-correcting codes is list-decodable from error fraction $p$ if, for every code in the family, the number of codewords in any Hamming ball of fractional radius $p$ is less than some integer $L$ that is independent of the code length. It is said to be list-recoverable for input $\ell$ if for every sufficiently large subset of codewords (of $L$ or more), there is a coordinate where the codewords take more than $\ell$ values. The parameter $L$ is said to be the list size in either case. The capacity, i.e., the largest possible rate for these notions as the $L \to \infty$, is known to be $1-h_q(p)$ for list-decoding, and $1-\log_q \ell$ for list-recovery, where $q$ is the alphabet of the code family. In this work, we study the of random linear codes for both list-decoding and list-recovery as the rate approaches capacity. We show the following claims hold with high probability over the choice of the code (below, $\epsilon > 0$ is the gap to capacity). (1) A random linear code of rate $1 - \log_q(\ell) - \epsilon$ requires $L \ge \ell^{\Omega(1/\epsilon)}$ for list-recovery from input $\ell$. This is surprisingly in contrast to completely random codes, where $L = O(\ell/\epsilon)$ suffices w.h.p. (2) A random linear code of rate $1 - h_q(p) - \epsilon$ requires $L \ge \lfloor h_q(p)/\epsilon+0.99 \rfloor$ for list-decoding from error fraction $p$, when $\epsilon$ is sufficiently small. (3) A random binary linear code of rate $1 - h_2(p) - \epsilon$ is list-decodable from average error fraction $p$ with with $L \leq \lfloor h_2(p)/\epsilon \rfloor + 2$. The second and third results together precisely pin down the sizes for binary random linear codes for both list-decoding and average-radius list-decoding to three possible values.