Nathan Keller

Itai Dinur, Nathan Keller, Avichai Marmor

The power of adaptivity in algorithms has been intensively studied in diverse areas of theoretical computer science. In this paper, we obtain a number of sharp lower bound results which show that adaptivity provides a significant extra power in cryptanalytic time-space tradeoffs with (possibly unlimited) preprocessing time. Most notably, we consider the discrete logarithm (DLOG) problem in a generic group of $N$ elements. The classical `baby-step giant-step' algorithm for the problem has time complexity $T=O(\sqrt{N})$, uses $O(\sqrt{N})$ bits of space (up to logarithmic factors in $N$) and achieves constant success probability. We examine a generalized setting where an algorithm obtains an advice string of $S$ bits and is allowed to make $T$ arbitrary non-adaptive queries that depend on the advice string (but not on the challenge group element). We show that in this setting, the $T=O(\sqrt{N})$ online time complexity of the baby-step giant-step algorithm cannot be improved, unless the advice string is more than $Ω(\sqrt{N})$ bits long. This lies in stark contrast with the classical adaptive Pollard's rho algorithm for DLOG, which can exploit preprocessing to obtain the tradeoff curve $ST^2=O(N)$. We obtain similar sharp lower bounds for several other cryptanalytic problems. To obtain our results, we present a new model that allows analyzing non-adaptive preprocessing algorithms for a wide array of search and decision problems in a unified way. Since previous proof techniques inherently cannot distinguish between adaptive and non-adaptive algorithms for the problems in our model, they cannot be used to obtain our results. Consequently, our proof uses a variant of Shearer's lemma for this setting, due to Barthe, Cordero-Erausquin, Ledoux, and Maurey (2011). This seems to be the first time a variant of Shearer's lemma for permutations is used in an algorithmic context.

Elette Boyle, Itai Dinur, Niv Gilboa et al.

Can we sense our location in an unfamiliar environment by taking a sublinear-size sample of our surroundings? Can we efficiently encrypt a message that only someone physically close to us can decrypt? To solve this kind of problems, we introduce and study a new type of hash functions for finding shifts in sublinear time. A function $h:\{0,1\}^n\to \mathbb{Z}_n$ is a $(d,δ)$ {\em locality-preserving hash function for shifts} (LPHS) if: (1) $h$ can be computed by (adaptively) querying $d$ bits of its input, and (2) $\Pr [ h(x) \neq h(x \ll 1) + 1 ] \leq δ$, where $x$ is random and $\ll 1$ denotes a cyclic shift by one bit to the left. We make the following contributions. * Near-optimal LPHS via Distributed Discrete Log: We establish a general two-way connection between LPHS and algorithms for distributed discrete logarithm in the generic group model. Using such an algorithm of Dinur et al. (Crypto 2018), we get LPHS with near-optimal error of $δ=\tilde O(1/d^2)$. This gives an unusual example for the usefulness of group-based cryptography in a post-quantum world. We extend the positive result to non-cyclic and worst-case variants of LPHS. * Multidimensional LPHS: We obtain positive and negative results for a multidimensional extension of LPHS, making progress towards an optimal 2-dimensional LPHS. * Applications: We demonstrate the usefulness of LPHS by presenting cryptographic and algorithmic applications. In particular, we apply multidimensional LPHS to obtain an efficient "packed" implementation of homomorphic secret sharing and a sublinear-time implementation of location-sensitive encryption whose decryption requires a significantly overlapping view.

Itai Dinur, Nathan Keller, Ohad Klein

An average-case variant of the $k$-SUM conjecture asserts that finding $k$ numbers that sum to 0 in a list of $r$ random numbers, each of the order $r^k$, cannot be done in much less than $r^{\lceil k/2 \rceil}$ time. On the other hand, in the dense regime of parameters, where the list contains more numbers and many solutions exist, the complexity of finding one of them can be significantly improved by Wagner's $k$-tree algorithm. Such algorithms for $k$-SUM in the dense regime have many applications, notably in cryptanalysis. In this paper, assuming the average-case $k$-SUM conjecture, we prove that known algorithms are essentially optimal for $k= 3,4,5$. For $k>5$, we prove the optimality of the $k$-tree algorithm for a limited range of parameters. We also prove similar results for $k$-XOR, where the sum is replaced with exclusive or. Our results are obtained by a self-reduction that, given an instance of $k$-SUM which has a few solutions, produces from it many instances in the dense regime. We solve each of these instances using the dense $k$-SUM oracle, and hope that a solution to a dense instance also solves the original problem. We deal with potentially malicious oracles (that repeatedly output correlated useless solutions) by an obfuscation process that adds noise to the dense instances. Using discrete Fourier analysis, we show that the obfuscation eliminates correlations among the oracle's solutions, even though its inputs are highly correlated.

Nathan Keller, Ohad Klein

We prove the following conjecture, raised by Aaronson and Ambainis in 2008: Let $f:\{-1,1\}^n \rightarrow [-1,1]$ be a multilinear polynomial of degree $d$. Then there exists a variable $x_i$ whose influence on $f$ is at least $\mathrm{poly}(\mathrm{Var}(f)/d)$. As was shown by Aaronson and Ambainis, this result implies the following well-known conjecture on the power of quantum computing, dating back to 1999: Let $Q$ be a quantum algorithm that makes $T$ queries to a Boolean input and let $ε,δ> 0$. Then there exists a deterministic classical algorithm that makes $\mathrm{poly}(T,1/ε,1/δ)$ queries to the input and that approximates $Q$'s acceptance probability to within an additive error $ε$ on a $1-δ$ fraction of inputs. In other words, any quantum algorithm can be simulated on most inputs by a classical algorithm which is only polynomially slower, in terms of query complexity.

Achiya Bar-On, Itai Dinur, Orr Dunkelman et al.

The problem of online checkpointing is a classical problem with numerous applications which had been studied in various forms for almost 50 years. In the simplest version of this problem, a user has to maintain $k$ memorized checkpoints during a long computation, where the only allowed operation is to move one of the checkpoints from its old time to the current time, and his goal is to keep the checkpoints as evenly spread out as possible at all times. Bringmann et al. studied this problem as a special case of an online/offline optimization problem in which the deviation from uniformity is measured by the natural discrepancy metric of the worst case ratio between real and ideal segment lengths. They showed this discrepancy is smaller than $1.59-o(1)$ for all $k$, and smaller than $\ln4-o(1)\approx1.39$ for the sparse subset of $k$'s which are powers of 2. In addition, they obtained upper bounds on the achievable discrepancy for some small values of $k$. In this paper we solve the main problems left open in the above-mentioned paper by proving that $\ln4$ is a tight upper and lower bound on the asymptotic discrepancy for all large $k$, and by providing tight upper and lower bounds (in the form of provably optimal checkpointing algorithms, some of which are in fact better than those of Bringmann et al.) for all the small values of $k \leq 10$. In the last part of the paper we describe some new applications of this online checkpointing problem.