Showing papers by "Johan Håstad published in 2017"

Quantum Algorithms for Computing Short Discrete Logarithms and Factoring RSA Integers

Abstract: We generalize the quantum algorithm for computing short discrete logarithms previously introduced by Ekera [2] so as to allow for various tradeoffs between the number of times that the algorithm need be executed on the one hand, and the complexity of the algorithm and the requirements it imposes on the quantum computer on the other hand. Furthermore, we describe applications of algorithms for computing short discrete logarithms. In particular, we show how other important problems such as those of factoring RSA integers and of finding the order of groups under side information may be recast as short discrete logarithm problems. This gives rise to an algorithm for factoring RSA integers that is less complex than Shor’s general factoring algorithm in the sense that it imposes smaller requirements on the quantum computer. In both our algorithm and Shor’s algorithm, the main hurdle is to compute a modular exponentiation in superposition. When factoring an n bit integer, the exponent is of length 2n bits in Shor’s algorithm, compared to slightly more than n/2 bits in our algorithm.

$(2+\varepsilon)$-Sat Is NP-hard

Abstract: We prove the following hardness result for a natural promise variant of the classical CNF-satisfiability problem: Given a CNF-formula where each clause has width $w$ and the guarantee that there exists an assignment satisfying at least $g = \lceil \frac{w}{2}\rceil -1$ literals in each clause, it is NP-hard to find a satisfying assignment to the formula (that sets at least one literal to true in each clause). On the other hand, when $g = \lceil \frac{w}{2}\rceil$, it is easy to find a satisfying assignment via simple generalizations of the algorithms for 2-Sat. Viewing 2-Sat $\in \mathrm{P}$ as tractability of Sat when 1 in 2 literals are true in every clause, and NP-hardness of 3-Sat as intractability of Sat when 1 in 3 literals are true, our result shows, for any fixed $\varepsilon > 0$, the difficulty of finding a satisfying assignment to instances of “$(2+\varepsilon)$-Sat” where the density of satisfied literals in each clause is guaranteed to exceed $\frac{1}{2+\varepsilon}$. We also strengthen the ...

An Average-Case Depth Hierarchy Theorem for Boolean Circuits

Abstract: We prove an average-case depth hierarchy theorem for Boolean circuits over the standard basis of AND, OR, and NOT gates. Our hierarchy theorem says that for every d ≥ 2, there is an explicit n-variable Boolean function f, computed by a linear-size depth-d formula, which is such that any depth-(d−1) circuit that agrees with f on (1/2 + on(1)) fraction of all inputs must have size exp(nΩ (1/d)). This answers an open question posed by Hastad in his Ph.D. thesis (Hastad 1986b).Our average-case depth hierarchy theorem implies that the polynomial hierarchy is infinite relative to a random oracle with probability 1, confirming a conjecture of Hastad (1986a), Cai (1986), and Babai (1987). We also use our result to show that there is no “approximate converse” to the results of Linial, Mansour, Nisan (Linial et al. 1993) and (Boppana 1997) on the total influence of bounded-depth circuits.A key ingredient in our proof is a notion of random projections which generalize random restrictions.

On Small-Depth Frege Proofs for Tseitin for Grids

Abstract: We prove a lower bound on the size of a small depth Frege refutation of the Tseitin contradiction on the grid. We conclude that polynomial size such refutations must use formulas of almost logarithmic depth.

Improved NP-Inapproximability for $2$-Variable Linear Equations

Abstract: An instance of the 2-Lin(2) problem is a system of equations of the form "x(i)+x(j)= b (mod 2)." Given such a system in which it is possible to satisfy all but an epsilon fraction of the equations, ...

Super-Polylogarithmic Hypergraph Coloring Hardness via Low-Degree Long Codes

Abstract: We prove improved inapproximability results for hypergraph coloring using the low-degree polynomial code (aka the “short code” of Barak et al. [SIAM J. Comput., 44 (2015), pp. 1287--1324]) and the techniques proposed by Dinur and Guruswami [Israel J. Math., 209 (2015), pp. 611--649] to incorporate this code for inapproximability results. In particular, we prove quasi NP-hardness of the following problems on $n$-vertex hypergraphs: coloring a 2-colorable 8-uniform hypergraph with $2^{2^{\Omega(\sqrt{\log \log n})}}$ colors; coloring a 4-colorable 4-uniform hypergraph with $2^{2^{\Omega(\sqrt{\log \log n})}}$ colors; and coloring a 3-colorable 3-uniform hypergraph with $(\log n)^{\Omega(1/\log\log\log n)}$ colors. For the first two cases, the hardness results obtained are superpolynomial in what was previously known, and in the last case it is an exponential improvement. In fact, prior to this result, $(\log n)^{O(1)}$ colors was the strongest quantitative bound on the number of colors ruled out by inapprox...

Quantum algorithms for computing short discrete logarithms and factoring RSA integers

Abstract: In this paper we generalize the quantum algorithm for computing short discrete logarithms previously introduced by Ekera so as to allow for various tradeoffs between the number of times that the algorithm need be executed on the one hand, and the complexity of the algorithm and the requirements it imposes on the quantum computer on the other hand. Furthermore, we describe applications of algorithms for computing short discrete logarithms. In particular, we show how other important problems such as those of factoring RSA integers and of finding the order of groups under side information may be recast as short discrete logarithm problems. This immediately gives rise to an algorithm for factoring RSA integers that is less complex than Shor's general factoring algorithm in the sense that it imposes smaller requirements on the quantum computer. In both our algorithm and Shor's algorithm, the main hurdle is to compute a modular exponentiation in superposition. When factoring an n bit integer, the exponent is of length 2n bits in Shor's algorithm, compared to slightly more than n/2 bits in our algorithm.