arXiv subjects

Avichai Marmor

Publications and source records attributed to Avichai Marmor.

4 recordsLinked to original sources

arXiv2025

Non-Adaptive Cryptanalytic Time-Space Lower Bounds via a Shearer-like Inequality for Permutations

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 $\Omega(\sqrt{N})$ 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.

cs.CR↗

arXiv2023

Bounds for Characters of the Symmetric Group: A Hypercontractive Approach

Finding upper bounds for character ratios is a fundamental problem in asymptotic group theory. Previous bounds in the symmetric group have led to remarkable applications in unexpected domains. The existing approaches predominantly relied on algebraic methods, whereas our approach combines analytic and algebraic tools. Specifically, we make use of a tool called `hypercontractivity for global functions' from the theory of Boolean functions. By establishing sharp upper bounds on the $L^p$-norms of characters of the symmetric group, we improve existing results on character ratios from the work of Larsen and Shalev [Larsen, M., Shalev, A. Characters of symmetric groups: sharp bounds and applications. Invent. math. 174, 645-687 (2008)]. We use our norm bounds to bound Fourier coefficients of class functions, product mixing of normal sets, mixing time of normal Cayley graphs, and Kronecker coefficients. Our approach bypasses the need for the $S_n$-specific Murnaghan--Nakayama rule. Instead we leverage more flexible representation theoretic tools, such as Young's branching rule, which potentially extend the applicability of our method to groups beyond $S_n$.

math.CO↗

arXiv2023

Schur-Positivity of Short Chords in Matchings

We prove that the set of matchings with a fixed number of unmatched vertices is Schur-positive with respect to the set of short chords. Two proofs are presented. The first proof applies a new combinatorial criterion for Schur-positivity, while the second is bijective. The coefficients in the Schur expansion are derived, and interpreted in terms of Bessel polynomials. We present a Knuth-like equivalence relation on matchings, and show that every equivalence class corresponds to an irreducible representation. We proceed to find various refined Schur-positive sets, including the set of matchings with a prescribed crossing number and the set of matchings with a given number of pairs of intersecting chords. Finally, we characterize all the matchings $m$ such that the set of matchings avoiding $m$ is Schur-positive.

math.CO↗

arXiv2022

Tight Lower Bound for Pattern Avoidance and Symmetric Functions

For a set of permutations (patterns) $\Pi$ in $S_k$, consider the set of permutations in $S_n$ that avoid all patterns in $\Pi$. In current algebraic combinatorics, a significant problem is to identify pattern sets $\Pi$ for which the corresponding quasisymmetric function is symmetric for all $n$. Recently, Bloom and Sagan proved that unless $\Pi \subseteq \{12\dots k, k \dots 21\}$, the size of such $\Pi$ must be at least $3$ for any $k \ge 4$. They also posed a general lower bound conjecture. In this work, we resolve this conjecture and give a tight lower bound, namely, the minimal size of such $\Pi$ is exactly $k - 1$. The proof relies on a novel generalization of Bose's theorem in extremal combinatorics, utilizing the multilinear polynomial approach introduced by Alon, Babai, and Suzuki.

math.CO↗