arXiv subjects

Ido Grayevsky

Publications and source records attributed to Ido Grayevsky.

5 recordsLinked to original sources

arXiv2025

Dehn functions: computations, lower bounds, and the quasiisometric rigidity of $\rm{Sol}_5$

We establish distortion estimates in completely solvable Lie groups, using a sublinear bilipschitz retraction constructed by Cornulier, and interpolating between two theorems of Osin. This provides new lower bounds on Dehn functions. Our second main result is the quasiisometric rigidity of $\rm{Sol}_5$ and its lattices. Together with a theorem of Peng, a key tool for the rigidity is the complete list of Dehn functions and dimensions of asymptotic cones of all simply connected solvable Lie groups of exponential growth up to dimension $5$, which we compute using Cornulier and Tessera's results.

math.GR↗

arXiv2024

Sublinear Bilipschitz Equivalence and the Quasiisometric Classification of Solvable Lie Groups

We prove a product theorem for sublinear bilipschitz equivalences which generalizes the classical work of Kapovich, Kleiner and Leeb on quasiisometries between product spaces. We employ our product theorem to distinguish up to quasiisometry certain families of solvable groups which share the same dimension, cone-dimension and Dehn function; actually we do this by distinguishing them up to sublinear bilipschitz equivalence, which is slightly stronger. As an application, we recover the fact, recently obtained by Bourdon and R\'emy with different groups, that there exists uncountably many quasiisometry classes of indecomposable, non-unimodular, high rank solvable Lie groups.

math.GR↗

arXiv2023

Sublinear Rigidity of Lattices in Semisimple Lie Groups

Let $G$ be a real centre-free semisimple Lie group without compact factors. I prove that irreducible lattices in $G$ are rigid under two types of sublinear distortions. The first result is that the class of lattices in groups that do not admit $\mathbb{R}$-rank $1$ factors is $\textit{SBE complete}$: if $Λ$ is an abstract finitely generated group that is Sublinearly BiLipschitz Equivalent (SBE) to a lattice $Γ\leq G$, then $Λ$ can be homomorphically mapped into $G$ with finite kernel and image a lattice in $G$. For such $G$ this generalizes the well known quasi-isometric completeness of lattices. The second result concerns sublinear distortions within $G$ itself, and holds without any restriction on the rank of the factors: if $Λ\leq G$ is a discrete subgroup that $\textit{sublinearly covers}$ a lattice $Γ\leq G$, then $Λ$ is itself a lattice.

math.GR↗

arXiv2020

Economically Viable Randomness

We study the problem of providing blockchain applications with \emph{economically viable randomness} (EVR), namely, randomness that has significant economic consequences. Applications of EVR include blockchain-based lotteries and gambling. An EVR source guarantees (i) secrecy, assuring that the random bits are kept secret until some predefined condition indicates that they are safe to reveal (e.g., the lottery's ticket sale closes), and (ii) robustness, guaranteeing that the random bits are published once the condition holds. We formalize the EVR problem and solve it on top of an Ethereum-like blockchain abstraction, which supports smart contracts and a transferable native coin. Randomness is generated via a distributed open commit-reveal scheme by game-theoretic agents who strive to maximize their coin holdings. Note that in an economic setting, such agents might profit from breaking secrecy or robustness, and may engage in side agreements (via smart contracts) to this end. Our solution creates an incentive structure that counters such attacks. We prove that following the protocol gives rise to a stable state, called Coalition-Proof Nash Equilibrium, from which no coalition comprised of a subset of the players can agree to deviate. In this stable state, robustness and secrecy are satisfied. Finally, we implement our EVR source over Ethereum.

cs.CR↗

arXiv2019

Rational Threshold Cryptosystems

We propose a framework for threshold cryptosystems under a permissionless-economic model in which the participants are rational profit-maximizing entities. To date, threshold cryptosystems have been considered under permissioned settings with a limited adversary. Our framework relies on an escrow service that slashes and redistributes deposits to incentivize participants to adhere desired behaviors. Today, more than ever, sophisticated escrow services can be implemented over public blockchains like Ethereum, without additional trust assumptions. The key threat to rational threshold cryptosystems is collusion---by cooperating `illegally', a subset of participants can reveal the cryptosystem's secret, which, in turn is translated to unfair profit. Our countermeasure to collusion is framing. If the escrow is notified of collusion, it rewards the framer and slashes the deposits of all other participants. We show that colluding parties find themselves in the prisoner's dilemma, where the dominant strategy is framing.

cs.CR↗