On the Impossibility of Tight Cryptographic Reductions

The existence of tight reductions in cryptographic security proofs is an important question, motivated by the theoretical search for cryptosystems whose security guarantees are truly independent of adversarial behavior and the practical necessity of concrete security bounds for the theoretically-sound selection of cryptographic parameters. At Eurocrypt 2002, Coron described a meta-reduction technique that allows to prove the impossibility of tight reductions for certain digital signature schemes. This seminal result has found many further interesting applications. However, due to a technical subtlety in the argument, the applicability of this technique beyond digital signatures in the single-user setting has turned out to be rather limited. We describe a new meta-reduction technique for proving such impossibility results, which improves on known ones in several ways. First, it enables interesting novel applications. This includes a formal proof that for certain cryptographic primitives (including public-key encryption/key encapsulation mechanisms and digital signatures), the security loss incurred when the primitive is transferred from an idealized single-user setting to the more realistic multi-user setting is impossible to avoid, and a lower tightness bound for non-interactive key exchange protocols. Second, the technique allows to rule out tight reductions from a very general class of non-interactive complexity assumptions. Third, the provided bounds are quantitatively and qualitatively better, yet simpler, than the bounds derived from Coron\u27s technique and its extensions

Tight bounds for classical and quantum coin flipping

Coin flipping is a cryptographic primitive for which strictly better protocols exist if the players are not only allowed to exchange classical, but also quantum messages. During the past few years, several results have appeared which give a tight bound on the range of implementable unconditionally secure coin flips, both in the classical as well as in the quantum setting and for both weak as well as strong coin flipping. But the picture is still incomplete: in the quantum setting, all results consider only protocols with perfect correctness, and in the classical setting tight bounds for strong coin flipping are still missing. We give a general definition of coin flipping which unifies the notion of strong and weak coin flipping (it contains both of them as special cases) and allows the honest players to abort with a certain probability. We give tight bounds on the achievable range of parameters both in the classical and in the quantum setting.Comment: 18 pages, 2 figures; v2: published versio

Separating Two-Round Secure Computation From Oblivious Transfer

We consider the question of minimizing the round complexity of protocols for secure multiparty computation (MPC) with security against an arbitrary number of semi-honest parties. Very recently, Garg and Srinivasan (Eurocrypt 2018) and Benhamouda and Lin (Eurocrypt 2018) constructed such 2-round MPC protocols from minimal assumptions. This was done by showing a round preserving reduction to the task of secure 2-party computation of the oblivious transfer functionality (OT). These constructions made a novel non-black-box use of the underlying OT protocol. The question remained whether this can be done by only making black-box use of 2-round OT. This is of theoretical and potentially also practical value as black-box use of primitives tends to lead to more efficient constructions. Our main result proves that such a black-box construction is impossible, namely that non-black-box use of OT is necessary. As a corollary, a similar separation holds when starting with any 2-party functionality other than OT. As a secondary contribution, we prove several additional results that further clarify the landscape of black-box MPC with minimal interaction. In particular, we complement the separation from 2-party functionalities by presenting a complete 4-party functionality, give evidence for the difficulty of ruling out a complete 3-party functionality and for the difficulty of ruling out black-box constructions of 3-round MPC from 2-round OT, and separate a relaxed "non-compact" variant of 2-party homomorphic secret sharing from 2-round OT

Pre-Reduction Graph Products: Hardnesses of Properly Learning DFAs and Approximating EDP on DAGs

The study of graph products is a major research topic and typically concerns the term $f(G*H)$, e.g., to show that $f(G*H)=f(G)f(H)$. In this paper, we study graph products in a non-standard form $f(R[G*H]$ where $R$ is a "reduction", a transformation of any graph into an instance of an intended optimization problem. We resolve some open problems as applications. (1) A tight $n^{1-\epsilon}$-approximation hardness for the minimum consistent deterministic finite automaton (DFA) problem, where $n$ is the sample size. Due to Board and Pitt [Theoretical Computer Science 1992], this implies the hardness of properly learning DFAs assuming $NP\neq RP$ (the weakest possible assumption). (2) A tight $n^{1/2-\epsilon}$ hardness for the edge-disjoint paths (EDP) problem on directed acyclic graphs (DAGs), where $n$ denotes the number of vertices. (3) A tight hardness of packing vertex-disjoint $k$-cycles for large $k$. (4) An alternative (and perhaps simpler) proof for the hardness of properly learning DNF, CNF and intersection of halfspaces [Alekhnovich et al., FOCS 2004 and J. Comput.Syst.Sci. 2008]

Limits to Non-Malleability

There have been many successes in constructing explicit non-malleable codes for various classes of tampering functions in recent years, and strong existential results are also known. In this work we ask the following question: When can we rule out the existence of a non-malleable code for a tampering class ?? First, we start with some classes where positive results are well-known, and show that when these classes are extended in a natural way, non-malleable codes are no longer possible. Specifically, we show that no non-malleable codes exist for any of the following tampering classes: - Functions that change d/2 symbols, where d is the distance of the code; - Functions where each input symbol affects only a single output symbol; - Functions where each of the n output bits is a function of n-log n input bits. Furthermore, we rule out constructions of non-malleable codes for certain classes ? via reductions to the assumption that a distributional problem is hard for ?, that make black-box use of the tampering functions in the proof. In particular, this yields concrete obstacles for the construction of efficient codes for NC, even assuming average-case variants of P ? NC

Minimal Complete Primitives for Secure Multi-Party Computation

The study of minimal cryptographic primitives needed to implement secure computation among two or more players is a fundamental question in cryptography. The issue of complete primitives for the case of two players has been thoroughly studied. However, in the multi-party setting, when there are n > 2 players and t of them are corrupted, the question of what are the simplest complete primitives remained open for t ≥ n/3. (A primitive is called complete if any computation can be carried out by the players having access only to the primitive and local computation.) In this paper we consider this question, and introduce complete primitives of minimal cardinality for secure multi-party computation. The cardinality issue (number of players accessing the primitive) is essential in settings where primitives are implemented by some other means, and the simpler the primitive the easier it is to realize. We show that our primitives are complete and of minimal cardinality possible for most case

Assisted Common Information: Further Results

We presented assisted common information as a generalization of G\'acs-K\"orner (GK) common information at ISIT 2010. The motivation for our formulation was to improve upperbounds on the efficiency of protocols for secure two-party sampling (which is a form of secure multi-party computation). Our upperbound was based on a monotonicity property of a rate-region (called the assisted residual information region) associated with the assisted common information formulation. In this note we present further results. We explore the connection of assisted common information with the Gray-Wyner system. We show that the assisted residual information region and the Gray-Wyner region are connected by a simple relationship: the assisted residual information region is the increasing hull of the Gray-Wyner region under an affine map. Several known relationships between GK common information and Gray-Wyner system fall out as consequences of this. Quantities which arise in other source coding contexts acquire new interpretations. In previous work we showed that assisted common information can be used to derive upperbounds on the rate at which a pair of parties can {\em securely sample} correlated random variables, given correlated random variables from another distribution. Here we present an example where the bound derived using assisted common information is much better than previously known bounds, and in fact is tight. This example considers correlated random variables defined in terms of standard variants of oblivious transfer, and is interesting on its own as it answers a natural question about these cryptographic primitives.Comment: 8 pages, 3 figures, 1 appendix; to be presented at the IEEE International Symposium on Information Theory, 201

Cryptology in the Crowd

Uhell skjer: Kanskje mistet du nøkkelen til huset, eller hadde PIN-koden til innbruddsalarmen skrevet på en dårlig plassert post-it lapp. Og kanskje endte de slik opp i hendene på feil person, som nå kan påføre livet ditt all slags ugagn: Sikkerhetssystemer gir ingen garantier når nøkler blir stjålet og PIN-koder lekket. Likevel burde naboen din, hvis nøkkel-og-PIN-kode rutiner er heller vanntette, kunne føle seg trygg i vissheten om at selv om du ikke evner å sikre huset ditt mot innbrudd, så forblir deres hjem trygt. Det er tilsvarende for kryptologi, som også lener seg på at nøkkelmateriale hemmeligholdes for å kunne garantere sikkerhet: Intuitivt forventer man at kjennskap til ett systems hemmelige nøkkel ikke burde være til hjelp for å bryte inn i andre, urelaterte systemer. Men det har vist seg overraskende vanskelig å sette denne intuisjonen på formell grunn, og flere konkurrerende sikkerhetsmodeller av varierende styrke har oppstått. Det blir dermed naturlig å spørre seg: Hvilken formalisme er den riktige når man skal modellere realistiske scenarioer med mange brukere og mulige lekkasjer? Eller: hvordan bygger man kryptografi i en folkemengde? Artikkel I begir seg ut på reisen mot et svar ved å sammenligne forskjellige flerbrukervarianter av sikkerhetsmodellen IND-CCA, med og uten evnen til å motta hemmelige nøkler tilhørende andre brukere. Vi finner et delvis svar ved å vise at uten denne evnen, så er noen modeller faktisk å foretrekke over andre. Med denne evnen, derimot, forblir situasjonen uavklart. Artikkel II tar et sidesteg til et sett relaterte sikkerhetsmodeller hvor, heller enn å angripe én enkelt bruker (ut fra en mengde av mulige ofre), angriperen ønsker å bryte kryptografien til så mange brukere som mulig på én gang. Man ser for seg en uvanlig mektig motstander, for eksempel en statssponset aktør, som ikke har problemer med å bryte kryptografien til en enkelt bruker: Målet skifter dermed fra å garantere trygghet for alle brukerne, til å gjøre masseovervåking så vanskelig som mulig, slik at det store flertall av brukere kan forbli sikret. Artikkel III fortsetter der Artikkel I slapp ved å sammenligne og systematisere de samme IND-CCA sikkerhetsmodellene med en større mengde med sikkerhetsmodeller, med det til felles at de alle modellerer det samme (eller lignende) scenarioet. Disse modellene, som går under navnene SOA (Selective Opening Attacks; utvalgte åpningsangrep) og NCE (Non-Committing Encryption; ikke-bindende kryptering), er ofte vesentlig sterkere enn modellene studert i Artikkel I. Med et system på plass er vi i stand til å identifisere en rekke hull i litteraturen; og dog vi tetter noen, etterlater vi mange som åpne problemer.Accidents happen: you may misplace the key to your home, or maybe the PIN to your home security system was written on an ill-placed post-it note. And so they end up in the hands of a bad actor, who is then granted the power to wreak all kinds of havoc in your life: the security of your home grants no guarantees when keys are stolen and PINs are leaked. Nonetheless your neighbour, whose key-and-pin routines leave comparatively little to be desired, should feel safe that just because you can’t keep your house safe from intruders, their home remains secured. It is likewise with cryptography, whose security also relies on the secrecy of key material: intuitively, the ability to recover the secret keys of other users should not help an adversary break into an uncompromised system. Yet formalizing this intuition has turned out tricky, with several competing notions of security of varying strength. This begs the question: when modelling a real-world scenario with many users, some of which may be compromised, which formalization is the right one? Or: how do we build cryptology in a crowd? Paper I embarks on the quest to answer the above questions by studying how various notions of multi-user IND-CCA compare to each other, with and without the ability to adaptively compromise users. We partly answer the question by showing that, without compromise, some notions of security really are preferable over others. Still, the situation is left largely open when compromise is accounted for. Paper II takes a detour to a related set of security notions in which, rather than attacking a single user, an adversary seeks to break the security of many. One imagines an unusually powerful adversary, for example a state-sponsored actor, for whom brute-forcing a single system is not a problem. Our goal then shifts from securing every user to making mass surveillance as difficult as possible, so that the vast majority of uncompromised users can remain secure. Paper III picks up where Paper I left off by comparing and systemizing the same security notions with a wider array of security notions that aim to capture the same (or similar) scenarios. These notions appear under the names of Selective Opening Attacks (SOA) and Non-Committing Encryption (NCE), and are typically significantly stronger than the notions of IND-CCA studied in Paper I. With a system in place, we identify and highlight a number of gaps, some of which we close, and many of which are posed as open problems.Doktorgradsavhandlin