233 research outputs found
An Improved Homomorphism Preservation Theorem From Lower Bounds in Circuit Complexity
Previous work of the author [Rossmann\u2708] showed that the Homomorphism Preservation Theorem of classical model theory remains valid when its statement is restricted to finite structures. In this paper, we give a new proof of this result via a reduction to lower bounds in circuit complexity, specifically on the AC0 formula size of the colored subgraph isomorphism problem. Formally, we show the following: if a first-order sentence of quantifier-rank k is preserved under homomorphisms on finite structures, then it is equivalent on finite structures to an existential-positive sentence of quantifier-rank poly(k). Quantitatively, this improves the result of [Rossmann\u2708], where the upper bound on quantifier-rank is a non-elementary function of k
Conditionals in Homomorphic Encryption and Machine Learning Applications
Homomorphic encryption aims at allowing computations on encrypted data
without decryption other than that of the final result. This could provide an
elegant solution to the issue of privacy preservation in data-based
applications, such as those using machine learning, but several open issues
hamper this plan. In this work we assess the possibility for homomorphic
encryption to fully implement its program without relying on other techniques,
such as multiparty computation (SMPC), which may be impossible in many use
cases (for instance due to the high level of communication required). We
proceed in two steps: i) on the basis of the structured program theorem
(Bohm-Jacopini theorem) we identify the relevant minimal set of operations
homomorphic encryption must be able to perform to implement any algorithm; and
ii) we analyse the possibility to solve -- and propose an implementation for --
the most fundamentally relevant issue as it emerges from our analysis, that is,
the implementation of conditionals (requiring comparison and selection/jump
operations). We show how this issue clashes with the fundamental requirements
of homomorphic encryption and could represent a drawback for its use as a
complete solution for privacy preservation in data-based applications, in
particular machine learning ones. Our approach for comparisons is novel and
entirely embedded in homomorphic encryption, while previous studies relied on
other techniques, such as SMPC, demanding high level of communication among
parties, and decryption of intermediate results from data-owners. Our protocol
is also provably safe (sharing the same safety as the homomorphic encryption
schemes), differently from other techniques such as
Order-Preserving/Revealing-Encryption (OPE/ORE).Comment: 14 pages, 1 figure, corrected typos, added introductory pedagogical
section on polynomial approximatio
Deterministic Identity Testing Paradigms for Bounded Top-Fanin Depth-4 Circuits
Polynomial Identity Testing (PIT) is a fundamental computational problem. The famous depth-4 reduction (Agrawal & Vinay, FOCS\u2708) has made PIT for depth-4 circuits, an enticing pursuit. The largely open special-cases of sum-product-of-sum-of-univariates (?^[k] ? ? ?) and sum-product-of-constant-degree-polynomials (?^[k] ? ? ?^[?]), for constants k, ?, have been a source of many great ideas in the last two decades. For eg. depth-3 ideas (Dvir & Shpilka, STOC\u2705; Kayal & Saxena, CCC\u2706; Saxena & Seshadhri, FOCS\u2710, STOC\u2711); depth-4 ideas (Beecken, Mittmann & Saxena, ICALP\u2711; Saha,Saxena & Saptharishi, Comput.Compl.\u2713; Forbes, FOCS\u2715; Kumar & Saraf, CCC\u2716); geometric Sylvester-Gallai ideas (Kayal & Saraf, FOCS\u2709; Shpilka, STOC\u2719; Peleg & Shpilka, CCC\u2720, STOC\u2721). We solve two of the basic underlying open problems in this work.
We give the first polynomial-time PIT for ?^[k] ? ? ?. Further, we give the first quasipolynomial time blackbox PIT for both ?^[k] ? ? ? and ?^[k] ? ? ?^[?]. No subexponential time algorithm was known prior to this work (even if k = ? = 3). A key technical ingredient in all the three algorithms is how the logarithmic derivative, and its power-series, modify the top ?-gate to ?
Preservation Theorems Through the Lens of Topology
In this paper, we introduce a family of topological spaces that captures the existence of preservation theorems. The structure of those spaces allows us to study the relativisation of preservation theorems under suitable definitions of surjective morphisms, subclasses, sums, products, topological closures, and projective limits. Throughout the paper, we also integrate already known results into this new framework and show how it captures the essence of their proofs
Fine-Grained Cryptography
Fine-grained cryptographic primitives are ones that are secure against adversaries with an a-priori bounded polynomial amount of resources (time, space or parallel-time), where the honest algorithms use less resources than the adversaries they are designed to fool. Such primitives were previously studied in the context of time-bounded adversaries (Merkle, CACM 1978), space-bounded adversaries (Cachin and Maurer, CRYPTO 1997) and parallel-time-bounded adversaries (HĂ„stad, IPL 1987). Our goal is come up with fine-grained primitives (in the setting of parallel-time-bounded adversaries) and to show unconditional security of these constructions when possible, or base security on widely believed separation of worst-case complexity classes. We show:
1. NCÂč-cryptography: Under the assumption that Open image in new window, we construct one-way functions, pseudo-random generators (with sub-linear stretch), collision-resistant hash functions and most importantly, public-key encryption schemes, all computable in NCÂč and secure against all NCÂč circuits. Our results rely heavily on the notion of randomized encodings pioneered by Applebaum, Ishai and Kushilevitz, and crucially, make non-black-box use of randomized encodings for logspace classes.
2. ACâ°-cryptography: We construct (unconditionally secure) pseudo-random generators with arbitrary polynomial stretch, weak pseudo-random functions, secret-key encryption and perhaps most interestingly, collision-resistant hash functions, computable in ACâ° and secure against all ACâ° circuits. Previously, one-way permutations and pseudo-random generators (with linear stretch) computable in ACâ° and secure against ACâ° circuits were known from the works of HĂ„stad and Braverman.United States. Defense Advanced Research Projects Agency (Contract W911NF-15-C-0226)United States. Army Research Office (Contract W911NF-15-C-0226
Universal Quantum Hamiltonians
Quantum many-body systems exhibit an extremely diverse range of phases and
physical phenomena. Here, we prove that the entire physics of any other quantum
many-body system is replicated in certain simple, "universal" spin-lattice
models. We first characterise precisely what it means for one quantum many-body
system to replicate the entire physics of another. We then show that certain
very simple spin-lattice models are universal in this very strong sense.
Examples include the Heisenberg and XY models on a 2D square lattice (with
non-uniform coupling strengths). We go on to fully classify all two-qubit
interactions, determining which are universal and which can only simulate more
restricted classes of models. Our results put the practical field of analogue
Hamiltonian simulation on a rigorous footing and take a significant step
towards justifying why error correction may not be required for this
application of quantum information technology.Comment: 78 pages, 9 figures, 44 theorems etc. v2: Trivial fixes. v3: updated
and simplified proof of Thm. 9; 82 pages, 47 theorems etc. v3: Small fix in
proof of time-evolution lemma (this fix not in published version
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