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Liao Tai-ning - One of the best experts on this subject based on the ideXlab platform.
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On the Compressed-Oracle Technique, and Post-Quantum Security of Proofs of Sequential Work
2021Co-Authors: Chung Kai-min, Fehr Serge, Huang Yu-hsuan, Liao Tai-ningAbstract:We revisit the so-called compressed oracle technique, introduced by Zhandry for analyzing quantum algorithms in the quantum random oracle model (QROM). To start off with, we offer a concise exposition of the technique, which easily extends to the parallel-query QROM, where in each query-round the considered algorithm may make several queries to the QROM in parallel. This variant of the QROM allows for a more fine-grained query-complexity analysis. Our main technical contribution is a framework that simplifies the use of (the parallel-query generalization of) the compressed oracle technique for proving query complexity results. With our framework in place, whenever applicable, it is possible to prove quantum query complexity lower bounds by means of purely Classical Reasoning. More than that, for typical examples the crucial Classical observations that give rise to the Classical bounds are sufficient to conclude the corresponding quantum bounds. We demonstrate this on a few examples, recovering known results (like the optimality of parallel Grover), but also obtaining new results (like the optimality of parallel BHT collision search). Our main target is the hardness of finding a $q$-chain with fewer than $q$ parallel queries, i.e., a sequence $x_0, x_1,\ldots, x_q$ with $x_i = H(x_{i-1})$ for all $1 \leq i \leq q$. The above problem of finding a hash chain is of fundamental importance in the context of proofs of sequential work. Indeed, as a concrete cryptographic application of our techniques, we prove that the "Simple Proofs of Sequential Work" proposed by Cohen and Pietrzak remains secure against quantum attacks. Such an analysis is not simply a matter of plugging in our new bound; the entire protocol needs to be analyzed in the light of a quantum attack. Thanks to our framework, this can now be done with purely Classical Reasoning
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On the compressed-oracle technique, and post-quantum security of proofs of sequential work
'Springer Science and Business Media LLC', 2021Co-Authors: Chung Kai-min, Fehr Serge, Huang Yu-hsuan, Liao Tai-ningAbstract:We revisit the so-called compressed oracle technique, introduced by Zhandry for analyzing quantum algorithms in the quantum random oracle model (QROM). To start off with, we offer a concise exposition of the technique, which easily extends to the parallel-query QROM, where in each query-round the considered algorithm may make several queries to the QROM in parallel. This variant of the QROM allows for a more fine-grained query-complexity analysis. Our main technical contribution is a framework that simplifies the use of (the parallel-query generalization of) the compressed oracle technique for proving query complexity results. With our framework in place, whenever applicable, it is possible to prove quantum query complexity lower bounds by means of purely Classical Reasoning. More than that, for typical examples the crucial Classical observations that give rise to the Classical bounds are sufficient to conclude the corresponding quantum bounds. We demonstrate this on a few examples, recovering known results but also obtaining new results. Our main target is the hardness of finding a q-chain with fewer than q parallel queries, i.e., a sequence x0, x1, …, xq with xi= H(xi-1) for all 1 ≤ i≤ q. The above problem of finding a hash chain is of fundamental importance in the context of proofs of sequential work. Indeed, as a concrete cryptographic application of our techniques, we prove quantum security of the “Simple Proofs of Sequential Work” by Cohen and Pietrzak
Marco Schaerf - One of the best experts on this subject based on the ideXlab platform.
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is intractability of nonmonotonic Reasoning a real drawback
Artificial Intelligence, 1996Co-Authors: Marco Cadoli, Francesco M Donini, Marco SchaerfAbstract:Abstract Several studies about computational complexity of nonmonotonic Reasoning (NMR) showed that nonmonotonic inference is significantly harder than Classical, monotonic inference. This contrasts with the general idea that NMR can be used to make knowledge representation and Reasoning simpler, not harder. In this paper we show that, to some extent, NMR fulfills the representation goal. In particular, we prove that nonmonotonic formalisms such as circumscription and default logic allow for a much more compact and natural representation of propositional knowledge than propositional calculus. Proofs are based on a suitable definition of a compilable inference problem, and on non-uniform complexity classes. Some results about intractability of circumscription and default logic can therefore be interpreted as the price one has to pay for having such an extra-compact representation. On the other hand, intractability of inference and compactness of representation are not equivalent notions: we exhibit intractable nonmonotonic formalisms whose nonmonotonic assumptions are representable by few propositional formulae. Finally, sometimes NMR really makes Reasoning simpler. We present prototypical scenarios where closed-world Reasoning and well-founded semantics account for a faster, complete and unsound approximation of Classical Reasoning.
Herbelin Hugo - One of the best experts on this subject based on the ideXlab platform.
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On the logical structure of choice and bar induction principles
HAL CCSD, 2021Co-Authors: Brede Nuria, Herbelin HugoAbstract:International audienceWe develop an approach to choice principles and their contrapositive bar-induction principles as extensionality schemes connecting an "intensional" or "effective" view of respectively ill-and well-foundedness properties to an "extensional" or "ideal" view of these properties. After classifying and analysing the relations between different intensional definitions of ill-foundedness and well-foundedness, we introduce, for a domain A, a codomain B and a "filter" T on finite approximations of functions from A to B, a generalised form GDC(A,B,T) of the axiom of dependent choice and dually a generalised bar induction principle GBI(A,B,T) such that:- GDC(A,B,T) intuitionistically captures the strength of• the general axiom of choice expressed as ∀a ∃b R(a, b) ⇒ ∃α ∀a R(a, α(a))) when T is a filter that derives point-wise from a relation R on A × B without introducing further constraints,• the Boolean Prime Filter Theorem / Ultrafilter Theorem if B is the two-element set Bool (for a constructive definition of prime filter),• the axiom of dependent choice if A = ℕ,• Weak König’s Lemma if A = ℕ and B = Bool (up to weak Classical Reasoning)- GBI(A,B,T) intuitionistically captures the strength of• Gödel’s completeness theorem in the form validity implies provability for entailment relations if B = Bool,• bar induction when A = ℕ,• the Weak Fan Theorem when A = ℕ and B = Bool.Contrastingly, even though GDC(A,B,T) and GBI(A,B,T) smoothly capture several variants of choice and bar induction, some instances are inconsistent, e.g. when A is Bool^ℕ and B is ℕ
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On the logical structure of choice and bar induction principles
2021Co-Authors: Brede Nuria, Herbelin HugoAbstract:We develop an approach to choice principles and their contrapositive bar-induction principles as extensionality schemes connecting an "intensional" or "effective" view of respectively ill-and well-foundedness properties to an "extensional" or "ideal" view of these properties. After classifying and analysing the relations between different intensional definitions of ill-foundedness and well-foundedness, we introduce, for a domain $A$, a codomain $B$ and a "filter" $T$ on finite approximations of functions from $A$ to $B$, a generalised form GDC$_{A,B,T}$ of the axiom of dependent choice and dually a generalised bar induction principle GBI$_{A,B,T}$ such that: GDC$_{A,B,T}$ intuitionistically captures the strength of $\bullet$ the general axiom of choice expressed as $\forall a\exists b R(a, b) \Rightarrow\exists\alpha\forall \alpha R(\alpha,\alpha(a))$ when $T$ is a filter that derives point-wise from a relation $R$ on $A \times B$ without introducing further constraints, $\bullet$ the Boolean Prime Filter Theorem / Ultrafilter Theorem if $B$ is the two-element set $\mathbb{B}$ (for a constructive definition of prime filter), $\bullet$ the axiom of dependent choice if $A = \mathbb{N}$, $\bullet$ Weak K{\"o}nig's Lemma if $A = \mathbb{N}$ and $B = \mathbb{B}$ (up to weak Classical Reasoning) GBI$_{A,B,T}$ intuitionistically captures the strength of $\bullet$ G{\"o}del's completeness theorem in the form validity implies provability for entailment relations if $B = \mathbb{B}$, $\bullet$ bar induction when $A = \mathbb{N}$, $\bullet$ the Weak Fan Theorem when $A = \mathbb{N}$ and $B = \mathbb{B}$. Contrastingly, even though GDC$_{A,B,T}$ and GBI$_{A,B,T}$ smoothly capture several variants of choice and bar induction, some instances are inconsistent, e.g. when $A$ is $\mathbb{B}^\mathbb{N}$ and $B$ is $\mathbb{N}$.Comment: LICS 2021 - 36th Annual Symposium on Logic in Computer Science, Jun 2021, Rome / Virtual, Ital
Chung Kai-min - One of the best experts on this subject based on the ideXlab platform.
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On the Compressed-Oracle Technique, and Post-Quantum Security of Proofs of Sequential Work
2021Co-Authors: Chung Kai-min, Fehr Serge, Huang Yu-hsuan, Liao Tai-ningAbstract:We revisit the so-called compressed oracle technique, introduced by Zhandry for analyzing quantum algorithms in the quantum random oracle model (QROM). To start off with, we offer a concise exposition of the technique, which easily extends to the parallel-query QROM, where in each query-round the considered algorithm may make several queries to the QROM in parallel. This variant of the QROM allows for a more fine-grained query-complexity analysis. Our main technical contribution is a framework that simplifies the use of (the parallel-query generalization of) the compressed oracle technique for proving query complexity results. With our framework in place, whenever applicable, it is possible to prove quantum query complexity lower bounds by means of purely Classical Reasoning. More than that, for typical examples the crucial Classical observations that give rise to the Classical bounds are sufficient to conclude the corresponding quantum bounds. We demonstrate this on a few examples, recovering known results (like the optimality of parallel Grover), but also obtaining new results (like the optimality of parallel BHT collision search). Our main target is the hardness of finding a $q$-chain with fewer than $q$ parallel queries, i.e., a sequence $x_0, x_1,\ldots, x_q$ with $x_i = H(x_{i-1})$ for all $1 \leq i \leq q$. The above problem of finding a hash chain is of fundamental importance in the context of proofs of sequential work. Indeed, as a concrete cryptographic application of our techniques, we prove that the "Simple Proofs of Sequential Work" proposed by Cohen and Pietrzak remains secure against quantum attacks. Such an analysis is not simply a matter of plugging in our new bound; the entire protocol needs to be analyzed in the light of a quantum attack. Thanks to our framework, this can now be done with purely Classical Reasoning
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On the compressed-oracle technique, and post-quantum security of proofs of sequential work
'Springer Science and Business Media LLC', 2021Co-Authors: Chung Kai-min, Fehr Serge, Huang Yu-hsuan, Liao Tai-ningAbstract:We revisit the so-called compressed oracle technique, introduced by Zhandry for analyzing quantum algorithms in the quantum random oracle model (QROM). To start off with, we offer a concise exposition of the technique, which easily extends to the parallel-query QROM, where in each query-round the considered algorithm may make several queries to the QROM in parallel. This variant of the QROM allows for a more fine-grained query-complexity analysis. Our main technical contribution is a framework that simplifies the use of (the parallel-query generalization of) the compressed oracle technique for proving query complexity results. With our framework in place, whenever applicable, it is possible to prove quantum query complexity lower bounds by means of purely Classical Reasoning. More than that, for typical examples the crucial Classical observations that give rise to the Classical bounds are sufficient to conclude the corresponding quantum bounds. We demonstrate this on a few examples, recovering known results but also obtaining new results. Our main target is the hardness of finding a q-chain with fewer than q parallel queries, i.e., a sequence x0, x1, …, xq with xi= H(xi-1) for all 1 ≤ i≤ q. The above problem of finding a hash chain is of fundamental importance in the context of proofs of sequential work. Indeed, as a concrete cryptographic application of our techniques, we prove quantum security of the “Simple Proofs of Sequential Work” by Cohen and Pietrzak
Russell Oconnor - One of the best experts on this subject based on the ideXlab platform.
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Classical mathematics for a constructive world
Mathematical Structures in Computer Science, 2011Co-Authors: Russell OconnorAbstract:Interactive theorem provers based on dependent type theory have the flexibility to support both constructive and Classical Reasoning. Constructive Reasoning is supported natively by dependent type theory, and Classical Reasoning is typically supported by adding additional non-constructive axioms. However, there is another perspective that views constructive logic as an extension of Classical logic. This paper will illustrate how Classical Reasoning can be supported in a practical manner inside dependent type theory without additional axioms. We will show several examples of how Classical results can be applied to constructive mathematics. Finally, we will show how to extend this perspective from logic to mathematics by representing Classical function spaces using a weak value monad.
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Classical mathematics for a constructive world
arXiv: Logic in Computer Science, 2010Co-Authors: Russell OconnorAbstract:Interactive theorem provers based on dependent type theory have the flexibility to support both constructive and Classical Reasoning. Constructive Reasoning is supported natively by dependent type theory and Classical Reasoning is typically supported by adding additional non-constructive axioms. However, there is another perspective that views constructive logic as an extension of Classical logic. This paper will illustrate how Classical Reasoning can be supported in a practical manner inside dependent type theory without additional axioms. We will see several examples of how Classical results can be applied to constructive mathematics. Finally, we will see how to extend this perspective from logic to mathematics by representing Classical function spaces using a weak value monad.