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Implicit Computational Complexity of Subrecursive Definitions and Applications to Cryptographic Proofs

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Abstract

We define a call-by-value variant of Gödel’s system \(\textsf {T} \) with references, and equip it with a linear dependent type and effect system, called \(\textsf {d}\ell \textsf {T} \), that can estimate the time complexity of programs, as a function of the size of their inputs. We prove that the type system is intentionally sound, in the sense that it over-approximates the complexity of executing the programs on a variant of the CEK abstract machine. Moreover, we define a sound and complete type inference algorithm which critically exploits the subrecursive nature of \(\textsf {d}\ell \textsf {T} \). Finally, we demonstrate the usefulness of \(\textsf {d}\ell \textsf {T} \) for analyzing the complexity of cryptographic reductions by providing an upper bound for the constructed adversary of the Goldreich–Levin theorem.

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Notes

  1. The original CEK machine comes from [16] and its name comes from the fact that its states have three components, a term, an environment and a continuation.

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Authors and Affiliations

  1. Univ Lyon, CNRS, EnsL, UCBL, LIP, 69342, Lyon Cedex 07, France

    Patrick Baillot

  2. IMDEA Software Institute, Madrid, Spain

    Gilles Barthe

  3. Università di Bologna, Bologna, Italy

    Ugo Dal Lago

  4. INRIA, Sophia Antipolis, France

    Ugo Dal Lago

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  1. Patrick Baillot
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  2. Gilles Barthe
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  3. Ugo Dal Lago
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Correspondence to Ugo Dal Lago.

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Baillot, P., Barthe, G. & Dal Lago, U. Implicit Computational Complexity of Subrecursive Definitions and Applications to Cryptographic Proofs. J Autom Reasoning 63, 813–855 (2019). https://doi.org/10.1007/s10817-019-09530-2

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