New (and old) proof systems for lattice problems. Alamati, N., Peikert, C., & Stephens-Davidowitz, N. In *PKC*, 2018.

Paper abstract bibtex 19 downloads

Paper abstract bibtex 19 downloads

$ \newcommand{Øtil}{\ensuremath{\widetilde{O}}} \newcommand{\problem}[1]{\ensuremath{\mathsf{#1}}} \newcommand{\gapspp}{\problem{GapSPP}} \newcommand{øNP}{\mathsf{coNP}} \newcommand{\eps}{ε} \newcommand{\ensuremath}[1]{#1} $We continue the study of statistical zero-knowledge (SZK) proofs, both interactive and noninteractive, for computational problems on point lattices. We are particularly interested in the problem $\gapspp$ of approximating the $ε$-smoothing parameter (for some $ε < 1/2$) of an $n$-dimensional lattice. The smoothing parameter is a key quantity in the study of lattices, and $\gapspp$ has been emerging as a core problem in lattice-based cryptography, e.g., in worst-case to average-case reductions. We show that $\gapspp$ admits SZK proofs for remarkably low approximation factors, improving on prior work by up to roughly $\sqrt{n}$. Specifically: $•$ There is a noninteractive SZK proof for $O(łog(n) \sqrt{łog (1/\eps)})$-approximate $\gapspp$. Moreover, for any negligible $ε$ and a larger approximation factor $Øtil(\sqrt{n łog(1/\eps)})$, there is such a proof with an efficient prover. $•$ There is an (interactive) SZK proof with an efficient prover for $O(łog n + \sqrt{łog(1/\eps)/łog n})$-approximate $\problem{coGapSPP}$. We show this by proving that $O(łog n)$-approximate $\gapspp$ is in $øNP$. In addition, we give an (interactive) SZK proof with an efficient prover for approximating the lattice covering radius to within an $O(\sqrt{n})$ factor, improving upon the prior best factor of $ω(\sqrt{n łog n})$.

@inproceedings{APSNewOld18, title = {New (and old) proof systems for lattice problems}, abstract = {$ \newcommand{\Otil}{\ensuremath{\widetilde{O}}} \newcommand{\problem}[1]{\ensuremath{\mathsf{#1}}} \newcommand{\gapspp}{\problem{GapSPP}} \newcommand{\oNP}{\mathsf{coNP}} \newcommand{\eps}{\epsilon} \newcommand{\ensuremath}[1]{#1} $We continue the study of statistical zero-knowledge (SZK) proofs, both interactive and noninteractive, for computational problems on point lattices. We are particularly interested in the problem $\gapspp$ of approximating the $\epsilon$-smoothing parameter (for some $\epsilon < 1/2$) of an $n$-dimensional lattice. The smoothing parameter is a key quantity in the study of lattices, and $\gapspp$ has been emerging as a core problem in lattice-based cryptography, e.g., in worst-case to average-case reductions. We show that $\gapspp$ admits SZK proofs for remarkably low approximation factors, improving on prior work by up to roughly $\sqrt{n}$. Specifically: $\bullet$ There is a noninteractive SZK proof for $O(\log(n) \sqrt{\log (1/\eps)})$-approximate $\gapspp$. Moreover, for any negligible $\epsilon$ and a larger approximation factor $\Otil(\sqrt{n \log(1/\eps)})$, there is such a proof with an efficient prover. $\bullet$ There is an (interactive) SZK proof with an efficient prover for $O(\log n + \sqrt{\log(1/\eps)/\log n})$-approximate $\problem{coGapSPP}$. We show this by proving that $O(\log n)$-approximate $\gapspp$ is in $\oNP$. In addition, we give an (interactive) SZK proof with an efficient prover for approximating the lattice covering radius to within an $O(\sqrt{n})$ factor, improving upon the prior best factor of $\omega(\sqrt{n \log n})$.}, urldate = {2018-05-30}, url = {https://eprint.iacr.org/2017/1226}, booktitle = {PKC}, author = {Alamati, Navid and Peikert, Chris and {Stephens-Davidowitz}, Noah}, year = {2018} }

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{"_id":"T4ia4qTWS3heehGnj","bibbaseid":"alamati-peikert-stephensdavidowitz-newandoldproofsystemsforlatticeproblems-2018","downloads":19,"creationDate":"2018-05-30T22:56:10.548Z","title":"New (and old) proof systems for lattice problems","author_short":["Alamati, N.","Peikert, C.","Stephens-Davidowitz, N."],"year":2018,"bibtype":"inproceedings","biburl":"https://dl.dropbox.com/s/26018h26wgh5c2o/bibbase.bib","bibdata":{"bibtype":"inproceedings","type":"inproceedings","title":"New (and old) proof systems for lattice problems","abstract":"$ \\newcommand{Øtil}{\\ensuremath{\\widetilde{O}}} \\newcommand{\\problem}[1]{\\ensuremath{\\mathsf{#1}}} \\newcommand{\\gapspp}{\\problem{GapSPP}} \\newcommand{øNP}{\\mathsf{coNP}} \\newcommand{\\eps}{ε} \\newcommand{\\ensuremath}[1]{#1} $We continue the study of statistical zero-knowledge (SZK) proofs, both interactive and noninteractive, for computational problems on point lattices. We are particularly interested in the problem $\\gapspp$ of approximating the $ε$-smoothing parameter (for some $ε < 1/2$) of an $n$-dimensional lattice. The smoothing parameter is a key quantity in the study of lattices, and $\\gapspp$ has been emerging as a core problem in lattice-based cryptography, e.g., in worst-case to average-case reductions. We show that $\\gapspp$ admits SZK proofs for remarkably low approximation factors, improving on prior work by up to roughly $\\sqrt{n}$. Specifically: $•$ There is a noninteractive SZK proof for $O(łog(n) \\sqrt{łog (1/\\eps)})$-approximate $\\gapspp$. Moreover, for any negligible $ε$ and a larger approximation factor $Øtil(\\sqrt{n łog(1/\\eps)})$, there is such a proof with an efficient prover. $•$ There is an (interactive) SZK proof with an efficient prover for $O(łog n + \\sqrt{łog(1/\\eps)/łog n})$-approximate $\\problem{coGapSPP}$. We show this by proving that $O(łog n)$-approximate $\\gapspp$ is in $øNP$. In addition, we give an (interactive) SZK proof with an efficient prover for approximating the lattice covering radius to within an $O(\\sqrt{n})$ factor, improving upon the prior best factor of $ω(\\sqrt{n łog n})$.","urldate":"2018-05-30","url":"https://eprint.iacr.org/2017/1226","booktitle":"PKC","author":[{"propositions":[],"lastnames":["Alamati"],"firstnames":["Navid"],"suffixes":[]},{"propositions":[],"lastnames":["Peikert"],"firstnames":["Chris"],"suffixes":[]},{"propositions":[],"lastnames":["Stephens-Davidowitz"],"firstnames":["Noah"],"suffixes":[]}],"year":"2018","bibtex":"@inproceedings{APSNewOld18,\n title = {New (and old) proof systems for lattice problems},\n abstract = {$\n\\newcommand{\\Otil}{\\ensuremath{\\widetilde{O}}}\n\\newcommand{\\problem}[1]{\\ensuremath{\\mathsf{#1}}}\n\\newcommand{\\gapspp}{\\problem{GapSPP}}\n\\newcommand{\\oNP}{\\mathsf{coNP}}\n\\newcommand{\\eps}{\\epsilon}\n\\newcommand{\\ensuremath}[1]{#1}\n$We continue the study of statistical zero-knowledge (SZK)\nproofs, both interactive and noninteractive, for computational\nproblems on point lattices. We are particularly interested in the\nproblem $\\gapspp$ of approximating the $\\epsilon$-smoothing\nparameter (for some $\\epsilon < 1/2$) of an $n$-dimensional\nlattice. The smoothing parameter is a key quantity in the study of\nlattices, and $\\gapspp$ has been emerging as a core problem in\nlattice-based cryptography, e.g., in worst-case to average-case\nreductions.\n\nWe show that $\\gapspp$ admits SZK proofs for remarkably low\napproximation factors, improving on prior work by up to\nroughly $\\sqrt{n}$. Specifically:\n\n $\\bullet$ There is a noninteractive SZK proof for\n$O(\\log(n) \\sqrt{\\log (1/\\eps)})$-approximate $\\gapspp$. Moreover,\nfor any negligible $\\epsilon$ and a larger approximation factor\n$\\Otil(\\sqrt{n \\log(1/\\eps)})$, there is such a proof with an\nefficient prover.\n $\\bullet$ There is an (interactive) SZK proof with an efficient prover for\n$O(\\log n + \\sqrt{\\log(1/\\eps)/\\log n})$-approximate\n$\\problem{coGapSPP}$. We show this by proving that\n$O(\\log n)$-approximate $\\gapspp$ is in $\\oNP$.\n\nIn addition, we give an (interactive) SZK proof with an efficient\nprover for approximating the lattice covering radius to within\nan $O(\\sqrt{n})$ factor, improving upon the prior best factor of\n$\\omega(\\sqrt{n \\log n})$.},\n urldate = {2018-05-30},\n url = {https://eprint.iacr.org/2017/1226},\n booktitle = {PKC},\n author = {Alamati, Navid and Peikert, Chris and {Stephens-Davidowitz}, Noah},\n year = {2018}\n}\n\n","author_short":["Alamati, N.","Peikert, C.","Stephens-Davidowitz, N."],"key":"APSNewOld18","id":"APSNewOld18","bibbaseid":"alamati-peikert-stephensdavidowitz-newandoldproofsystemsforlatticeproblems-2018","role":"author","urls":{"Paper":"https://eprint.iacr.org/2017/1226"},"metadata":{"authorlinks":{"stephens-davidowitz, 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