# Collision Resistant Hashing for Paranoids: Dealing with Multiple Collisions

Ontology type: schema:Chapter

### Chapter Info

DATE

2018-03-31

AUTHORS ABSTRACT

A collision resistant hash (CRH) function is one that compresses its input, yet it is hard to find a collision, i.e. a x1≠x2\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$x_1 \ne x_2$$\end{document} s.t. h(x1)=h(x2)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$h(x_1) = h(x_2)$$\end{document}. Collision resistant hash functions are one of the more useful cryptographic primitives both in theory and in practice and two prominent applications are in signature schemes and succinct zero-knowledge arguments.In this work we consider a relaxation of the above requirement that we call Multi-CRH: a function where it is hard to find x1,x2,…,xk\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$x_1, x_2, \ldots , x_k$$\end{document} which are all distinct, yet h(x1)=h(x2)=⋯=h(xk)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$h(x_1) = h(x_2) = \cdots = h(x_k)$$\end{document}. We show that for some of the major applications of CRH functions it is possible to replace them by the weaker notion of a Multi-CRH, albeit at the price of adding interaction: we show a constant-round statistically-hiding commitment scheme with succinct interaction (committing to poly(n)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\mathsf {poly}(n)$$\end{document} bits requires exchanging O~(n)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\tilde{O}(n)$$\end{document} bits) that can be opened locally (without revealing the full string). This in turn can be used to provide succinct arguments for any NP\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\textsf {NP}}$$\end{document} statement.We formulate four possible worlds of hashing-related assumptions (in the spirit of Impagliazzo’s worlds). They are (1) Nocrypt, where no one-way functions exist, (2) Unihash, where one-way functions exist, and hence also UOWHFs and signature schemes, but no Multi-CRH functions exist, (3) Minihash, where Multi-CRH functions exist but no CRH functions exist, and (4) Hashomania, where CRH functions exist. We show that these four worlds are distinct in a black-box model: we show a separation of CRH from Multi-CRH and a separation of Multi-CRH from one-way functions. More... »

PAGES

162-194

### Book

TITLE

Advances in Cryptology – EUROCRYPT 2018

ISBN

978-3-319-78374-1
978-3-319-78375-8

### Identifiers

URI

http://scigraph.springernature.com/pub.10.1007/978-3-319-78375-8_6

DOI

http://dx.doi.org/10.1007/978-3-319-78375-8_6

DIMENSIONS

https://app.dimensions.ai/details/publication/pub.1101874799

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