Short collision search in arbitrary SL2 homomorphic hash functions.

Ciaran Mullan; Boaz Tsaban

doi:10.48550/arXiv.1306.5646

Short collision search in arbitrary SL2 homomorphic hash functions.

Ciaran Mullan, Boaz Tsaban

Department of Mathematics

Research output: Working paper / Preprint › Preprint

Abstract

We study homomorphic hash functions into SL(2,q), the 2x2 matrices with determinant 1 over the field with q elements. Modulo a well supported number theoretic hypothesis, which holds in particular for concrete homomorphisms proposed thus far, we provide a worst case to average case reduction for these hash functions: upto a logarithmic factor, a random homomorphism is as secure as _any_ concrete homomorphism. For a family of homomorphisms containing several concrete proposals in the literature, we prove that collisions of length O(log(q)) can be found in running time O(sqrt(q)). For general homomorphisms we offer an algorithm that, heuristically and according to experiments, in running time O(sqrt(q)) finds collisions of length O(log(q)) for q even, and length O(log^2(q)/loglog(q))$ for arbitrary q. While exponetial time, our algorithms are faster in practice than all earlier generic algorithms, and produce much shorter collisions.
Related DOI:
https://doi.org/10.1007/s10623-015-0129-8

Original language	English
Publisher	Cornell University Library, arXiv.org
Volume	abs/1306.5646
DOIs	https://doi.org/10.48550/arXiv.1306.5646
State	Published - 2013

Bibliographical note

DBLP's bibliographic metadata records provided through http://dblp.org/search/publ/api are distributed under a Creative Commons CC0 1.0 Universal Public Domain Dedication. Although the bibliographic metadata records are provided consistent with CC0 1.0 Dedication, the content described by the metadata records is not. Content may be subject to copyright, rights of privacy, rights of publicity and other restrictions.

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10.48550/arXiv.1306.5646License: Unspecified

https://dblp.org/rec/journals/corr/MullanT13

1 Article

SL ₂ homomorphic hash functions: worst case to average case reduction and short collision search
Mullan, C. & Tsaban, B., 1 Oct 2016, In: Designs, Codes, and Cryptography. 81, 1, p. 83-107 25 p.
Research output: Contribution to journal › Article › peer-review
8 Scopus citations

Cite this

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title = "Short collision search in arbitrary SL2 homomorphic hash functions.",

abstract = "We study homomorphic hash functions into SL(2,q), the 2x2 matrices with determinant 1 over the field with q elements. Modulo a well supported number theoretic hypothesis, which holds in particular for concrete homomorphisms proposed thus far, we provide a worst case to average case reduction for these hash functions: upto a logarithmic factor, a random homomorphism is as secure as _any_ concrete homomorphism. For a family of homomorphisms containing several concrete proposals in the literature, we prove that collisions of length O(log(q)) can be found in running time O(sqrt(q)). For general homomorphisms we offer an algorithm that, heuristically and according to experiments, in running time O(sqrt(q)) finds collisions of length O(log(q)) for q even, and length O(log^2(q)/loglog(q))$ for arbitrary q. While exponetial time, our algorithms are faster in practice than all earlier generic algorithms, and produce much shorter collisions.Related DOI:https://doi.org/10.1007/s10623-015-0129-8",

author = "Ciaran Mullan and Boaz Tsaban",

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N2 - We study homomorphic hash functions into SL(2,q), the 2x2 matrices with determinant 1 over the field with q elements. Modulo a well supported number theoretic hypothesis, which holds in particular for concrete homomorphisms proposed thus far, we provide a worst case to average case reduction for these hash functions: upto a logarithmic factor, a random homomorphism is as secure as _any_ concrete homomorphism. For a family of homomorphisms containing several concrete proposals in the literature, we prove that collisions of length O(log(q)) can be found in running time O(sqrt(q)). For general homomorphisms we offer an algorithm that, heuristically and according to experiments, in running time O(sqrt(q)) finds collisions of length O(log(q)) for q even, and length O(log^2(q)/loglog(q))$ for arbitrary q. While exponetial time, our algorithms are faster in practice than all earlier generic algorithms, and produce much shorter collisions.Related DOI:https://doi.org/10.1007/s10623-015-0129-8

AB - We study homomorphic hash functions into SL(2,q), the 2x2 matrices with determinant 1 over the field with q elements. Modulo a well supported number theoretic hypothesis, which holds in particular for concrete homomorphisms proposed thus far, we provide a worst case to average case reduction for these hash functions: upto a logarithmic factor, a random homomorphism is as secure as _any_ concrete homomorphism. For a family of homomorphisms containing several concrete proposals in the literature, we prove that collisions of length O(log(q)) can be found in running time O(sqrt(q)). For general homomorphisms we offer an algorithm that, heuristically and according to experiments, in running time O(sqrt(q)) finds collisions of length O(log(q)) for q even, and length O(log^2(q)/loglog(q))$ for arbitrary q. While exponetial time, our algorithms are faster in practice than all earlier generic algorithms, and produce much shorter collisions.Related DOI:https://doi.org/10.1007/s10623-015-0129-8

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DO - 10.48550/arXiv.1306.5646

M3 - פרסום מוקדם

VL - abs/1306.5646

BT - Short collision search in arbitrary SL2 homomorphic hash functions.

PB - Cornell University Library, arXiv.org

ER -

Short collision search in arbitrary SL2 homomorphic hash functions.

Abstract

Bibliographical note

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Research output

SL 2 homomorphic hash functions: worst case to average case reduction and short collision search

Cite this

SL ₂ homomorphic hash functions: worst case to average case reduction and short collision search