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Groups Complexity Cryptology

Managing Editor: Shpilrain, Vladimir / Weil, Pascal

Editorial Board Member: Blackburn, Simon R. / Conder, Marston / Dehornoy, Patrick / Eick, Bettina / Fine, Benjamin / Gilman, Robert / Grigoriev, Dima / Ko, Ki Hyoung / Kreuzer, Martin / Mikhalev, Alexander V. / Myasnikov, Alexei / Roman'kov, Vitalii / Rosenberger, Gerhard / Sapir, Mark / Schäge, Sven / Thomas, Rick / Tsaban, Boaz / Capell, Enric Ventura

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Compositions of linear functions and applications to hashing

Vladimir Shpilrain
  • Department of Mathematics, The City College of New York, New York, NY 10031, United States of America
  • Email:
/ Bianca Sosnovski
  • Corresponding author
  • Queensborough Community College and Graduate Center, City University of New York, United States of America
  • Email:
Published Online: 2016-10-14 | DOI: https://doi.org/10.1515/gcc-2016-0016


Cayley hash functions are based on a simple idea of using a pair of (semi)group elements, A and B, to hash the 0 and 1 bit, respectively, and then to hash an arbitrary bit string in the natural way, by using multiplication of elements in the (semi)group. In this paper, we focus on hashing with linear functions of one variable over 𝔽p. The corresponding hash functions are very efficient. In particular, we show that hashing a bit string of length n with our method requires, in general, at most 2n multiplications in 𝔽p, but with particular pairs of linear functions that we suggest, one does not need to perform any multiplications at all. We also give explicit lower bounds on the length of collisions for hash functions corresponding to these particular pairs of linear functions over 𝔽p.

Keywords: Cayley hash functions; semigroups; linear functions

MSC 2010: 20M05; 94A60; 68P30


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About the article

Received: 2016-05-08

Published Online: 2016-10-14

Published in Print: 2016-11-01

Funding Source: National Science Foundation

Award identifier / Grant number: CNS-1117675

Funding Source: Office of Naval Research

Award identifier / Grant number: N000141210758

Research of the first author was partially supported by the NSF grant CNS-1117675 and by the ONR (Office of Naval Research) grant N000141210758.

Citation Information: Groups Complexity Cryptology, ISSN (Online) 1869-6104, ISSN (Print) 1867-1144, DOI: https://doi.org/10.1515/gcc-2016-0016. Export Citation

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