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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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1869-6104
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Volume 8, Issue 2 (Nov 2016)

# 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

## Abstract

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 ${𝔽}_{p}$. 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 ${𝔽}_{p}$, 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 ${𝔽}_{p}$.

MSC 2010: 20M05; 94A60; 68P30

## References

• [1]

Bromberg L., Shpilrain V. and Vdovina A., Navigating in the Cayley graph of ${\mathrm{SL}}_{2}\left({𝔽}_{p}\right)$ and applications to hashing, preprint 2014 http://arxiv.org/abs/1409.4478; to appear in Semigroup Forum.

• [2]

Caldwell C., The primes pages, https://primes.utm.edu.

• [3]

Cassaigne J., Harju T. and Karhumäki J., On the undecidability of freeness of matrix semigroups, Internat. J. Algebra Comput. 9 (1999), 295–305.

• [4]

Contini S., Lenstra A. K. and Steinfeld R., VSH, an efficient and provable collision-resistant hash function, Advances in Cryptology – EUROCRYPT 2006, Lecture Notes in Comput. Sci. 4004, Springer, Berlin (2006), 165–182.

• [5]

Dai W., Crypto++ 5.6.0 benchmarks, http://www.cryptopp.com/benchmarks.html.

• [6]

Grassl M., Ilić I., Magliveras S. and Steinwandt R., Cryptanalysis of the Tillich–Zémor hash function, J. Cryptology 24 (2011), 148–156.

• [7]

Menezes A., van Oorschot P. and Vanstone S., Handbook of Applied Cryptography, CRC Press, Boca Raton, 1997.

• [8]

Mullan C. and Tsaban B., ${\mathrm{SL}}_{2}$ homomorphic hash functions: Worst case to average case reduction and short collision search, Des. Codes Cryptogr. 81 (2016), 83–107.

• [9]

Petit C., On graph-based cryptographic hash functions, PhD thesis, Université Catholique de Louvain, 2009.

• [10]

Petit C. and Quisquater J., Preimages for the Tillich–Zémor hash function, Selected Areas in Cryptography (SAC ’10), Lecture Notes in Comput. Sci. 6544, Springer, Berlin (2010), 282–301.

• [11]

Petit C. and Quisquater J.-J., Rubik’s for cryptographers, Notices Amer. Math. Soc. 60 (2013), 733–739.

• [12]

Rukhin A., Soto J., Nechvatal J., Smid M., Barker E., Leigh S., Levenson M., Vangel M., Banks D., Heckert A., Dray J. and S. Vo , A statistical test suite for random and pseudorandom number generators for cryptographic applications, SP 800-22 Rev. 1a, National Institute of Standards & Technology Gaithersburg, 2010.

• [13]

Tillich J.-P. and Zémor G., Group-theoretic hash functions, Proceedings of the First French–Israeli Workshop on Algebraic Coding, Lecture Notes in Comput. Sci. 781, Springer, Berlin (1993), 90–110.

• [14]

Tillich J.-P. and Zémor G., Hashing with ${\mathrm{SL}}_{2}$, Advances in cryptology – CRYPTO ’94, Lecture Notes in Comput. Sci. 839, Springer, Berlin (1994), 40–49.

• [15]

National Institute of Standards and Technology – NIST , NIST Statistical Test Suite, http://csrc.nist.gov/groups/ST/toolkit/rng/documentation\_software.html.

## 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, Export Citation

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