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Groups Complexity Cryptology

Managing Editor: Shpilrain, Vladimir / Weil, Pascal

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Pseudo-free families of finite computational elementary abelian p-groups

Mikhail Anokhin
Published Online: 2017-04-19 | DOI: https://doi.org/10.1515/gcc-2017-0001


We initiate the study of (weakly) pseudo-free families of computational elementary abelian p-groups, where p is an arbitrary fixed prime. We restrict ourselves to families of computational elementary abelian p-groups Gd such that for every index d, each element of Gd is represented by a single bit string of length polynomial in the length of d. First, we prove that pseudo-freeness and weak pseudo-freeness for families of computational elementary abelian p-groups are equivalent. Second, we give some necessary and sufficient conditions for a family of computational elementary abelian p-groups to be pseudo-free (provided that at least one of two additional conditions holds). Third, we establish some necessary and sufficient conditions for the existence of pseudo-free families of computational elementary abelian p-groups.

Keywords: Family of computational groups; pseudo-free family of computational groups; weakly pseudo-free family of computational groups; collision-intractable family of functions; one-way family of functions

MSC 2010: 68Q17; 94A60


  • [1]

    M. Anokhin, Constructing a pseudo-free family of finite computational groups under the general integer factoring intractability assumption, Groups Complex. Cryptol. 5 (2013), no. 1, 53–74. Google Scholar

  • [2]

    K. Azimian, Breaking Diffie–Hellman is no easier than root finding, Electronic Colloquium on Computational Complexity ECCC TR05-124, 2005, https://eccc.weizmann.ac.il/.

  • [3]

    Y. Dodis, S. Goldwasser, Y. T. Kalai, C. Peikert and V. Vaikuntanathan, Public-key encryption schemes with auxiliary inputs, Proceedings of the 7th Theory of Cryptography Conference (TCC 2010), Lecture Notes in Comput. Sci. 5978, Springer, Berlin (2010), 361–381. Google Scholar

  • [4]

    M. Fukumitsu, Pseudo-free groups and cryptographic assumptions, Ph.D. thesis, Tohoku University, 2014. Google Scholar

  • [5]

    O. Goldreich, Foundations of Cryptography. Volume 1: Basic Tools, Cambridge University Press, Cambridge, 2001. Google Scholar

  • [6]

    S. Goldwasser and M. Bellare, Lecture notes on cryptography, lecture notes (2008), http://cseweb.ucsd.edu/~mihir/papers/gb.html.

  • [7]

    S. Hasegawa, S. Isobe, H. Shizuya and K. Tashiro, On the pseudo-freeness and the CDH assumption, Int. J. Inf. Secur. 8 (2009), no. 5, 347–355. Web of ScienceCrossrefGoogle Scholar

  • [8]

    S. R. Hohenberger, The cryptographic impact of groups with infeasible inversion, Master’s thesis, Massachusetts Institute of Technology, 2003. Google Scholar

  • [9]

    R. Impagliazzo and M. Naor, Efficient cryptographic schemes provably as secure as subset sum, J. Cryptology 9 (1996), no. 4, 199–216. Google Scholar

  • [10]

    M. Luby, Pseudorandomness and Cryptographic Applications, Princeton University Press, Princeton, 1996. Google Scholar

  • [11]

    D. Micciancio, The RSA group is pseudo-free, J. Cryptology 23 (2010), no. 2, 169–186. Web of ScienceGoogle Scholar

  • [12]

    D. Micciancio and P. Mol, Pseudorandom knapsacks and the sample complexity of LWE search-to-decision reductions, Advances in Cryptology – Crypto 2011, Lecture Notes in Comput. Sci. 6841, Springer, Berlin (2011), 465–484. Google Scholar

  • [13]

    R. L. Rivest, On the notion of pseudo-free groups, Proceedings of the 1st Theory of Cryptography Conference (TCC 2004), Lecture Notes in Comput. Sci. 2951, Springer, Berlin (2004), 505–521. Google Scholar

  • [14]

    R. L. Rivest, On the notion of pseudo-free groups, presentation (2004), https://people.csail.mit.edu/rivest/pubs/Riv04e.slides.pdf.

  • [15]

    Z. Shmuely, Composite Diffie–Hellman public-key generating systems are hard to break, Technical Report 356, Technion – Israel Institute of Technology, Haifa, 1985. Google Scholar

  • [16]

    V. Shoup, A Computational Introduction to Number Theory and Algebra, 2nd ed., Cambridge University Press, Cambridge, 2008. Google Scholar

About the article

Received: 2015-11-29

Published Online: 2017-04-19

Published in Print: 2017-05-01

Funding Source: Russian Foundation for Basic Research

Award identifier / Grant number: 13-01-00183

This research was supported in part by the Russian Foundation for Basic Research (13-01-00183).

Citation Information: Groups Complexity Cryptology, Volume 9, Issue 1, Pages 1–18, ISSN (Online) 1869-6104, ISSN (Print) 1867-1144, DOI: https://doi.org/10.1515/gcc-2017-0001.

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