Analysis of secret sharing schemes based on Nielsen transformations
-
Matvei Kotov
Abstract
We investigate security properties of two secret-sharing protocols
proposed by Fine, Moldenhauer, and Rosenberger
in Sections 4 and 5 of [B. Fine, A. Moldenhauer and G. Rosenberger,
Cryptographic protocols based on Nielsen transformations,
J. Comput. Comm. 4 2016, 63–107]
(Protocols I and II resp.).
For both protocols, we consider a one missing share challenge.
We show that Protocol I can be reduced to a system of polynomial equations
and (for most randomly generated instances)
solved by the computer algebra system Singular.
Protocol II is approached using the technique of Stallings’ graphs.
We show that knowledge of
Funding source: National Science Foundation
Award Identifier / Grant number: DMS-1318716
Funding statement: The third author has been partially supported by NSF grant DMS-1318716.
References
[1] J. Birman, An inverse function theorem for free groups, Proc. Amer. Math. Soc. 41 (1973), 634–638. 10.1090/S0002-9939-1973-0330295-8Search in Google Scholar
[2] G. Blakley, Safeguarding cryptographic keys, Proceedings of the 1979 AFIPS National Computer Conference, Texas A&M University, College Station (1979), 313–317. 10.1109/MARK.1979.8817296Search in Google Scholar
[3] C. S. Chum, B. Fine, A. I. S. Moldenhauer, G. Rosenberger and X. Zhang, On secret sharing protocols, Algebra and Computer Science, Contemp. Math. 677, American Mathematical Society, Providence (2016), 51–78. 10.1090/conm/677/13621Search in Google Scholar
[4] W. Decker, G.-M. Greuel, G. Pfister and H. Schönemann, Singular 4-0-2 – A computer algebra system for polynomial computations, preprint (2015), http://www.singular.uni-kl.de. Search in Google Scholar
[5] B. Fine, A. Moldenhauer and G. Rosenberger, Cryptographic protocols based on Nielsen transformations, J. Comput. Comm. 4 (2016), 63–107. 10.4236/jcc.2016.412004Search in Google Scholar
[6] I. Kapovich and A. G. Miasnikov, Stallings foldings and subgroups of free groups, J. Algebra 248 (2002), 608–668. 10.1006/jabr.2001.9033Search in Google Scholar
[7] M. Kotov and A. Ushakov, Implementation of attacks on secret sharing schemes based on Nielsen transformations, https://github.com/mkotov/nielsen. Search in Google Scholar
[8] A. G. Miasnikov, V. Shpilrain and A. Ushakov, Non-Commutative Cryptography and Complexity of Group-Theoretic Problems, Math. Surveys Monogr., American Mathematical Society, Providence, 2011. 10.1090/surv/177Search in Google Scholar
[9] A. Moldenhauer, Secret sharing protocols based on the closest vector theorem and Nielsen transformation, preprint (2014), www.math.uni-hamburg.de/home/moldenhauer/Moscow.pdf. Search in Google Scholar
[10] J. Moldenhauer, Cryptographic protocols based on inner product spaces and group theory with a special focus on use of Nielsen transformations, Ph.D. thesis, University of Hamburg, 2016. Search in Google Scholar
[11] D. Panagopoulos, A secret sharing scheme using groups, preprint (2010), http://arxiv.org/abs/1009.0026. Search in Google Scholar
[12] V. Roman’kov, Cryptanalysis of a combinatorial public key cryptosystem, Groups Complex. Cryptol. 9 (2017), 125–135. 10.1515/gcc-2017-0013Search in Google Scholar
[13] V. Roman’kov, Essays in Algebra and Cryptology. Solvable Groups, Omsk State University, Omsk, 2017. Search in Google Scholar
[14] A. Shamir, How to share a secret, Commun. ACM 22 (1979), no. 11, 612–613. 10.1145/359168.359176Search in Google Scholar
[15] J. Stallings, Topology of finite graphs, Invent. Math. 71 (1983), 551–565. 10.1007/BF02095993Search in Google Scholar
[16] U. Umirbaev, Primitive elements of free groups, Russian Math. Surveys 49 (1994), 184–185. 10.1070/RM1994v049n02ABEH002233Search in Google Scholar
[17] The GAP Group, GAP – Groups, Algorithms, and Programming, Version 4.7.7, 2015. Search in Google Scholar
© 2018 Walter de Gruyter GmbH, Berlin/Boston
Articles in the same Issue
- Frontmatter
- Analysis of secret sharing schemes based on Nielsen transformations
- The word problem of ℤn is a multiple context-free language
- Practical private-key fully homomorphic encryption in rings
- More secure version of a Cayley hash function
- Certifying numerical estimates of spectral gaps
- Orderable groups, elementary theory, and the Kaplansky conjecture
Articles in the same Issue
- Frontmatter
- Analysis of secret sharing schemes based on Nielsen transformations
- The word problem of ℤn is a multiple context-free language
- Practical private-key fully homomorphic encryption in rings
- More secure version of a Cayley hash function
- Certifying numerical estimates of spectral gaps
- Orderable groups, elementary theory, and the Kaplansky conjecture
