Abstract
The use of quasigroups in cryptography is increasingly popular. One method to find quasigroups suitable for cryptographic purposes is to use identity sieves, i.e. to find appropriate identities and check candidate quasigroups against them. We propose the use of functional equation approach to this problem. Namely, every identity can be considered as a functional equation and solutions to these equations as models of given identities. The identity i.e. functional equation can be transformed into related generalized functional equation which is suitable for algebraic treatment. A new method of solution of quadratic and parastrophically uncancelablle equations is given, using trees and dichotomies (a special equivalence relations). General solution is given by closed formulas. The quasigroups obtained can be further filtered using much simpler conditions.
This is a preview of subscription content, log in via an institution to check access.
Access this chapter
Tax calculation will be finalised at checkout
Purchases are for personal use only
Preview
Unable to display preview. Download preview PDF.
References
Belousov, V.D.: Foundations of the Theory of Quasigroups and Loops, Nauka, Moscow (1967) (Russian) (MR 36-1569, Zbl 163:01801)
Chein, O., Pflugfelder, H.O., Smith, J.D.H.: Quasigroups and Loops: Theory and Applications. Sigma Series in Pure Math., vol. 9. Heldermann Verlag, Berlin (1990) (MR 93g:21033, Zbl 0719.20036)
Kechris, A.S.: Classical Descriptive Set Theory. Graduate Texts in Mathematics, vol. 156. Springer, New York (1995)
Krapež, A.: Strictly Quadratic Functional Equations on Quasigroups I. Publ. Inst. Math. N. S. 29(43), 125–138 (1981) (Belgrade) (Zbl 0489.39006)
Krapež, A.: An Application of Quasigroups in Cryptology. Math. Maced. 8, 47–52 (2010)
Krapež, A., Živković, D.: Parastrophically Equivalent Quasigroup Equations. Publ. Inst. Math. N. S 87(101), 39–58 (2010), doi:10.2298/PIM1001039K (Belgrade)
Markovski, S., Gligoroski, D., Bakeva, V.: Quasigrouop String Processing: Part 1. Contributions. Sec. Math. Tech. Sci., MANU XX(1-2), 13–28 (1999)
Markovski, S., Dimitrova, V., Samardziska, S.: Identities Sieves for Quasigroups. Quasigroups and Related Systems 18(2), 149–164 (2010)
Petrescu, A.: n–quasigroups Cryptographic Primitives: Stream Ciphers. Studia Univ. Babeş–Bolyai, Informatica LV (2), 27–34 (2010)
Pflugfelder, H.O.: Quasigroups and Loops: Introduction. Sigma Series in Pure Math., vol. 8. Heldermann Verlag, Berlin (1990) (MR 93g:20132, Zbl 719.20036)
Schauffler, R.: Die Associvität im Ganzen besonders bei Quasigruppen. Math. Zeitschr. 67(5), 428–435 (1957)
Shcherbacov, V.A.: Quasigroup Based Crypto–algorithms (2012) arXiv:1201.3572
Sokhats’kyi, F.M.: On the Classification of Functional Equations on Quasigroups. Ukrain. Mat. Zh. 56(9), 1259–1266 (2004) (Ukrainian)
Sokhats’kyi, F.M.: On the Classification of Functional Equations on Quasigroups. Ukrainian Math. J. 56(9), 1499–1508 (2004); English translation of [13]
Author information
Authors and Affiliations
Corresponding author
Editor information
Editors and Affiliations
Rights and permissions
Copyright information
© 2013 Springer-Verlag Berlin Heidelberg
About this paper
Cite this paper
Krapež, A. (2013). Cryptographically Suitable Quasigroups via Functional Equations. In: Markovski, S., Gusev, M. (eds) ICT Innovations 2012. ICT Innovations 2012. Advances in Intelligent Systems and Computing, vol 207. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-37169-1_26
Download citation
DOI: https://doi.org/10.1007/978-3-642-37169-1_26
Publisher Name: Springer, Berlin, Heidelberg
Print ISBN: 978-3-642-37168-4
Online ISBN: 978-3-642-37169-1
eBook Packages: EngineeringEngineering (R0)