Polak, M.

Romanczuk-Polubiec, U.

Ustimenko, V. A.

Wroblewska, A.

BSU Digital Library

ON THE NEW APPLICATIONS OF ALGEBRAIC GRAPH THEORY TO
MULTIVARIATE CRYPTOGRAPHY
V.A. Ustimenko1, U. Roman´czuk-Polubiec2, A. Wro´blewska1, M. Polak1
1 The University of Maria Curie Sklodowska, Institute of Mathematics,
pl. Marii Curie-Skiodowskiej 1, 20-031 Lublin, Poland,
vasyl@hektor.umcs.lublin.pl, awroblewska@hektor.umcs.lublin.pl, monika.katarzyna.plak@gmail.com
2 The Independent Polish Researcher, urszula_romanczuk@yahoo.pl
Presented research of authors is partially supported through the project ”Scientiﬁc fellowships for PhD students
working in research teams”, which is realized by Self-Ggovernment of the of Lubelskie Voivodeship and Regional
Operational Programme Department of the Marshal’s Oﬃce of Lubelskie Voivodeship in Lublin, Poland, within
the framework of Sub-measure 8.2.2 Regional Innovation Strategies, Measure 8.2 Transfer of knowledge, Priority
VIII Regional human resources for the economy Human Capital Operational Programme co-ﬁnanced by European
Social Fund and state budget. The authors team together with the ﬁrm LabSQL.pl collaborate on the above project.
The RSA is one of the most popular cryptosystems. It is based on number factorisation problem
and Euler Theorem. Peter Shor discovered that factorisation problem can be eﬀectively solved with
the usage of theoretical quantum computer. It means that RSA could not be a security tool in the
future postquantum era. One of the research directions which can lead to a postquantum secure
public key is the Multivariate Cryptography which uses a polynomial maps of aﬃne space Kn
deﬁned over a ﬁnite commutative ring into itself as encryption tools (see [1]). This is a young
promising research area with the current lack of known cryptosystems with the proven resistence
against attacks with the use of Turing machines. Other important direction of Postquantum
Cryptography is the studies of Elliptic Curves cryptosystems.
Applications of Algebraic Graph Theory to Multivariate Cryptography were observed in our
talks at Erdos Centennial (2013, Budapest) and Central European Conference on Cryptology 2014
(Alfred Renyi Institute, Budapest) [2, 3]. This talk was devoted to algorithms based on bijective
maps of aﬃne spaces into itself. Applications of algebraic graphs to cryptography started from
symmetric algorithms based on explicit constructions of extremal graph theory and their directed
analogue (see survey [4, 5]). The main idea is to convert an algebraic graph in ﬁnite automaton
and use the preudorandom walks on the graph as encryption tools. This approach can be also
used for the key exchange protocols. Nowadays the idea of ”symbolic walks” on algebraic graphs
when the walk on the graph depends on parameters given as special multivariate polynomials in
variables depending from plainspace vector brings several public key cryptosystems. Other source
of graphs suitable for cryptography is connected with ﬁnite geometries and their ﬂag system (see
[4], [11] and further references).
Our presentation at DIMA 2015 includes new cryptoalgorithms in terms of Algebraic Combinatorics
which use non bijective transformations of Kn.
Multivariate cryptography started from studies of potential for the special quadratic encryption
multivariate bijective map of Kn, where K is an extention os ﬁnite ﬁeld Fq of characteristic 2.
One of the ﬁrst such cryptosystems were proposed by Imai and Matsumoto, cryptanalysis for this
system was invented by J. Patarin. The survey on various modiﬁcations of this algorithm and
corresponding cryptanalysis the reader can ﬁnd in [1]. Bijective multivariate sparse encryption
maps of rather high degree based on walks in algebraic graphs were proposed in [6].
One of the ﬁrst usage of non bijective map of multivariate cryptography was in oil and vinegar
crytosystem proposed in [7] and analysed in [8]. Nowadays this general idea is strongly supported
by publication [9] devoted to security analysis of direct attacks on modiﬁed unbalanced oil and
vinegar systems. It looks like such systems and rainbow signatures schemes may lead to promising
Public Key Schemes of Multivariate Encryption deﬁned over ﬁnite ﬁelds. Non bijective multivariate
sparse encryption maps of degree 3 and  3 based on walks on algebraic graphs D(n;K) deﬁned
over general commutative ring and their homomorphic images were proposed in [10].
The new cryptosystems with non bijective multivariate encryption maps on the aﬃne space
Zm
n into itself will be presented. It uses the plainspace Zm
n, where n = k(k   1)=2, k  2 can
be arbitrary natural number. The private key space is formed by sequence of general multivariate
polynomials from Zm[x1; x2; : : : ; xk 1] and sequence of parameters li, i = 1; 2; : : : ; k  1 which are
mutually prime with (m). The properties of the encryption map depends heavily on the prime
factorisation of m. This non bijective encryption map is the deformation of special computation
generated by Schubert automaton of ”k   1 dimensional projective geometry” over Zm. This
method does not use the partition of variables into groups, non bijective nature of the map caused
by zero devisors of composite integer m. In fact the idea of multiple ”hidden RSA” is used.
This algorithm is a modiﬁcation of public key cryptosystem based on the computation of Tits
automaton in the case of ﬁnite projective geometry [11], which were presented at the conference
ALCOMA 2015.
The talk is dedicated to the memory of D. A. Suprunenko whose research is an inspirational
example of multifaceted work in Pure and Applied Mathematics in areas of Algebra and Discrete
Mathematics.
References
1. Ding J., Gower J. E., Schmidt D. S. Multivariate Public Key Cryptosystems. Springer. Advances in
Information Security. V. 25. 2006.
2. Polak M., Roman´czuk U., Ustimenko V., Wro´blewska A. On the applications of Extremal Graph
Theory to Coding Theory and Cryptography // Erdo˝s Centennial, Proceedings of Erdo˝s Centennial (EP
100). Electronic Notes in Discrete Mathematics. 2013. V. 43. P. 329–342.
3. Ustimenko V. A. Explicit constructions of extremal graphs and new multivariate cryptosystems
// Studia Scientiarum Mathematicarum Hungarica, Special issue “Proceedings of The Central European
Conference, 2014, Budapest” (to appear in 2015).
4. Ustimenko V. A. Graphs with Special Arcs and Cryptograph // Acta Applicandae Mathematicae.
2002. V. 71. No 2. P. 117–153.
5. Ustimenko V. On the extremal graph theory for directed graphs and its cryptographical applications
// In: T. Shaska, W.C. Huﬀman, D. Joener and V. Ustimenko. Advances in Coding Theory and Cryptography.
Series on Coding and Cryptology. 2007. V. 3. P. 181–200.
6. Ustimenko V. OnMultivariate Cryptosystems Based on Computable Maps with Invertible Decompositions
// Annales of UMCS. Informatica (special issue “Proceedings of International Conference Cryptography
and Security Systems”). 2014. V. 14. P. 7–18.
7. Patarin J. The Oil i Vinegar digital signatures // Dagstuhl Workshop on Cryptography. 1997.
8. Kipnis A., Shamir A. Cryptanalisis of the Oil and Vinegar Signature Scheme // Advances in
Cryptology, Crypto 96. Lecture Notes in Computer Science. 1996. V. 1462. P. 257–266.
9. Bulygin S., Petzoldt A., Buchmann J. Towards provable security of the unbalanced oil and vinegar
signature scheme under direct attacks // In “Progress in Cryptology, INDOCRYPT”, eds.: Guang Gong,
KishanChand Gupta editors, Lecture notes in Computer Science. 2010. V. 6498. P. 17–32.
10. Roman´czuk-Polubiec U., Ustimenko V. On two windows multivariate cryptosystem depending on
random parameters // Algebra and Discrete Mathematics. 2015. V. 19. No. 1. P. 101–129.
11. Ustimenko V. A. On the ﬂag geometry of simple group of Lie type and Multivariate Cryptography
// Algebra and Discrete Mathematics. 2015. V. 19. No 1. P. 130–144.


On the new applications of algebraic graph theory to multivariate cryptography

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On the new applications of algebraic graph theory to multivariate cryptography

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