A computational algebraic geometry approach to analyze pseudo-random sequences based on Latin squares

  • Raúl M. FalcónEmail author
  • Víctor Álvarez
  • Félix Gudiel


Latin squares are used as scramblers on symmetric-key algorithms that generate pseudo-random sequences of the same length. The robustness and effectiveness of these algorithms are respectively based on the extremely large key space and the appropriate choice of the Latin square under consideration. It is also known the importance that isomorphism classes of Latin squares have to design an effective algorithm. In order to delve into this last aspect, we improve in this paper the efficiency of the known methods on computational algebraic geometry to enumerate and classify partial Latin squares. Particularly, we introduce the notion of affine algebraic set of a partial Latin square L = (lij) of order n over a field \(\mathbb {K}\) as the set of zeros of the binomial ideal \(\langle x_{i}x_{j}-x_{l_{ij}}\colon (i,j) \text { is a non-empty cell in} L \rangle \subseteq \mathbb {K}[x_{1},\ldots ,x_{n}]\). Since isomorphic partial Latin squares give rise to isomorphic affine algebraic sets, every isomorphism invariant of the latter constitutes an isomorphism invariant of the former. In particular, we deal computationally with the problem of deciding whether two given partial Latin squares have either the same or isomorphic affine algebraic sets. To this end, we introduce a new pair of equivalence relations among partial Latin squares: being partial transpose and being partial isotopic.


Symmetric-key algorithm Image pattern Partial Latin square Affine algebraic set Isomorphism 

Mathematics Subject Classification (2010)

05B15 20N05 14G50 


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.



  1. 1.
    Adams, W., Loustaunau, P.: An Introduction to Gröbner Bases Graduate Studies in Mathematics, vol. 3. American Mathematical Society, Providence (1994)zbMATHGoogle Scholar
  2. 2.
    Bayer, D.: The Division Algorithm and the Hilbert Scheme. PhD Thesis, Harvard University (1982)Google Scholar
  3. 3.
    Buchberger, B.: An algorithm for finding the basis elements of the residue class ring of a zero dimensional polynomial ideal. J. Symbolic Comput. 41, 475–511 (2006)MathSciNetCrossRefGoogle Scholar
  4. 4.
    Chum, C.S., Zhang, X.: The Latin squares and the secret sharing schemes. Groups Complex Cryptol. 2, 175–202 (2010)MathSciNetzbMATHGoogle Scholar
  5. 5.
    Cooper, J., Donovan, D., Seberry, J.: Secret sharing schemes arising from Latin squares. Bull. Inst. Combin. Appl. 12, 33–43 (1994)MathSciNetzbMATHGoogle Scholar
  6. 6.
    Danan, E., Falcon, R.M., Kotlar, D., Marbach, T.G., Stones, R.J.: Refining invariants for computing autotopism groups of partial Latin rectangles. Submitted (2018)Google Scholar
  7. 7.
    Decker, W., Greuel, G.M., Pfister, G., Schönemann, H.: Singular 4-1-1. A computer algebra system for polynomial computations. (2018). Accessed 30 September 2018
  8. 8.
    Dénes, J., Keedwell, A.D.: Latin Squares and Their Applications. Academic Press, New York-London (1974)zbMATHGoogle Scholar
  9. 9.
    Dimitrova, V., Markovski, J.: On Quasigroup Pseudo Random Sequence Generator. In: Manolopoulos, Y., Spirakis, P. (eds.) Proceedings of the First Balkan Conference in Informatics, pp 393–401, Thessaloniki (2004)Google Scholar
  10. 10.
    Dimitrova, V., Markovski, S.: Classification of quasigroups by image patterns. In: Proceedings of the Fifth International Conference for Informatics and Information Technology, pp 152–160. Bitola, Macedonia (2007)Google Scholar
  11. 11.
    Dimitrova, V., Markovski, S., Mileva, A.: Periodic Quasigroup String Transformations. Quasigroups Related Systems 17, 191–204 (2009) On Quasigroup Pseudo Random Sequence Generator. In: Manolopoulos, Y., Spirakis, P. (eds.) Proceedings of the First Balkan Conference in Informatics, pp 393–401, Thessaloniki (2004)Google Scholar
  12. 12.
    Falcón, R.M.: Latin squares associated to principal autotopisms of long cycles. Application in Cryptography. In: Dumas, J. (ed.) Proceedings of Transgressive Computing 2006, a conference in honor of Jean Della Dora, pp 213–230. Universidad de Granada, Granada (2006)Google Scholar
  13. 13.
    Falcón, R.M.: The set of autotopisms of partial Latin squares. Discret. Math. 313, 1150–1161 (2013)MathSciNetCrossRefGoogle Scholar
  14. 14.
    Falcón, R.M.: Enumeration and classification of self-orthogonal partial Latin rectangles by using the polynomial method. Eur. J. Combin. 48, 215–223 (2015)MathSciNetCrossRefGoogle Scholar
  15. 15.
    Falcón, R.M., Falcón, O.J., Núñez, J.: Counting and enumerating partial Latin rectangles by means of computer algebra systems and CSP solvers. Math. Methods Appl. Sci. (2018). Accessed 30 September 2018
  16. 16.
    Falcón, R.M., Martín-Morales, J.: Gröbner bases and the number of Latin squares related to autotopisms of order 7. J. Symbolic Comput. 42, 1142–1154 (2007)MathSciNetCrossRefGoogle Scholar
  17. 17.
    Falcón, R.M., Stones, R.J.: Classifying partial Latin rectangles. Electron. Notes Discret. Math. 49, 765–771 (2015)CrossRefGoogle Scholar
  18. 18.
    Falcón, R.M., Stones, R.J.: Partial Latin rectangle graphs and autoparatopism groups of partial Latin rectangles with trivial autotopism groups. Discret. Math. 340, 1242–1260 (2017)MathSciNetCrossRefGoogle Scholar
  19. 19.
    Falcón, R.M., Stones, R.J.: Enumerating partial Latin rectangles. Submitted (2018)Google Scholar
  20. 20.
    Gao, S.: Counting Zeros over Finite Fields Using Gröbner Bases. Carnegie Mellon University (2009)Google Scholar
  21. 21.
    Hashemi, A.: Nullstellensätze for zero-dimensional Gröbner bases. Comput. Complex. 18, 155–168 (2009)CrossRefGoogle Scholar
  22. 22.
    Hashemi, A., Lazard, D.: Sharper complexity bounds for zero-dimensional Gröbner bases and polynomial system solving. Internat. J. Algebra Comput. 21, 703–713 (2011)MathSciNetCrossRefGoogle Scholar
  23. 23.
    Hillebrand, D.: Triangulierung Nulldimensionaler Ideale - Implementierung Und Vergleich Zweier Algorithmen. Universitaet Dortmund, Fachbereich Mathematik (1999)Google Scholar
  24. 24.
    Hulpke, A., Kaski, P., ÖStergård, P.R.J.: The number of Latin squares of order 11. Math. Comp. 80, 1197–1219 (2011)MathSciNetCrossRefGoogle Scholar
  25. 25.
    Kreuzer, M., Robbiano, L.: Computational commutative algebra 1. Springer, Berlin (2000)CrossRefGoogle Scholar
  26. 26.
    Koscielny, C.: A method of constructing quasigroup-based stream-ciphers. Int. J. Appl. Math. Comput. Sci. 6, 109–121 (1996)MathSciNetzbMATHGoogle Scholar
  27. 27.
    Koscielny, C., Mullen, G.L.: A quasigroup-based public-key cryptosystem. Int. Int. J. Appl. Math. Comput. Sci. 9, 955–963 (1999)MathSciNetzbMATHGoogle Scholar
  28. 28.
    Lakshman, Y.N.: On the complexity of computing a Gröbner basis for the radical of a zero dimensional ideal. In: Ortiz, H. (ed.) Proceedings of the Twenty-Second annual ACM Symposium on Theory of Computing, STOC’90, pp 555–563. ACM, New York (1990)Google Scholar
  29. 29.
    Lazard, D.: Solving zero-dimensional algebraic systems. J. Symbolic Comput. 13, 117–131 (1992)MathSciNetCrossRefGoogle Scholar
  30. 30.
    Johnson, K.W.: Latin Square Determinants. In: Algebraic, Extremal and Metric Combinatorics 1986, pp. 146–154. London Math. Soc. Lecture Note Ser. 131 (1988)Google Scholar
  31. 31.
    Kolesova, G., Lam, C.W.H., Thiel, L.: On the number of 88 Latin squares. J. Combin. Theory Ser. A 54, 143–148 (1990)MathSciNetCrossRefGoogle Scholar
  32. 32.
    Markovski, S., Dimitrova, V., Samardjiska, S.: Identity sieves for quasigroups. Quasigroups Relat. Syst. 18, 149–163 (2010)MathSciNetzbMATHGoogle Scholar
  33. 33.
    Markovski, S., Gligoroski, D., Andova, S.: Using quasigroups for one-one secure encoding. In: Proceedings of Eight Conference Logic and Computer Science (LIRA), pp. 157–162. Novi Sad (1997)Google Scholar
  34. 34.
    Markovski, S., Gligoroski, D., Bakeva, V.: Quasigroup string processing: Part 1. In: Proceedings of Macedonian Academy of Sciences and Arts for Mathematical and Technical Sciences XX, 1-2, pp. 13–28 (1999)Google Scholar
  35. 35.
    Markovski, S., Gligoroski Markovski, J.: Classification of quasigroups by random walk on torus. J. Appl. Math. Comput. 19, 57–75 (2005)MathSciNetCrossRefGoogle Scholar
  36. 36.
    Markovski, S., Kusakatov, V.: Quasigroup string processing: Part 2. In: Proceedings of Macedonian Academy of Sciences and Arts for Mathematical and Technical Sciences XXI, 1-2, pp. 15–32 (2000)Google Scholar
  37. 37.
    McKay, B.D., Meynert, A., Myrvold, W.: Small Latin squares, quasigroups, and loops. J. Combin. Des. 15, 98–119 (2007)MathSciNetCrossRefGoogle Scholar
  38. 38.
    Moldovyan, N.A., Shcherbacov, A.V., Shcherbacov, V.: On Some Applications of Quasigroups in Cryptography. In: Workshop on Foundations of Informatics, pp 331–340. Acad. Sci. Moldova, Inst. Math. Comput. Sci., Chişinău (2015)Google Scholar
  39. 39.
    Moldovyan, N.A., Shcherbacov, A.V., Shcherbacov, V.A.: Some applications of quasigroups in cryptology. Comput. Sci. J. Moldova 24, 55–67 (2016)MathSciNetzbMATHGoogle Scholar
  40. 40.
    Möller, H.M.: On decomposing systems of polynomial equations with finitely many solutions. Appl. Algebra Engrg. Comm. Comput. 4, 217–230 (1993)MathSciNetCrossRefGoogle Scholar
  41. 41.
    Shcherbacov, V.: Elements of quasigroup theory and applications. Monographs and Research Notes in Mathematics. CRC Press, Boca Raton (2017)CrossRefGoogle Scholar
  42. 42.
    Stones, D.S.: Symmetries of partial Latin squares. Eur. J. Combin. 34, 1092–1107 (2013)MathSciNetCrossRefGoogle Scholar
  43. 43.
    Stones, R.J., Falcón, R.M., Kotlar, D., Marbach, T.G.: Computing autotopism groups of partial Latin rectangles: a pilot study. Submitted (2018)Google Scholar
  44. 44.
    Stones, R.J., Su, M., Liu, X., Wang, G., Lin, S.: A Latin square autotopism secret sharing scheme. Des. Codes Cryptogr. 80, 635–650 (2015)MathSciNetCrossRefGoogle Scholar

Copyright information

© Springer Science+Business Media, LLC, part of Springer Nature 2018

Authors and Affiliations

  1. 1.Department Applied Mathematics IUniversity of SevilleSevilleSpain

Personalised recommendations