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Solving Multivariate Polynomial Systems and an Invariant from Commutative Algebra

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Arithmetic of Finite Fields (WAIFI 2020)

Part of the book series: Lecture Notes in Computer Science ((LNTCS,volume 12542))

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Abstract

The complexity of computing the solutions of a system of multivariate polynomial equations by means of Gröbner bases computations is upper bounded by a function of the solving degree. In this paper, we discuss how to rigorously estimate the solving degree of a system, focusing on systems arising within public-key cryptography. In particular, we show that it is upper bounded by, and often equal to, the Castelnuovo-Mumford regularity of the ideal generated by the homogenization of the equations of the system, or by the equations themselves in case they are homogeneous. We discuss the underlying commutative algebra and clarify under which assumptions the commonly used results hold. In particular, we discuss the assumption of being in generic coordinates (often required for bounds obtained following this type of approach) and prove that systems that contain the field equations or their fake Weil descent are in generic coordinates. We also compare the notion of solving degree with that of degree of regularity, which is commonly used in the literature. We complement the paper with some examples of bounds obtained following the strategy that we describe.

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References

  1. Bardet, M.: Étude des systémes algébriques surdéterminés. Applications aux codes correcteurs et á la cryptographie, Ph.D. thesis, Université Paris 6 (2004)

    Google Scholar 

  2. Bardet, M., Faugère, J.-C., Salvy, B.: On the complexity of Gröbner basis computation of semi-regular overdetermined algebraic equations. In: ICPPSS International Conference on Polynomial System Solving (2004)

    Google Scholar 

  3. Bardet, M., Faugère, J.-C., Salvy, B.: On the complexity of the \(F_5\) Gröbner basis algorithm. J. Symb. Comput. 70, 49–70 (2015)

    Article  Google Scholar 

  4. Bayer, D., Stillman, M.: A criterion for detecting m-regularity. Invent. Math. 87(1), 1–11 (1987). https://doi.org/10.1007/BF01389151

    Article  MathSciNet  MATH  Google Scholar 

  5. Bianco, G.: Trace-zero subgroups of elliptic and twisted Edwards curves: a study for cryptographic applications, Ph.D. thesis (2017). https://doi.org/10.35662/unine-thesis-2631

  6. Bianco, G., Gorla, E.: Index calculus in trace-zero subgroups and generalized summation polynomials, preprint (2018)

    Google Scholar 

  7. Bigdeli, M., De Negri, E., Dizdarevic, M.M., Gorla, E., Minko, R., Tsakou, S.: Semi-regular sequences and other random systems of equations, preprint (2020)

    Google Scholar 

  8. Bosma, W., Cannon, J., Playoust, C.: The Magma algebra system. I. The user language. J. Symb. Comput. 24, 235–265 (1997)

    Article  MathSciNet  Google Scholar 

  9. Bruns, W., Herzog, J.: Cohen-Macaulay rings. Revised ed. Cambridge Studies in Advanced Mathematics, vol. 39. Cambridge University Press, Cambridge (1998)

    Google Scholar 

  10. Budur, N., Casanellas, M., Gorla, E.: Hilbert functions of irreducible arithmetically Gorenstein schemes. J. Algebra 272(1), 292–310 (2004)

    Article  MathSciNet  Google Scholar 

  11. Caminata, A., Gorla, E.: The complexity of MinRank. In: Cojocaru, A., Ionica, S., Lorenzo Garcia, E. (eds.) Women in Numbers Europe III: Research Directions in Number Theory. Springer (to appear)

    Google Scholar 

  12. Chardin, M.: Some results and questions on Castelnuovo-Mumford regularity. In: Syzygies and Hilbert Functions. Lecture Notes in Pure and Applied Mathematics, vol. 254, pp. 1–40 (2007)

    Google Scholar 

  13. Conca, A., De Negri, E., Gorla, E.: Universal Gröbner bases for maximal minors. In: International Mathematics Research Notices, IMRN 2015, no. 11, pp. 3245–3262 (2015)

    Google Scholar 

  14. Conca, A., De Negri, E., Gorla, E.: Universal Gröbner bases and Cartwright-Sturmfels ideals. International Mathematics Research Notices, IMRN 2020, no. 7, pp. 1979–1991 (2020)

    Google Scholar 

  15. Courtois, N., Klimov, A., Patarin, J., Shamir, A.: Efficient algorithms for solving overdefined systems of multivariate polynomial equations. In: Preneel, B. (ed.) EUROCRYPT 2000. LNCS, vol. 1807, pp. 392–407. Springer, Heidelberg (2000). https://doi.org/10.1007/3-540-45539-6_27

    Chapter  Google Scholar 

  16. Cox, D., Little, J., O’Shea, D.: Ideals, Varieties, and Algorithms. An Introduction to Computational Algebraic Geometry and Commutative Algebra. Springer, Cham (2015). https://doi.org/10.1007/978-0-387-35651-8

    Book  MATH  Google Scholar 

  17. Ding, J., Buchmann, J., Mohamed, M.S.E., Moahmed, W.S.A.E., Weinmann, R.-P.: MutantXL. In: Proceedings of the 1st international conference on Symbolic Computation and Cryptography (SCC08), Beijing, China, LMIB, pp. 16–22 (2008)

    Google Scholar 

  18. Ding, J., Schmidt, D.: Solving degree and degree of regularity for polynomial systems over a finite fields. In: Fischlin, M., Katzenbeisser, S. (eds.) Number Theory and Cryptography. LNCS, vol. 8260, pp. 34–49. Springer, Heidelberg (2013). https://doi.org/10.1007/978-3-642-42001-6_4

    Chapter  Google Scholar 

  19. Ding, J., Yang, B.-Y.: Degree of regularity for HFEv and HFEv-. In: Gaborit, P. (ed.) PQCrypto 2013. LNCS, vol. 7932, pp. 52–66. Springer, Heidelberg (2013). https://doi.org/10.1007/978-3-642-38616-9_4

    Chapter  Google Scholar 

  20. Dubois, V., Gama, N.: The degree of regularity of HFE systems. In: Abe, M. (ed.) ASIACRYPT 2010. LNCS, vol. 6477, pp. 557–576. Springer, Heidelberg (2010). https://doi.org/10.1007/978-3-642-17373-8_32

    Chapter  Google Scholar 

  21. Eisenbud, D.: Commutative Algebra. With a View Toward Algebraic Geometry. Graduate Texts in Mathematics, vol. 150. Springer, New York (1994). https://doi.org/10.1007/978-1-4612-5350-1

    Book  MATH  Google Scholar 

  22. Eisenbud, D.: The Geometry of Syzygies A Second Course in Algebraic Geometry and Commutative Algebra. Graduate Texts in Mathematics, vol. 229. Springer, New York (2005). https://doi.org/10.1007/b137572

    Book  MATH  Google Scholar 

  23. Eagon, J.A., Northcott, D.G.: Ideals defined by matrices and a certain complex associated with them. In: Proceedings of the Royal Society of London. Series A, Mathematical and Physical Sciences, vol. 269, no. 1337, pp. 188–204 (1962)

    Google Scholar 

  24. Faugère, J.-C.: A new efficient algorithm for computing Gröbner bases (F4). J. Pure Appl. Algebra 139, 61–88 (1999)

    Article  MathSciNet  Google Scholar 

  25. Faugère, J.-C.: A new efficient algorithm for computing Gröbner bases without reduction to zero (F5). In: Proceedings of the 2002 International Symposium on Symbolic and Algebraic Computation, ISSAC 2002, New York, NY, USA, pp. 75–83 (2002)

    Google Scholar 

  26. Faugère, J.-C., Gianni, P.M., Lazard, D., Mora, T.: Efficient computation of zero-dimensional Gröbner bases by change of ordering. J. Symb. Comput. 16(4), 329–344 (1993)

    Article  Google Scholar 

  27. Faugère, J.-C., El Din, M.S., Spaenlehauer, P.-J.: Computing loci of rank defects of linear matrices using Gröbner bases and applications to cryptology. In: Proceedings of the 2010 International Symposium on Symbolic and Algebraic Computation, ISSAC 2010, Munich, Germany, pp. 257–264 (2010)

    Google Scholar 

  28. Faugère, J.-C., Din, M.S.E., Spaenlehauer, P.-J.: On the complexity of the generalized MinRank problem. J. Symb. Comput. 55, 30–58 (2013)

    Article  MathSciNet  Google Scholar 

  29. Galligo, A.: A propos du théorème de préparation de weierstrass. In: Norguet, F. (ed.) Fonctions de Plusieurs Variables Complexes. LNM, vol. 409, pp. 543–579. Springer, Heidelberg (1974). https://doi.org/10.1007/BFb0068121

    Chapter  Google Scholar 

  30. Gaudry, P.: Index calculus for abelian varieties of small dimension and the elliptic curve discrete logarithm problem. J. Symb. Comput. 44(12), 1690–1702 (2009)

    Article  MathSciNet  Google Scholar 

  31. Goubin, L., Courtois, N.T.: Cryptanalysis of the TTM cryptosystem. In: Okamoto, T. (ed.) ASIACRYPT 2000. LNCS, vol. 1976, pp. 44–57. Springer, Heidelberg (2000). https://doi.org/10.1007/3-540-44448-3_4

    Chapter  Google Scholar 

  32. Gorla, E., Massierer, M.: Index calculus in the trace zero variety. Adv. Math. Commun. 9(4), 515–539 (2015)

    Article  MathSciNet  Google Scholar 

  33. Kipnis, A., Shamir, A.: Cryptanalysis of the HFE public key cryptosystem by relinearization. In: Wiener, M. (ed.) CRYPTO 1999. LNCS, vol. 1666, pp. 19–30. Springer, Heidelberg (1999). https://doi.org/10.1007/3-540-48405-1_2

    Chapter  Google Scholar 

  34. Kreuzer, M., Robbiano, L.: Computational Commutative Algebra 1. Springer, Heidelberg (2000). https://doi.org/10.1007/978-3-540-70628-1

    Book  MATH  Google Scholar 

  35. Kreuzer, M., Robbiano, L.: Computational Commutative Algebra 2. Springer, Heidelberg (2005). https://doi.org/10.1007/3-540-28296-3

    Book  MATH  Google Scholar 

  36. Kreuzer, M., Robbiano, L.: Computational Linear and Commutative Algebra. Springer, Heidelberg (2016). https://doi.org/10.1007/978-3-319-43601-2

    Book  MATH  Google Scholar 

  37. Lazard, D.: Gröbner bases, Gaussian elimination and resolution of systems of algebraic equations. In: Computer Algebra (London, 1983). Lecture Notes in Computer Science, vol. 162, pp. 146–156. Springer, Berlin (1983)

    Google Scholar 

  38. National Institute of Standards, Post-Quantum Cryptography, Round 3 Submissions. https://csrc.nist.gov/projects/post-quantum-cryptography/round-3-submissions

  39. Tao, C., Diene, A., Tang, S., Ding, J.: Simple matrix scheme for encryption. In: Gaborit, P. (ed.) PQCrypto 2013. LNCS, vol. 7932, pp. 231–242. Springer, Heidelberg (2013). https://doi.org/10.1007/978-3-642-38616-9_16

    Chapter  Google Scholar 

  40. Tao, C., Xiang, H., Petzoldt, A., Ding, J.: Simple matrix - a multivariate public key cryptosystem (MPKC) for encryption. Finite Fields Appl. 35, 352–368 (2015)

    Article  MathSciNet  Google Scholar 

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Acknowledgements

The authors are grateful to Albrecht Petzoldt for help with MAGMA computations, to Wouter Castryck for pointing out some typos in an earlier version of this paper, and to Marc Chardin, Teo Mora, Christophe Petit, and Pierre-Jean Spaenlehauer for useful discussions on the material of this paper. This work was made possible thanks to funding from Armasuisse.

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Authors and Affiliations

  1. Dipartimento di Matematica, Università degli Studi di Genova, via Dodecaneso 35, 16146, Genova, Italy

    Alessio Caminata

  2. Institut de Mathématiques, Université de Neuchâtel, Rue Emile-Argand 11, 2000, Neuchâtel, Switzerland

    Elisa Gorla

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  1. Alessio Caminata
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Correspondence to Elisa Gorla .

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Editors and Affiliations

  1. Sorbonne Paris, Paris, France

    Jean Claude Bajard

  2. Sabancı University, Istanbul, Turkey

    Alev Topuzoğlu

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Caminata, A., Gorla, E. (2021). Solving Multivariate Polynomial Systems and an Invariant from Commutative Algebra. In: Bajard, J.C., Topuzoğlu, A. (eds) Arithmetic of Finite Fields. WAIFI 2020. Lecture Notes in Computer Science(), vol 12542. Springer, Cham. https://doi.org/10.1007/978-3-030-68869-1_1

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  • DOI: https://doi.org/10.1007/978-3-030-68869-1_1

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