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Gröbner Bases, Coding, and Cryptography pp 239–261Cite as

Gröbner Bases over Commutative Rings and Applications to Coding Theory

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Abstract

We give a survey of results and applications relating to the theory of Gröbner bases of ideals and modules where the coefficient ring is a finite commutative ring. For applications, we specialize to the case of a finite chain ring. We discuss and compare the main algorithms that may be implemented to compute Gröbner and (in the case of a chain ring) Szekeres-like bases. We give an account of a number of decoding algorithms for alternant codes over commutative finite chain rings.

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References

  • J. Apel, Computational ideal theory in finitely generated extension rings, Theoret. Comput. Sci. 244 (2000), nos. 1–2, 1–33.

    Article  MathSciNet  MATH  Google Scholar 

  • M. A. Armand, Improved list decoding of generalized Reed–Solomon and alternant codes over Galois rings, IEEE Trans. on Inf. Th. 51 (2005a), no. 2, 728–733.

    Article  MathSciNet  Google Scholar 

  • M. A. Armand, List decoding of generalized Reed–Solomon codes over commutative rings, IEEE Trans. on Inf. Th. 51 (2005b), no. 1, 411–419.

    Article  MathSciNet  Google Scholar 

  • A. Assi, Effective constructions in commutative algebra, Ph.D. thesis, Grenoble, 1991.

    Google Scholar 

  • T. Becker and V. Weispfenning, Gröbner bases, Graduate texts in mathematics, vol. 141, Springer, Berlin, 1993. A computational approach to commutative algebra, In cooperation with Heinz Kredel.

    Book  MATH  Google Scholar 

  • O. Billet and J. Ding, Overview of cryptanalysis techniques in multivariate public key cryptography, this volume, 2009, pp. 263–283.

    Google Scholar 

  • M. Brickenstein, Gröbner bases with slim polynomials, Reports in Comp. Alg. 35, Univ. Kaiserslautern, Kaiserslautern, 2005, http://www.mathematik.uni-kl.de/.

  • B. Buchberger, Ein Algorithmus zum Auffinden der Basiselemente des Restklassenringes nach einem nulldimensionalen Polynomideal, Ph.D. thesis, Innsbruck, 1965.

    Google Scholar 

  • B. Buchberger, Ein algorithmisches Kriterium für die Lösbarkeit eines algebraischen Gleichungssystems, Aequationes Math. 4 (1970), 374–383.

    Article  MathSciNet  MATH  Google Scholar 

  • B. Buchberger, Gröbner-bases: An algorithmic method in polynomial ideal theory, Multidimensional systems theory, Reidel, Dordrecht, 1985, pp. 184–232.

    Chapter  Google Scholar 

  • B. Buchberger, An algorithmical criterion for the solvability of algebraic systems of equations, London Math. Soc. LNS 251 (1998), 535–545.

    MathSciNet  Google Scholar 

  • B. Buchberger, Bruno Buchberger’s PhD thesis 1965: An algorithm for finding the basis elements of the residue class ring of a zero dimensional polynomial ideal, J. Symb. Comput. 41 (2006), nos. 3–4, 475–511.

    Article  MathSciNet  MATH  Google Scholar 

  • E. Byrne, Lifting decoding schemes over a Galois ring, LNCS, vol. 2227, Springer, Berlin, 2001, pp. 323–332.

    Google Scholar 

  • E. Byrne, Decoding a class of Lee metric codes over a Galois ring, IEEE Trans. on Inf. Th, 48 (2002), no. 4, 966–975.

    Article  MathSciNet  MATH  Google Scholar 

  • E. Byrne and P. Fitzpatrick, Gröbner bases and alternant codes over Galois rings, Proc. of ISIT 2000, 2000.

    Google Scholar 

  • E. Byrne and P. Fitzpatrick, Gröbner bases over Galois rings with an application to decoding alternant codes, J. Symb. Comput. 31 (2001), no. 5, 565–584.

    Article  MathSciNet  MATH  Google Scholar 

  • E. Byrne and P. Fitzpatrick, Hamming metric decoding of alternant codes over Galois rings, IEEE Trans. on Inf. Th. 48 (2002), no. 3, 683–694.

    Article  MathSciNet  MATH  Google Scholar 

  • E. Byrne, M. Greferath, and M. E. O’Sullivan, The linear programming bound for codes over finite Frobenius rings, Des. Codes Cryptogr. 42 (2007), no. 3, 289–301.

    Article  MathSciNet  MATH  Google Scholar 

  • E. Byrne, M. Greferath, and T. Honold, Ring geometries, two-weight codes, and strongly regular graphs, Des. Codes Cryptogr. 48 (2008), no. 1, 1–16.

    Article  MathSciNet  MATH  Google Scholar 

  • D. Cox, J. Little, and D. O’Shea, Ideals, varieties, and algorithms, Springer, Berlin, 1992, An introduction to computational algebraic geometry and commutative algebra.

    MATH  Google Scholar 

  • P. Elias, List decoding for noisy channels, Tech. Rep. 335, MIT, Cambridge, 1957.

    Google Scholar 

  • J. C. Faugère, A new efficient algorithm for computing Gröbner bases (F 4), J. Pure Appl. Algebra 139 (1999), nos. 1–3, 61–88.

    Article  MathSciNet  MATH  Google Scholar 

  • J. C. Faugère, A new efficient algorithm for computing Gröbner bases without reduction to zero (F 5), Proc. of ISSAC 2002, ACM, New York, 2002, pp. 75–83.

    Google Scholar 

  • J. C. Faugère and A. Joux, Algebraic cryptanalysis of Hidden Field Equation (HFE) cryptosystems using Gröbner bases, LNCS, vol. 2729, Springer, Berlin, 2003, pp. 44–60.

    Google Scholar 

  • J. C. Faugère, P. Gianni, D. Lazard, and T. Mora, Efficient computation of zero-dimensional Gröbner bases by change of ordering, J. Symb. Comput. 16 (1993), no. 4, 329–344.

    Article  MATH  Google Scholar 

  • P. Fitzpatrick, On the key equation, IEEE Trans. on Inf. Th. 41 (1995), no. 5, 1290–1302.

    Article  MathSciNet  MATH  Google Scholar 

  • P. Fitzpatrick, On the scalar rational interpolation problem, Math. Control Sign. Sys. 9 (1996), no. 4, 352–369.

    Article  MathSciNet  MATH  Google Scholar 

  • P. Fitzpatrick, Solving a multivariable congruence by change of term order, J. Symb. Comput. 11 (1997), 505–510.

    Google Scholar 

  • P. Fitzpatrick and S. M. Jennings, Comparison of two algorithms for decoding alternant codes, AAECC 9 (1998), no. 3, 211–220.

    Article  MathSciNet  MATH  Google Scholar 

  • G. D. Forney, On decoding BCH codes, IEEE Trans. on Inf. Th. 11 (1965), 549–557.

    Article  MathSciNet  MATH  Google Scholar 

  • R. Gebauer and H. M. Möller, On an installation of Buchberger’s algorithm, J. Symb. Comput. 6 (1988), nos. 2–3, 275–286.

    Article  MATH  Google Scholar 

  • R. Gilmer, Multiplicative ideal theory, Pure and applied mathematics, vol. 12, Dekker, New York, 1972.

    MATH  Google Scholar 

  • A. Giovini, T. Mora, G. Niesi, L. Robbiano, and C. Traverso, “One Sugar cube, please” or selection strategies in the Buchberger algorithm, Proceedings of ISSAC 1991, ACM, New York, 1991, pp. 49–54.

    Google Scholar 

  • M. Greferath, An introduction to ring-linear coding theory, this volume, 2009, pp. 219–238.

    Google Scholar 

  • M. Greferath and M. E. O’Sullivan, On bounds for codes over Frobenius rings under homogeneous weights, Discrete Math. 289 (2004), nos. 1–3, 11–24.

    Article  MathSciNet  MATH  Google Scholar 

  • M. Greferath and S. E. Schmidt, Finite-ring combinatorics and MacWilliams’ equivalence theorem, J. Combin. Theory Ser. A 92 (2000), no. 1, 17–28.

    Article  MathSciNet  MATH  Google Scholar 

  • M. Greferath, A. Nechaev, and R. Wisbauer, Finite quasi-Frobenius modules and linear codes, J. Algebra Appl. 3 (2004), no. 3, 247–272.

    Article  MathSciNet  MATH  Google Scholar 

  • E. Guerrini and A. Rimoldi, FGLM-like decoding: from Fitzpatrick’s approach to recent developments, this volume, 2009, pp. 197–218.

    Google Scholar 

  • V. Guruswami and M. Sudan, Improved decoding of Reed–Solomon and algebraic geometric codes, IEEE Trans. on Inf. Th. 45 (1999), no. 6, 1757–1767.

    Article  MathSciNet  MATH  Google Scholar 

  • A. R. Hammons Jr., P. V. Kumar, A. R. Calderbank, N. J. A. Sloane, and P. Solé, The Z 4 -linearity of Kerdock, Preparata, Goethals, and related codes, IEEE Trans. on Inf. Th. 40 (1994), no. 2, 301–319.

    Article  MATH  Google Scholar 

  • J. C. Interlando and R. J. Palazzo, Multisequence generation and decoding of cyclic codes over Z q , Proc. of ISIT 1995, 1995, pp. 1–6.

    Google Scholar 

  • J. C. Interlando, R. Palazzo Jr., and M. Elia, On the decoding of Reed–Solomon and BCH codes over integer residue rings, IEEE Trans. on Inf. Th. 43 (1997), no. 3, 1013–1021.

    Article  MathSciNet  MATH  Google Scholar 

  • A. Kandri-Rody and D. Kapur, Computing a Gröbner basis of a polynomial ideal over a Euclidean domain, J. Symb. Comput. 6 (1988), no. 1, 37–57.

    Article  MathSciNet  MATH  Google Scholar 

  • R. Kötter and A. Vardy, Algebraic soft-decision decoding of Reed–Solomon codes, Trans. on Inf. Th. 49 (2003), no. 11, 2809–2825.

    Article  Google Scholar 

  • V. Kurakin, A. Kuzmin, V. Markov, A. Mikhalev, and A. Nechaev, Linear codes and polylinear recurrences over finite rings and modules (a survey), LNCS, vol. 1719, Springer, Berlin, 1999, pp. 365–391.

    Google Scholar 

  • M. Lauer, Canonical representative for residue classes of a polynomial ideal, Proc. of ACM SSAC1976, 1976, pp. 339–345.

    Google Scholar 

  • B. R. McDonald, Finite rings with identity, Dekker, New York, 1974.

    MATH  Google Scholar 

  • H. M. Möller, On the construction of Gröbner bases using syzygies, J. Symb. Comput. 6 (1988), nos. 2–3, 345–359.

    Article  MATH  Google Scholar 

  • T. Mora, Solving polynomial equation systems. II, Macaulay’s paradigm and Gröbner technology, Encyclopedia of mathematics and its applications, vol. 99, Cambridge University Press, Cambridge, 2005.

    MATH  Google Scholar 

  • T. Mora, Gröbner technology, this volume, 2009, pp. 11–25.

    Google Scholar 

  • T. Mora and E. Orsini, Decoding cyclic codes: the Cooper philosophy, this volume, 2009, pp. 69–91.

    Google Scholar 

  • T. Mora, E. Orsini, and M. Sala, General error locator polynomials for binary cyclic codes with t<=2 and n<63, BCRI preprint, www.bcri.ucc.ie 43, UCC, Cork, Ireland, 2006.

  • A. A. Nechaev, Kerdock codes in cyclic form, Discrete Math. Appl. 1 (1991), no. 4, 365–384.

    Article  MathSciNet  MATH  Google Scholar 

  • G. Norton, On minimal realization over a finite chain ring, Des. Codes Cryptogr. 16 (1999), no. 2, 161–178.

    Article  MathSciNet  MATH  Google Scholar 

  • G. H. Norton and A. Sălăgean, On the key equation over a commutative ring, Des. Codes Cryptogr. 20 (2000), no. 2, 125–141.

    Article  MathSciNet  MATH  Google Scholar 

  • G. H. Norton and A. Sălăgean, Cyclic codes and minimal strong Gröbner bases over a principal ideal ring, Finite Fields Appl. 9 (2003), no. 2, 237–249.

    Article  MathSciNet  MATH  Google Scholar 

  • H. O’Keeffe and P. Fitzpatrick, Gröbner bases solutions of constrained interpolation problems, Linear Algebra and Its Applications 351–352 (2002), 533–551.

    Article  MathSciNet  Google Scholar 

  • E. Orsini and M. Sala, Correcting errors and erasures via the syndrome variety, J. Pure Appl. Algebra 200 (2005), 191–226.

    Article  MathSciNet  MATH  Google Scholar 

  • E. Orsini and M. Sala, General error locator polynomials for binary cyclic codes with t≤2 and n<63, IEEE Trans. on Inf. Th. 53 (2007), 1095–1107.

    Article  MathSciNet  Google Scholar 

  • L. Pan, On the D-bases of polynomial ideals over principal ideal domains, J. Symb. Comput. 7 (1989), no. 1, 55–69.

    Article  MATH  Google Scholar 

  • F. L. Pritchard, The ideal membership problem in non-commutative polynomial rings, J. Symb. Comput. 22 (1996), no. 1, 27–48.

    Article  MathSciNet  MATH  Google Scholar 

  • N. Ratnakar and R. Kötter, Exponential error bounds for algebraic soft-decision decoding of Reed–Solomon codes, IEEE Trans. on Inf. Th. 51 (2005), no. 11, 3899–3917.

    Article  Google Scholar 

  • R. M. Roth and G. Ruckenstein, Efficient decoding of Reed–Solomon codes beyond half the minimum distance, IEEE Trans. on Inf. Th. 46 (2000), no. 1, 246–257.

    Article  MathSciNet  MATH  Google Scholar 

  • R. M. Roth and P. H. Siegel, Lee-metric BCH codes and their application to constrained and partial-response channels, IEEE Trans. on Inf. Th. 40 (1994), 1083–1096.

    Article  MathSciNet  MATH  Google Scholar 

  • D. A. Spear, A constructive approach to commutative ring theory, Proc. of the 1977 MACSYMA Users’ Conference, NASA CP-2012, 1977, pp. 369–376.

    Google Scholar 

  • M. Sudan, Decoding of Reed–Solomon codes beyond the error correction bound, J. of Complexity 13 (1997), 180–193.

    Article  MathSciNet  MATH  Google Scholar 

  • G. Szekeres, A canonical basis for the ideals of a polynomial domain, Amer. Math. Monthly 59 (1952), 379–386.

    Article  MathSciNet  MATH  Google Scholar 

  • C. Traverso and L. Donato, Experimenting the Gröbner basis algorithm with AlPI system, Proc. of ISSAC 1989, ACM, New York, 1989, pp. 192–198.

    Google Scholar 

  • P. Udaya and A. Bonnecaze, Decoding of cyclic codes over F 2+uF 2, IEEE Trans. on Inf. Th. 45 (1999), no. 6, 2148–2157.

    Article  MathSciNet  MATH  Google Scholar 

  • J. F. Voloch and J. L. Walker, Codes over rings from curves of higher genus, IEEE Trans. on Inf. Th. 45 (1999), no. 6, 1768–1776.

    Article  MathSciNet  MATH  Google Scholar 

  • J. A. Wood, Duality for modules over finite rings and applications to coding theory, Amer. J. Math. 121 (1999), no. 3, 555–575.

    Article  MathSciNet  MATH  Google Scholar 

  • G. Zacharias, Generalized Gröbner bases in commutative polynomial rings, Ph.D. thesis, MIT, 1978.

    Google Scholar 

  • O. Zariski and P. Samuel, Commutative algebra, vol. I, Van Nostrand, Princeton, 1958.

    MATH  Google Scholar 

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Authors and Affiliations

  1. School of Mathematical Sciences, University College, Dublin, Ireland

    Eimear Byrne

  2. Department of Information Theory, University of Genoa, Genoa, Italy

    Teo Mora

Authors
  1. Eimear Byrne
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  2. Teo Mora
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Correspondence to Eimear Byrne .

Editor information

Editors and Affiliations

  1. Boole Centre for Research in Informatics, University College Cork, Crosses Green 17 South Bank, Cork, Ireland

    Massimiliano Sala

  2. Faculty of Engineering Dept. Production Systems Engineering, Toyohashi University of Technology, 1-1 Hibarigaoka Toyohashi, Aichi, Tempaku-cho, 441-8580, Japan

    Shojiro Sakata

  3. Dipto. Matematica, Università Genova, Via Dodecaneso, 35, 16146, Genova, Italy

    Teo Mora

  4. Dipto. Matematica, Università Pisa, Largo Bruno Pontecorvo, 5, 56127, Pisa, Italy

    Carlo Traverso

  5. LIP6, University of Paris VI, 104 avenue du Président Kennedy, 75016, Paris, France

    Ludovic Perret

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Byrne, E., Mora, T. (2009). Gröbner Bases over Commutative Rings and Applications to Coding Theory. In: Sala, M., Sakata, S., Mora, T., Traverso, C., Perret, L. (eds) Gröbner Bases, Coding, and Cryptography. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-540-93806-4_14

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