Étude des systèmes algébriques surdéterminés. Applications aux codes correcteurs et à la cryptographie

Les bases de Grobner constituent un outil important pour la resolution de systemes d'equations algebriques, et leur calcul est souvent la partie difficile de la resolution. Cette these est consacree a des analyses de complexite de calculs de bases de Grobner pour des systemes surdetermines (le nombre m d'equations est superieur au nombre n d'inconnues). Dans le cas generique (”aleatoire”), des outils existent pour analyser la complexite du calcul de base de Grobner pour un systeme non surdetermine (suites regulieres, borne de Macaulay). Nous etendons ces resultats au cas surdetermine, en definissant les suites semi-regulieres et le degre de regularite dont nous donnons une analyse asymptotique precise. Par exemple des que m > n nous gagnons un facteur 2 sur la borne de Macaulay, et un facteur 11,65 quand m = 2n (ces facteurs se repercutent sur l'exposant de la complexite globale). Nous determinons la complexite de l'algorithme F5 (J-C. Faugere) de calcul de base de Grobner. Ces resultats sont appliques en protection de l'information, ou les systemes sont alors consideres modulo 2 : analyse de la complexite des attaques algebriques sur des cryptosystemes, algorithmes de decodage des codes cycliques. Dans ce dernier cas, une remise en equation complete du probleme conduit a utiliser des systemes de dimension positive dont la resolution est de maniere surprenante plus rapide. Nous obtenons ainsi un algorithme de decodage efficace de codes precedemment indecodables, permettant un decodage en liste et applicable a tout code cyclique.

[1]  Guillermo Moreno-Socías,et al.  Degrevlex Gröbner bases of generic complete intersections , 2003 .

[2]  J. Faugère,et al.  On the complexity of Gröbner basis computation of semi-regular overdetermined algebraic equations , 2004 .

[3]  Miles Reid,et al.  Commutative Ring Theory , 1989 .

[4]  Ralf Fröberg,et al.  An inequality for Hilbert series of graded algebras. , 1985 .

[5]  Aviezri S. Fraenkel,et al.  Complexity of problems in games, graphs and algebraic equations , 1979, Discret. Appl. Math..

[6]  Mohab Safey El Din,et al.  Strong bi-homogeneous Bézout theorem and its use in effective real algebraic geometry , 2006, ArXiv.

[7]  Philippe Loustaunau,et al.  On the Decoding of Cyclic Codes Using Gröbner Bases , 1997, Applicable Algebra in Engineering, Communication and Computing.

[8]  Daniel Lazard,et al.  Gröbner-Bases, Gaussian elimination and resolution of systems of algebraic equations , 1983, EUROCAL.

[9]  Maria Grazia Marinari,et al.  The shape of the Shape Lemma , 1994, ISSAC '94.

[10]  Daniel Augot.  Description of Minimum Weight Codewords of Cyclic Codes by Algebraic Systems , 1996 .

[11]  Hideki Imai,et al.  Public Quadratic Polynominal-Tuples for Efficient Signature-Verification and Message-Encryption , 1988, EUROCRYPT.

[12]  Antoine Joux,et al.  Algebraic Cryptanalysis of Hidden Field Equation (HFE) Cryptosystems Using Gröbner Bases , 2003, CRYPTO.

[13]  Neal Koblitz,et al.  Algebraic aspects of cryptography , 1998, Algorithms and computation in mathematics.

[14]  Alex Biryukov,et al.  Block Ciphers and Systems of Quadratic Equations , 2003, FSE.

[15]  Robert J. McEliece,et al.  A public key cryptosystem based on algebraic coding theory , 1978 .

[16]  Don Coppersmith,et al.  Matrix multiplication via arithmetic progressions , 1987, STOC.

[17]  Frederik Armknecht,et al.  Algebraic Attacks on Combiners with Memory , 2003, CRYPTO.

[18]  Massimo Caboara,et al.  The Chen-Reed-Helleseth-Truong Decoding Algorithm and the Gianni-Kalkbrenner Gröbner Shape Theorem , 2002, Applicable Algebra in Engineering, Communication and Computing.

[19]  Barry M. Trager,et al.  Degree reduction under specialization , 2001 .

[20]  D. Coppersmith.  Solving homogeneous linear equations over GF (2) via block Wiedemann algorithm , 1994 .

[21]  Jacques Patarin,et al.  Hidden Fields Equations (HFE) and Isomorphisms of Polynomials (IP): Two New Families of Asymmetric Algorithms , 1996, EUROCRYPT.

[22]  Patrizia M. Gianni,et al.  Properties of Gröbner bases under specializations , 1987, EUROCAL.

[23]  Xuemin Chen,et al.  Decoding the (47, 24, 11) quadratic residue code , 2001, IEEE Trans. Inf. Theory.

[24]  O. Antoine,et al.  Theory of Error-correcting Codes , 2022 .

[25]  Daniel Lazard,et al.  Solving systems of algebraic equations , 2001, SIGS.

[26]  Patrick Fitzpatrick,et al.  Comparison of Two Algorithms for Decoding Alternant Codes , 1998, Applicable Algebra in Engineering, Communication and Computing.

[27]  Michael Kalkbrener,et al.  Converting Bases with the Gröbner Walk , 1997, J. Symb. Comput..

[28]  Lorenzo Robbiano,et al.  Computational aspects of commutative algebra , 1989 .

[29]  Trieu-Kien Truong,et al.  Algebraic decoding of the (32, 16, 8) quadratic residue code , 1990, IEEE Trans. Inf. Theory.

[30]  EIMEAR BYRNE,et al.  Gröbner Bases over Galois Rings with an Application to Decoding Alternant Codes , 2001, J. Symb. Comput..

[31]  John F. Humphreys.  Algebraic decoding of the ternary (13, 7, 5) quadratic residue code , 1992, IEEE Trans. Inf. Theory.

[32]  Russel J. Higgs,et al.  Decoding the ternary Golay code , 1993, IEEE Trans. Inf. Theory.

[33]  Jean-Charles Faugère,et al.  Complexity of Gröbner basis computation for Semi-regular Overdetermined sequences over F_2 with solutions in F_2 , 2002 .

[34]  Massimiliano Sala,et al.  On the Gröbner bases of some symmetric systems and their application to coding theory , 2003, J. Symb. Comput..

[35]  K. Conrad,et al.  Finite Fields , 2018, Series and Products in the Development of Mathematics.

[36]  F. S. Macaulay,et al.  The Algebraic Theory of Modular Systems , 1972 .

[37]  Michael Eugene Stillman,et al.  On the Complexity of Computing Syzygies , 1988, J. Symb. Comput..

[38]  Patrizia M. Gianni,et al.  Algebraic Solution of Systems of Polynomial Equations Using Groebner Bases , 1987, AAECC.

[39]  C. Chester,et al.  An extension of the method of steepest descents , 1957, Mathematical Proceedings of the Cambridge Philosophical Society.

[40]  Daniel Augot.  Étude algèbrique des mots de poids minimum des codes cycliques, méthodes d'algèbre linéaire sur les corps finis , 1993 .

[41]  Matthew J. B. Robshaw,et al.  Essential Algebraic Structure within the AES , 2002, CRYPTO.

[42]  David A. Cox,et al.  Ideals, Varieties, and Algorithms , 1997 .

[43]  Elwyn R. Berlekamp,et al.  On the inherent intractability of certain coding problems (Corresp.) , 1978, IEEE Trans. Inf. Theory.

[44]  A. Meyer,et al.  The complexity of the word problems for commutative semigroups and polynomial ideals , 1982 .

[45]  Ralf Fröberg,et al.  An introduction to Gröbner bases , 1997, Pure and applied mathematics.

[46]  Aviezri S. Fraenkel,et al.  Complexity of Solving Algebraic Equations , 1980, Inf. Process. Lett..

[47]  Tor Helleseth,et al.  General principles for the algebraic decoding of cyclic codes , 1994, IEEE Trans. Inf. Theory.

[48]  J. Faugère,et al.  Efficient decoding of (binary) cyclic codes above the correction capacity of the code using grobner bases , 2003, IEEE International Symposium on Information Theory, 2003. Proceedings..

[49]  Tor Helleseth,et al.  Algebraic decoding of cyclic codes: A polynomial ideal point of view , 1993 .

[50]  D. Anick,et al.  Thin algebras of embedding dimension three , 1986 .

[51]  T. T. Moh,et al.  A public key system with signature and master key functions , 1999 .

[52]  Willi Meier,et al.  Fast Algebraic Attacks on Stream Ciphers with Linear Feedback , 2003, CRYPTO.

[53]  Milton Abramowitz,et al.  Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables , 1964 .

[54]  J. Faugère.  A new efficient algorithm for computing Gröbner bases (F4) , 1999 .

[55]  Jacques Patarin,et al.  Cryptanalysis of the Matsumoto and Imai Public Key Scheme of Eurocrypt'88 , 1995, CRYPTO.

[56]  Peter L. Montgomery,et al.  A Block Lanczos Algorithm for Finding Dependencies Over GF(2) , 1995, EUROCRYPT.

[57]  Heinz Kredel,et al.  Gröbner Bases: A Computational Approach to Commutative Algebra , 1993 .

[58]  Elwyn R. Berlekamp,et al.  Algebraic coding theory , 1984, McGraw-Hill series in systems science.

[59]  Steve Szabo,et al.  Complexity Issues in Coding Theory , 1997 .

[60]  Josef Pieprzyk,et al.  Cryptanalysis of Block Ciphers with Overdefined Systems of Equations , 2002, ASIACRYPT.

[61]  Carlo Traverso,et al.  “One sugar cube, please” or selection strategies in the Buchberger algorithm , 1991, ISSAC '91.

[62]  Jean-Charles Faugère,et al.  Efficient Computation of Zero-Dimensional Gröbner Bases by Change of Ordering , 1993, J. Symb. Comput..

[63]  Bruno Buchberger,et al.  A note on the complexity of constructing Gröbner-Bases , 1983, EUROCAL.

[64]  Marc Giusti,et al.  Some Effectivity Problems in Polynomial Ideal Theory , 1984, EUROSAM.

[65]  Jean Charles Faugère,et al.  A new efficient algorithm for computing Gröbner bases without reduction to zero (F5) , 2002, ISSAC '02.

[66]  G Moreno-Socias,et al.  Autour De La Fonction De Hilbert-samuel (Escaliers D'ideaux Polynomiaux) , 1991 .

[67]  Tor Helleseth,et al.  Use of Grobner bases to decode binary cyclic codes up to the true minimum distance , 1994, IEEE Trans. Inf. Theory.

[68]  Jean-Charles Faugère.  Algebraic cryptanalysis of HFE using Gröbner bases , 2002 .

[69]  Philippe Gimenez,et al.  Computing the Castelnuovo–Mumford regularity of some subschemes of PKn using quotients of monomial ideals , 2001 .

[70]  Carlo Traverso,et al.  Effective methods in algebraic geometry , 1991 .