Analyse numérique et réduction de réseaux

L'algorithmique des reseaux euclidiens est un outil frequemment utilise en informatique et en mathematiques. Elle repose essentiellement sur la reduction LLL qu'il est donc important de rendre aussi efficace que possible. Une approche initiee par Schnorr consiste a effectuer des calculs approches pour estimer les orthogonalisations de Gram-Schmidt sous-jacentes. Sans approximations, ces calculs dominent le cout de la reduction. Recemment, des outils classiques d'analyse numerique ont ete revisites et ameliores, pour exploiter plus systematiquement l'idee de Schnorr et reduire les couts. Nous decrivons ces developpements, notamment comment l'algorithmique en nombres flottants peut etre introduite a plusieurs niveaux dans la reduction.

[1]  László Lovász,et al.  Factoring polynomials with rational coefficients , 1982 .

[2]  Damien Stehlé,et al.  Perturbation Analysis of the QR factor R in the context of LLL lattice basis reduction , 2012, Math. Comput..

[3]  Damien Stehlé,et al.  LLL on the Average , 2006, ANTS.

[4]  Siegfried M. Rump.  10. Computer-Assisted Proofs and Self-Validating Methods , 2005, Accuracy and Reliability in Scientific Computing.

[5]  C. P. Schnorr,et al.  A Hierarchy of Polynomial Time Lattice Basis Reduction Algorithms , 1987, Theor. Comput. Sci..

[6]  Martin E. Hellman,et al.  Hiding information and signatures in trapdoor knapsacks , 1978, IEEE Trans. Inf. Theory.

[7]  Claus Fieker,et al.  Dependency of units in number fields , 2006, Math. Comput..

[8]  Damien Stehlé,et al.  Algorithmique de la réduction de réseaux et application à la recherche de pires cas pour l'arrondi defonctions mathématiques , 2005 .

[9]  Ji-guang Sun.  Componentwise perturbation bounds for some matrix decompositions , 1992 .

[10]  Stephen P. Boyd,et al.  Integer parameter estimation in linear models with applications to GPS , 1998, IEEE Trans. Signal Process..

[11]  Claus-Peter Schnorr,et al.  A More Efficient Algorithm for Lattice Basis Reduction , 1988, J. Algorithms.

[12]  Damien Stehlé,et al.  An LLL Algorithm with Quadratic Complexity , 2009, SIAM J. Comput..

[13]  Denyse Baillargeon,et al.  Bibliographie , 1929 .

[14]  Guillaume Hanrot,et al.  LLL: A Tool for Effective Diophantine Approximation , 2010, The LLL Algorithm.

[15]  Claus-Peter Schnorr,et al.  Segment LLL-Reduction of Lattice Bases , 2001, CaLC.

[16]  Subhash Khot.  Hardness of Approximating the Shortest Vector Problem in Lattices , 2004, FOCS.

[17]  Sanjay Mehrotra,et al.  Segment LLL Reduction of Lattice Bases Using Modular Arithmetic , 2010, Algorithms.

[18]  Úlfar Erlingsson,et al.  Generic Gram-Schmidt orthogonalization by exact division , 1996, ISSAC '96.

[19]  Damien Stehlé,et al.  Floating-Point LLL: Theoretical and Practical Aspects , 2010, The LLL Algorithm.

[20]  Nicholas J. Higham,et al.  INVERSE PROBLEMS NEWSLETTER , 1991 .

[21]  Vincent Lefèvre,et al.  MPFR: A multiple-precision binary floating-point library with correct rounding , 2007, TOMS.

[22]  Henri Cohen,et al.  A course in computational algebraic number theory , 1993, Graduate texts in mathematics.

[23]  Arnold Schönhage.  Factorization of Univariate Integer Polynomials by Diophantine Aproximation and an Improved Basis Reduction Algorithm , 1984, ICALP.

[24]  Nicolas Gama,et al.  Finding short lattice vectors within mordell's inequality , 2008, STOC.

[25]  Claus-Peter Schnorr,et al.  Lattice basis reduction: Improved practical algorithms and solving subset sum problems , 1991, FCT.

[26]  Damien Stehlé,et al.  From an LLL-reduced basis to another , 2009, ACCA.

[27]  Siegfried M. Rump,et al.  Fast verification of solutions of matrix equations , 2002, Numerische Mathematik.

[28]  Mark van Hoeij,et al.  Factoring Polynomials and 0-1 Vectors , 2001, CaLC.

[29]  Shafi Goldwasser,et al.  Complexity of lattice problems - a cryptographic perspective , 2002, The Kluwer international series in engineering and computer science.

[30]  Damien Stehlé,et al.  Floating-Point LLL Revisited , 2005, EUROCRYPT.

[31]  Johannes A. Buchmann,et al.  Reducing lattice bases by means of approximations , 1994, ANTS.

[32]  Damien Stehlé,et al.  Rigorous and Efficient Short Lattice Vectors Enumeration , 2008, ASIACRYPT.

[33]  Claus-Peter Schnorr,et al.  Fast LLL-type lattice reduction , 2006, Inf. Comput..

[34]  Alexander May,et al.  Using LLL-Reduction for Solving RSA and Factorization Problems , 2010, The LLL Algorithm.

[35]  Subhash Khot,et al.  Hardness of approximating the shortest vector problem in lattices , 2004, 45th Annual IEEE Symposium on Foundations of Computer Science.

[36]  Andrew M. Odlyzko,et al.  Cryptanalytic attacks on the multiplicative knapsack cryptosystem and on Shamir's fast signature scheme , 1984, IEEE Trans. Inf. Theory.

[37]  Erich Kaltofen,et al.  On the complexity of finding short vectors in integer lattices , 1983, EUROCAL.

[38]  A. Storjohann.  Faster algorithms for integer lattice basis reduction , 1996 .

[39]  Gilles Villard,et al.  Certification of the QR factor R and of lattice basis reducedness , 2007, ISSAC '07.

[40]  C. Caldwell.  Mathematics of Computation , 1999 .

[41]  Nicolas Gama,et al.  Rankin's Constant and Blockwise Lattice Reduction , 2006, CRYPTO.

[42]  Mark van Hoeij,et al.  Factoring univariate polynomials over the rationals , 2009, ACCA.

[43]  Wai Ho Mow.  Maximum likelihood sequence estimation from the lattice viewpoint , 1994, IEEE Trans. Inf. Theory.

[44]  Brian A. LaMacchia.  Basis Reduction Algorithms and Subset Sum Problems , 1991 .

[45]  Siegfried M. Rump,et al.  Verification of Positive Definiteness , 2006 .

[46]  László Lovász,et al.  Polynomial factorization and nonrandomness of bits of algebraic and some transcendental numbers , 1984, STOC '84.