Faster Algorithms for Multivariate Interpolation With Multiplicities and Simultaneous Polynomial Approximations

The interpolation step in the Guruswami-Sudan algorithm is a bivariate interpolation problem with multiplicities commonly solved in the literature using either structured linear algebra or basis reduction of polynomial lattices. This problem has been extended to three or more variables; for this generalization, all fast algorithms proposed so far rely on the lattice approach. In this paper, we reduce this multivariate interpolation problem to a problem of simultaneous polynomial approximations, which we solve using fast structured linear algebra. This improves the best known complexity bounds for the interpolation step of the list-decoding of Reed-Solomon codes, Parvaresh-Vardy codes, and folded Reed-Solomon codes. In particular, for Reed-Solomon list-decoding with re-encoding, our approach has complexity O~(ℓω-1m2(n - k)), where ℓ, m, n, and k are the list size, the multiplicity, the number of sample points, and the dimension of the code, and ω is the exponent of linear algebra; this accelerates the previously fastest known algorithm by a factor of ℓ/m.

[1]  Erich Kaltofen Asymptotically fast solution of Toeplitz-like singular linear systems , 1994, ISSAC '94.

[2]  V. Pan Structured Matrices and Polynomials , 2001 .

[3]  Alexander Vardy,et al.  The Re-Encoding Transformation in Algebraic List-Decoding of Reed–Solomon Codes , 2011, IEEE Transactions on Information Theory.

[4]  Alexander Vardy,et al.  Algebraic soft-decision decoding of Reed-Solomon codes , 2003, IEEE Trans. Inf. Theory.

[5]  Nadia Heninger,et al.  Approximate common divisors via lattices , 2011, IACR Cryptol. ePrint Arch..

[6]  Peter Beelen,et al.  Key equations for list decoding of Reed-Solomon codes and how to solve them , 2010, J. Symb. Comput..

[7]  Yingquan Wu,et al.  On Rational Interpolation-Based List-Decoding and List-Decoding Binary Goppa Codes , 2012, IEEE Transactions on Information Theory.

[8]  Kwankyu Lee,et al.  List decoding of Reed-Solomon codes from a Gröbner basis perspective , 2008, J. Symb. Comput..

[9]  Victor Shoup,et al.  A fast deterministic algorithm for factoring polynomials over finite fields of small characteristic , 1991, ISSAC '91.

[10]  Peter Beelen,et al.  Interpolation and List Decoding of Algebraic Codes , 2010 .

[11]  Alexander Vardy,et al.  Correcting errors beyond the Guruswami-Sudan radius in polynomial time , 2005, 46th Annual IEEE Symposium on Foundations of Computer Science (FOCS'05).

[12]  M. G. Bruin,et al.  A uniform approach for the fast computation of Matrix-type Padé approximants , 1996 .

[13]  P. Busse MULTIVARIATE LIST DECODING OF EVALUATION CODES WITH A GRÖBNER BASIS PERSPECTIVE , 2008 .

[14]  Daniel Augot,et al.  An Interpolation Procedure for List Decoding Reed–Solomon Codes Based on Generalized Key Equations , 2011, IEEE Transactions on Information Theory.

[15]  Martin Morf,et al.  Doubling algorithms for Toeplitz and related equations , 1980, ICASSP.

[16]  Ron M. Roth,et al.  Efficient decoding of Reed-Solomon codes beyond half the minimum distance , 2000, IEEE Trans. Inf. Theory.

[17]  B. Anderson,et al.  Asymptotically fast solution of toeplitz and related systems of linear equations , 1980 .

[18]  Bernhard Beckermann,et al.  A reliable method for computing M-Pade´ approximants on arbitrary staircases , 1992 .

[19]  Jacob T. Schwartz,et al.  Fast Probabilistic Algorithms for Verification of Polynomial Identities , 1980, J. ACM.

[20]  Gui Liang Feng,et al.  A generalization of the Berlekamp-Massey algorithm for multisequence shift-register synthesis with applications to decoding cyclic codes , 1991, IEEE Trans. Inf. Theory.

[21]  Jean-René Reinhard Algorithme LLL polynomial et applications , 2003 .

[22]  Johan Sebastian Rosenkilde Nielsen,et al.  List Decoding of Algebraic Codes , 2013 .

[23]  Venkatesan Guruswami,et al.  Improved decoding of Reed-Solomon and algebraic-geometry codes , 1999, IEEE Trans. Inf. Theory.

[24]  Nadia Heninger,et al.  Ideal forms of Coppersmith's theorem and Guruswami-Sudan list decoding , 2010, Adv. Math. Commun..

[25]  Daniel J. Bernstein,et al.  Simplified High-Speed High-Distance List Decoding for Alternant Codes , 2011, PQCrypto.

[26]  Soumojit Sarkar,et al.  Triangular x-basis decompositions and derandomization of linear algebra algorithms over K[x] , 2012, J. Symb. Comput..

[27]  Yingquan Wu,et al.  New List Decoding Algorithms for Reed–Solomon and BCH Codes , 2007, IEEE Transactions on Information Theory.

[28]  Arne Storjohann Notes on computing minimal approximant bases , 2006, Challenges in Symbolic Computation Software.

[29]  Philippe Gaborit,et al.  Improved Hermite multivariate polynomial interpolation , 2006, 2006 IEEE International Symposium on Information Theory.

[30]  Erich Kaltofen,et al.  On fast multiplication of polynomials over arbitrary algebras , 1991, Acta Informatica.

[31]  Richard Zippel,et al.  Probabilistic algorithms for sparse polynomials , 1979, EUROSAM.

[32]  Madhu Sudan,et al.  Decoding of Reed Solomon Codes beyond the Error-Correction Bound , 1997, J. Complex..

[33]  François Le Gall,et al.  Powers of tensors and fast matrix multiplication , 2014, ISSAC.

[34]  Michael Alekhnovich Linear diophantine equations over polynomials and soft decoding of Reed-Solomon codes , 2005, IEEE Transactions on Information Theory.

[35]  Soumojit Sarkar,et al.  Normalization of row reduced matrices , 2011, ISSAC '11.

[36]  Peter Trifonov Efficient Interpolation in the Guruswami–Sudan Algorithm , 2010, IEEE Transactions on Information Theory.

[37]  Bruno Buchberger,et al.  The Construction of Multivariate Polynomials with Preassigned Zeros , 1982, EUROCAM.

[38]  A. J. Stothers On the complexity of matrix multiplication , 2010 .

[39]  Claude-Pierre Jeannerod,et al.  On the complexity of polynomial matrix computations , 2003, ISSAC '03.

[40]  Tom Høholdt,et al.  Decoding Reed-Solomon Codes Beyond Half the Minimum Distance , 2000 .

[41]  R. Kotter Fast generalized minimum-distance decoding of algebraic-geometry and Reed-Solomon codes , 1996 .

[42]  Ralf Koetter Fast generalized minimum-distance decoding of algebraic-geometry and Reed-Solomon codes , 1996, IEEE Trans. Inf. Theory.

[43]  Yuan Zhou Introduction to Coding Theory , 2010 .

[44]  Virginia Vassilevska Williams,et al.  Multiplying matrices faster than coppersmith-winograd , 2012, STOC '12.

[45]  Alexander Zeh,et al.  Algebraic soft- and hard-decision decoding of generalized reed-solomon and cyclic codes , 2013 .

[46]  Erich Kaltofen,et al.  On Wiedemann's Method of Solving Sparse Linear Systems , 1991, AAECC.

[47]  Richard J. Lipton,et al.  A Probabilistic Remark on Algebraic Program Testing , 1978, Inf. Process. Lett..

[48]  Claude-Pierre Jeannerod,et al.  Solving structured linear systems of large displacement rank , 2006, ACCA.

[49]  Venkatesan Guruswami,et al.  Explicit Codes Achieving List Decoding Capacity: Error-Correction With Optimal Redundancy , 2005, IEEE Transactions on Information Theory.

[50]  R. McEliece The Guruswami-Sudan Decoding Algorithm for Reed-Solomon Codes , 2003 .

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

[52]  Alexander Vardy,et al.  A complexity reducing transformation in algebraic list decoding of Reed-Solomon codes , 2003, Proceedings 2003 IEEE Information Theory Workshop (Cat. No.03EX674).

[53]  Helmut Hasse Theorie der höheren Differentiale in einem algebraischen Funktionenkörper mit vollkommenem Konstantenkörper bei beliebiger Charakteristik. , 1936 .

[54]  R. Gregory Taylor,et al.  Modern computer algebra , 2002, SIGA.

[55]  Amin Shokrollahi,et al.  A displacement approach to efficient decoding of algebraic-geometric codes , 1999, STOC '99.